Fermat's Library | Toolbox for reconstructing quantum theory from rules on information acquisition annotated/explained version.

arXiv:1412.8323v8 [quant-ph] 13 Dec 2017

Toolbox for reconstructing quantum theory from

rules on information acquisition

Philipp Andres Höhn

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

December 14, 2017

We develop an operational approach

for reconstructing the quantum theory of

qubit systems from elementary rules on in-

formation acquisition. The f ocus lies on an

observer O interrogating a system S with

binary questions and S’s state is taken as

O’s ‘catalogue of knowledge’ about S. The

mathematical tools of the framework are

simple and we attempt to highlight all un-

derlying assumptions. Four rules are im-

posed, asserting (1) a limit on the amount

of information available to O; (2) the mere

existence of complementary information;

(3) O’s total amount of information to be

preserved in-betwee n interrogations; and,

(4) O’s ‘catalogue of knowledge’ to change

continuously in time in-between interroga-

tions a nd every consistent such evolution

to be possible. T his approach permits a

constructive derivation of quantum theory,

elucidating how the ensuing independence,

complementarity and compatibility struc-

ture of O’s questions matches that of pro-

jective measurements in quantum theory,

how entanglement and monogamy of en-

tanglement, non-locality and, more gener-

ally, how the correlation structure of arbi-

trarily many qubits a nd rebits arises. The

rules yield a reversible time evolution and

a quadratic measure, quantifying O’s infor-

mation about S. Finally, it is shown that

the four rules admit two solutions for the

simplest case of a single elementary sys-

tem: the Bloch ball and disc as state spaces

for a qubit and rebit, respectively, together

with their symmetries as time evolution

Philipp Andres Höhn:

hoephil@gmail.com, Present address:

Institute for Quantum Optics & Quantum Information, Aus-

trian Academy of Sciences; and Vienna Center for Quan-

tum Science and Technology, University of Vienna, Boltz-

manngasse 3, 1090 Vienna, Austria

groups. The reconstruction for arbitrarily

many qubits is completed in a companion

paper [

1] where an additional rule elimi-

nates the rebit case. This approach is in-

spired by (but does not rely on) the re-

lational interpretation and yields a novel

formulation of quantum theory in te rms of

questions.

1 Introduction

Tools and concepts from information theory have

seen an ever growing number of applications in

modern physics, often proving useful for under-

standing and interpreting speciﬁc physical phe-

nomena. Among a vast number of examples,

black hole entropy, or more generally space-time

horizon entropies, can be understood in terms

of entanglement entropy [

3–7], thermodynamics

naturally adheres to entropic and thus informa-

tional perspectives [8–12], and, above all, the en-

tire ﬁeld of quantum information and computa-

tion is the natural phys ical arena for applications

of information theoretic tools [

13].

This manuscript, by contrast, is motivated by

the question whether information theoretic con-

cepts, apart from their useful applications to con-

crete physical s ituations, can also tell us s ome-

thing deeper about physics, namely about the

physical content and architecture of theories. The

overriding idea is that elementary rules or restric-

tions of certain informational activities, e.g. infor-

mation acquisition or communication, should be

deeply intertwined with the structure of the ap-

propriate theory. In this article, we shall address

this question by means of the concrete example

of quantum theory. Our ambition is to develop

a novel informational framework for deriving the

formalism and structure of quantum theory for

systems of arbitrarily many qubits from elemen-

Accepted in Quantum 2017-11-27, click title to verify 1

tary operational postulates – a task which is com-

pleted in the companion paper [

1]. While neither

this question nor the fact that one can recon-

struct quantum theory from elementary axioms is

new and has been extensively explored before in

various contexts [

14–25], we shall approach both

from a novel constructive perspective and with

a stronger emphasis on the conceptual content

of the theory. The ultimate goal of this work is

therefore very rudimentary: to redo a well estab-

lished theory – albeit in a novel way which is es-

pecially engineered for exposing its informational

and logical structure, physical content and dis-

tinctive phenomena more clearly. In other words,

we shall attempt to rebuild quantum theory for

qubit systems from scratch.

In such an information based context it is nat-

ural to follow an operational approach, describing

physics from the perspective of an observer. Ac-

cordingly, we shall work under the premise that

we may only speak about the information an ob-

server has access to in an experiment. Our ap-

proach will thus be purely operational and epis-

temic (i.e. knowledge based) by construction and

shall survive without ontic statements (i.e. ref-

erences to ‘reality’). We shall thus say nothing

about whether or not ‘hidden variables’ could give

rise to the experiences of the observer. Under

these circumstances we adopt an ‘inside view of

physics’, holding properties of systems as being

relationally, rather than absolutely deﬁned.

Indeed, more generally the replacement of ab-

solute by relational concepts goes in hand with

the establishment of universal (i.e. observer in-

dependent) limits. For instance, the crucial step

from Galilean to special relativity is the realiza-

tion that the speed of light c constitutes a univer-

sal limit for information communication among

observers. The fact that all observers agree on

this limit is the origin of the relativity of space

and time. Similarly, the crucial step from classi-

cal to quantum mechanics is the recognition that

the Planck constant ~ establishes a universal limit

on how much simultaneous information is acces-

sible to an observer. While less explicit than in

the case of special relativity, this simple observa-

tion suggests a relational character of a system’s

quantum properties. More precisely, the process

of information acquisition through measurement

establishes an informational relation between the

observer and system. Only if there was no limit

on the acquisition of information would it make

sense to speak about an absolute state of a sys-

tem within a purely operational approach (unless

one accepts the existence of an omniscient and

absolute observer as an external standard). But

thanks to the existence of complementarity, im-

plied by ~, an observer may not access all conceiv-

able properties of the system at once. Further-

more, the observer can choose the experimental

setting and thereby which property of the system

she would like to reveal (although, clearly, she

cannot choose the experimental outcome). Under

our purely operational premise, we shall treat the

situation as if the system does not have any other

properties than those accessible to the observer at

any moment of time. In particular, the system’s

state is naturally interpreted as representing the

observer’s state of information about the system.

These ideas are in agreement with earlier propos-

als in the literature [

26,27] and, most speciﬁcally,

with the relational interpretation of qu antum me-

chanics [28 , 29].

Of course, in order for diﬀerent observers who

may communicate (by physical interaction) to

have a basis for agreeing on the description of

a system, some of its attributes must be observer

independent such as its state space, the set of

possible measurements on it and possibly a limit

on its information content. But without adhering

to an external standard against which measure-

ment outcomes and states could be deﬁned, it is

as meaningless to assert a sy stem’s physical state

to be independent of its relations to other sys-

tems as it is to relate a system’s dynamics to an

absolute Newtonian background time.

It is worthwhile to investigate what we can

learn about physics from such an operational

and informational approach. For this endeavour

we shall adopt the general conviction, which has

been voiced in many diﬀerent (even conﬂicting)

ways before in the literature [

26, 28, 30–41], that

quantum theory is best understood as an opera-

tional framework governing an observer’s acquisi-

tion of information about a system. While most

earlier works take quantum theory as given and

attempt to characterize and interpret its physical

content with an emphasis on information infer-

ence, here and in [

1] we take a step back and show

that one can actually derive quantum theory from

this perspective. This will require a focus on the

informational relation between an observer and a

Accepted in Quantum 2017-11-27, click title to verify 2

system and the rules governing the observer’s ac-

quisition of information. More precisely, our ap-

proach will be formulated in terms of the observer

interrogating a sys tem with elementary questions.

While the present work has been inspired by re-

lational ideas, we emphasize that the sequel does

not actually rely on them s o this should not dis-

courage a reader unsympathetic with relational

interpretations of quantum theory. Our approach

will reconstruct and produce a novel formulation

of quantum theory, but will clearly not single out

the relational interpretation as ‘the right one’.

However, it will support a partial interpretation,

namely that quantum theory is a law book gov-

erning an observer’s acquisition of information

about physical systems.

This is clearly not the only physical situation

to which such a relational approach applies; it

likewise opens up a novel perspective on elemen-

tary s pace-time structure which is encoded in the

informational relations among diﬀerent observers

and can be exposed by a communication game.

For instance, without presupposing a particular

space-time structure – and thus without assuming

an externally given transformation group between

diﬀerent reference frames – one can also derive

the Lorentz group as the minimal group trans-

lating between diﬀerent observer’s descriptions

of physics from their informational relations, es-

tablished by communication with quantum sys-

tems [

42].

More fundamentally, relational ideas are actu-

ally required and commonly employed in the con-

text of background independent quantum gravity

approaches where the notion of coordinates dis-

appears together with a classical notion of space-

time within which a dynamics could be deﬁned.

Instead, one has to resort to dynamical degrees

of freedom to deﬁne physical reference frames

(i.e., dynamical ‘rods’ and ‘clocks’) relative to

which a meaningful dynamics can be formulated

in the ﬁrst place. This constitutes the relational

paradigm of dynamics [

43–51] but goes beyond

a purely informational and operational approach

and thus clearly beyond the scope of this work.

The remainder of this manuscript is organized

as follows.

Contents

1 Introduction

1

2 Why a(nother) reconstruction of quantum theory? 5

3 Landscape of inference theories 5

3.1 The standard landscape of generalized probabilistic theories . . . . . . . . . . . . . . . 6

3.2 A novel landscape of inference theories . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Questions and answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.2 From information to probabilities: the state of S relative to O . . . . . . . . . . 9

3.2.3 ‘Belief’ updating and ‘collapse’ of the state . . . . . . . . . . . . . . . . . . . . 10

3.2.4 Elementary structure on Σ and Q . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.5 Logical compositions of questions and rules of inference . . . . . . . . . . . . . 14

3.2.6 Parametrization of S’s state and tomography . . . . . . . . . . . . . . . . . . . 16

3.2.7 Compos ite s ystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.8 Time evolution of S’s state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.9 The theory landscape L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Principles for the quantum theory of qubits as rules on information acquisition 21

4.1 The rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Strategy for building the necessary tools and proving the claim . . . . . . . . . . . . . 25

5 Question structure and correlations 26

5.1 A single gbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2 Two gbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2.1 Logical connectives of s ingle gbit questions . . . . . . . . . . . . . . . . . . . . . 26

Accepted in Quantum 2017-11-27, click title to verify 3

5.2.2 Independence, complementarity and entanglement . . . . . . . . . . . . . . . . 28

5.2.3 A logical argument for the dimensionality of the Bloch-sphere . . . . . . . . . . 31

5.2.4 An informationally complete set for N = 2 qubits . . . . . . . . . . . . . . . . . 32

5.2.5 An informationally complete set for N = 2 rebits . . . . . . . . . . . . . . . . . 33

5.2.6 Entanglement and non-local tomography for rebits . . . . . . . . . . . . . . . . 34

5.2.7 A Bell scenario with questions: ruling out local hidden variables . . . . . . . . . 36

5.3 Three gbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.3.1 Three qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3.2 Independence and compatibility for three qubits . . . . . . . . . . . . . . . . . 40

5.3.3 An informationally complete set for three qubits . . . . . . . . . . . . . . . . . 42

5.3.4 Entanglement of three qubits and monogamy . . . . . . . . . . . . . . . . . . . 43

5.3.5 Maximal entanglement for three qubits . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.6 Three rebits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3.7 Independence and compatibility for three rebits . . . . . . . . . . . . . . . . . . 45

5.3.8 An informationally complete set for three rebits . . . . . . . . . . . . . . . . . . 47

5.3.9 Monogamy and maximal entanglement for three rebits . . . . . . . . . . . . . . 48

5.4 Correlation structure for N = 2 gbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4.1 The logical mirror image of an inference theory . . . . . . . . . . . . . . . . . . 49

5.4.2 Collecting the results: odd and even correlation structure for N = 2 . . . . . . 52

5.5 The general case of N > 3 gbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.5.1 An informationally complete set and entanglement for N > 3 qubits . . . . . . 54

5.5.2 An informationally complete set and entanglement for N > 3 rebits . . . . . . . 58

6 Information measure, time evolution and state space 59

6.1 Why the Shannon entropy does not apply here . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Elementary conditions on the measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.3 The state of no information is an interior state . . . . . . . . . . . . . . . . . . . . . . 61

6.4 Time evolution of the ‘Bloch vector’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.5 Time evolution is injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.6 Time evolution is also reversible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.7 Time evolution deﬁnes a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.8 The squared length of the Bloch vector as information meas ure . . . . . . . . . . . . . 65

6.9 The set of all time evolutions is a subgroup of SO(D

N

) . . . . . . . . . . . . . . . . . . 67

6.10 Pure and mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 The N = 1 case and the Bloch ball 68

7.1 A single qubit and the Bloch ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.2 A single rebit and the Bloch disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8 Discussion and outlook 71

8.1 An operational alternative to the wave function of the universe . . . . . . . . . . . . . 73

Acknowledgements 74

References 74

The ﬁrst part of the article up to and including

section

4 includes a substantial amount of con-

ceptual elaborations. The second part, by con-

trast, will become more technical upon putting

the novel postulates to use in sections

5-7. The

reconstruction for arbitrarily many qubits is per-

formed in the companion article [

1] where an ad-

ditional postulate eliminates rebits (two-level sys-

tems over real Hilbert spaces) in favour of qubit

quantum theory. The rebit case is considered sep-

arately in [

2].

Accepted in Quantum 2017-11-27, click title to verify 4

2 Why a(nother) reconstruction o f

quantum theory?

Given that we have a beautifully working theory,

one may wonder why one should bother to re-

construct quantum theory from operational state-

ments. There are various motivations for this en-

deavour:

1. To equip the standard, physically obscure

textbook axioms for quantum theory with

an operational sense. In particular, in ad-

dition to its empirical success, a derivation

from operational statements can conceptu-

ally justify the f ormulation of the theory in

terms of Hilbert spaces, complex numbers,

tensor product rule for composite systems,

etc.

2. To better understand quantum theory within

a larger context. By singling out quan-

tum theory with operational s tatements one

can answer the q uestion “what makes quan-

tum theory special?”, thereby establishing a

bird’s-eye perspective on the formalism and

conceivable alternatives.

3. It may help to understand why or why not

quantum theory in its present form should

be fundamental and thus why it should or

should not be modiﬁed in view of attempt-

ing to construct fundamental theories. By

dropping or modifying some of its deﬁning

physical principles, one obtains a handle for

systematic generalizations of quantum the-

ory. This may also be interesting in view of

quantum gravity phenomenology (away from

the deep quantum regime). More fundamen-

tally, the question arises whether an infor-

mational perspective could be beneﬁcial for

quantum gravity in general.

4. The hop e has been voiced that a clear inter-

pretation of the theory may ﬁnally emerge

from a successful reconstruction, in analogy

to how the interpretation of special and gen-

eral relativity follows naturally from its un-

derlying principles [

28,31,36].

It is fair to say that the hope alluded to un-

der point 4 has not been realized thus far because

the existing successful reconstructions [14–25] are

fairly neutral as far as an interpretation is con-

cerned. While most of them emphasize the op-

erational character of the theory, a particular in-

terpretation of quantum theory is not strongly

suggested.

The language and concepts of the present re-

construction are diﬀerent. It will emphasize and

concretize the view (or partial interpretation)

that quantum theory is a framework governing

an observer’s acquisition of information ab out the

observed s ystem [26, 28, 30– 41]. The new postu-

lates are simple and conceptually comprehensible,

concerning only the relation between an observer

and the system. Due to the simplicity, the ensu-

ing derivation is mathematically quite elementary

and, in contrast to previous derivations, yields

the f ormalism, state s paces and time evolution

groups explicitly and in a more constructive man-

ner. The disadvantage, compared to other recon-

structions, is that a large number of detailed steps

is required. The advantage, on the other hand, is

the simplicity of the principles and mathematical

tools, and the fact that the reconstruction aﬀords

natural explanations for many quantum phenom-

ena, including entanglement and monogamy, and

elucidates the origin of the unitary group.

3 Landscape of inference theories

The ambition of a (re-)construction of quantum

theory is to derive its formalism, state spaces,

time evolution groups and permissible operations

from physical principles – in some rough analogy

to the construction of relativity theory from the

principle of relativity and the eq uivalence prin-

ciple. But in order to formulate physical postu-

lates, we clearly have to presuppose some mathe-

matical s tructure within which a precise meaning

can be given to them. (For example, also the

construction of special and general relativity cer-

tainly presupposed a substantial amount of me-

chanical structure.)

The procedure is thus to ﬁrstly deﬁne some

landscape of theories, which hopefully contains

quantum theory and classical information theory,

but within which theories are generally not for-

mulated in terms of the usual complex Hilbert

spaces, tensor product rules, etc. The mathe-

matical formulation of the landscape must there-

fore be more elementary and, in particular, op-

erational. That is, for the time being, we have

Accepted in Quantum 2017-11-27, click title to verify 5

to forget about the usual – mathematically crisp

but physically rather obscure – textbook axioms

of quantum theory. While diﬀerent theories will

have the mathematical and physical structure of

the landscape in common, they may have oth-

erwise very diﬀerent physical and informational

properties; e.g., they may admit much stronger

correlations than quantum theory [

52], or weaker

correlations as classical probability theory, they

may allow exotic communication and information

processing tasks to be accomplished [

53,54], and

so on. Secondly, given the language of this land-

scape, one can attempt to formulate comprehen-

sible physical statements which single out q uan-

tum theory from within it. Going to a larger the-

ory landscape and beyond the language of Hilbert

spaces is precisely what allows us to ask the ques-

tion “what makes quantum theory special?” and,

ultimately, to ﬁnd an operational and physical

justiﬁcation for the usual textbook axioms and

the s tandard Hilbert space formulations.

The goal of this section is precisely to build

such an appropriate landscape of inference theo-

ries both conceptually and mathematically from

scratch within which we shall subsequently for-

mulate those elementary rules, governing an ob-

server’s acquisition of information about a sys-

tem, that single out qubit quantum theory. This

novel landscape of inference theories employs a

diﬀerent language and is conceptually distinct

from the by now standard landscape of general-

ized probabilistic theories (GPTs) which are com-

monly employed f or characterizations (or gener-

alizations) of quantum theory. For contrast and

to put the new tools into a larger perspective, we

begin with a brief synopsis of the GPT language

before we establish the landscape of inference the-

ories underlying this manuscript.

3.1 The standard landscape of generalized

pro babilistic theories

It has become a standard in the literature to

employ the formalism of generalized probabilistic

theories (GPTs) for operational characterizations

or derivations of quantum theory [

14–23, 55–61].

The setup of GPTs is exclusively operational and

one considers three kinds of operation devices (s ee

ﬁgure

1): (1) a preparation device which can spit

out systems in some set of states deﬁned by a

vector of probabilities for the outcome of ﬁducial

measurements. The state spaces of the systems

are necessarily required to be convex to permit

convex mixtures of states. (2) A transformation

device can perform physical operations on the

prepared systems (e.g., a rotation) which may

change the state of the system (e.g., by some

group action on the state vector), but must al-

low it to continue its journey to (3), a measure-

ment device, which detects certain experimental

outcomes. The measurement devices are math-

ematically described by so-called ‘eﬀects’ which

are ass umed to be dual to, i.e. linear functionals

on, the states. (For the interested reader we note,

however, that Holevo has shown how the math-

ematical incarnation of measurements as linear

functionals on states follows from other simple

operational as sumptions [

62].)

PSfrag rep lacements

preparation

transformation

measurement

systems

convex state sp aces

cbit square bit rebit qubit

‘eﬀects’

dual t o states

Figure 1: The standard op e rational setup of generalized

probabilistic theories with examples of allowed convex

state spaces of elementary two-level systems.

Measurements and states are at the heart of

GPTs; ‘eﬀects’ directly determine the outcome

probabilities of measurements and thereby, when

a complete set is engaged, reveal the probabilistic

state of a system. An observer assumes a support-

ing role, giving intuitive meaning to the notion

of preparation, transformation and measurements

of sy stems. The reconstructions of quantum the-

ory within the GPT formalism depart from rather

global operational axioms, restricting the sets of

possible preparations (i.e. state spaces), transfor-

mations and measurements. The concrete acqui-

sition of information of the observer about the

systems is otherwise not accentuated. In par-

ticular, since the outcome probabilities are the

primary concept of GPTs, these axioms say lit-

tle about what an observer experiences in indi-

vidual experimental runs, but instead f ocus on

the totality of a large number of experimental

runs. This has led to a whole wave of s uccessful

Accepted in Quantum 2017-11-27, click title to verify 6

quantum theory reconstructions, employing GPT

concepts in one way or another [

14–23], most of

which, however, are fairly abstract on account of

the rather global axioms.

It is perhaps one of the great strengths of GPTs

that they constitute a functional and purely op-

erational framework which is interpretationally

fairly neutral. It is not the ambition of this frame-

work to elucidate the measurement problem, to

clarify what happens to a state during a mea-

surement, what probabilities are or, ultimately,

how to interpret quantum mechanics (except that

it highlights its operational character). As such,

this framework is compatible with most interpre-

tations of quantum theory.

3.2 A novel landscape of inference theories

The success of G PTs notwithstanding, we shall

now change semantics and perspective to deﬁne a

new landscape of theories and a novel framework

for (re)constructing and understanding quantum

theory. Henceforth, we shall fully engage the

observer and give primacy to his acquisition of

fundamentally limited information from observed

systems. This will include being more explicit

about what general sort of information will be

available to the observer even in individual exper-

imental runs. Probabilities, on the other hand,

can be viewed as secondary and as a consequence

of the limited information available to the ob-

server – although, clearly, probabilities will as-

sume a pivotal role too (after all we want to re-

construct quantum theory).

3.2.1 Questions and answers

As schematically depicted in ﬁgure

2, we shall

consider an observer O who can only interact with

a system S through interrogation via questions

Q

i

from some set of questions Q which we shall

further constrain below. (At this stage we make

no assumption about whether Q is continuous or

discrete.) The only information which we allow

O to acquire about S is by asking questions from

this set Q.

In principle, of course, O could conceive of and

ask S all kinds of questions, but S could not al-

ways give a meaningful answer; S may simply

not have the desired properties or carry the in-

formation O is inquiring about (e.g., S may not

be complex enough, or S does not interact with

other systems carrying the information in ques-

tion), or p ossibly the question is not a senseful

one in the ﬁrst place. We shall call a question Q

physically implementable on S if O can acquire a

‘meaningful’ answer from S to Q. An elementary

restriction on Q is that any question Q

i

from this

set be implementable on S and that, whenever O

asks Q

i

to S, S will give an answer to O. Clearly,

Q depends on S.

But how does O know whether S will give an

answer and how can he judge whether the latter

is ‘meaningful’? This requires O, like any experi-

menter, to have developed, from previous experi-

ences, a theoretical model by means of which he

describes and interprets his interactions with S.

An answer can only be ‘meaningful’ in the con-

text of this model such that our notion of imple-

mentability is actually dependent on O’s model.

We shall come back to this model frequently.

The central ingredients of this framework will

thus be questions and answers – and O’s informa-

tion ab out answer outcomes and their relations.

This will lead to a novel question calculus from

which many crucial q uantum properties will be

derived. That is, rather than focusing mostly

on probabilistic properties as in GPTs, this novel

framework will connect more directly to what can

be measured in experiments through its empha-

sis on questions and their relations. As a conse-

quence, this framework will also produce opera-

tionally more compelling explanations of typical

quantum phenomena.

In the s eq uel, we shall solely speak about the

information O has about S and, correspondingly,

about the state that O assigns to S based on

this information. Such a state of S is then de-

ﬁned relative to O (whether or not some hidden

variables give rise to this state is a question we

shall not address). The act of information ac-

quisition establishes a relation between O and S

and this will be the center of our attention. (A

priori, a diﬀerent observer O

′

may establish a

diﬀerent relation with S.) Although this frame-

work will also not give rise to a unique interpre-

tation, it will support the partial interpretation

that quantum theory is a law book governing an

observer’s acquisition of information about phy s-

ical systems and might therefore be considered

interpretationally less neutral than GPTs. While

this connects with general ideas underlying, e.g.,

the relational [

28,29], Brukner-Zeilinger informa-

Accepted in Quantum 2017-11-27, click title to verify 7

PSfrag rep lacements

Preparation Interrogation

S

O

Q

i

?

Figure 2: Schematic representation of an observer O interrogating a system S.

tional [

31–35], or QBist [36–39] interpretations of

quantum mechanics, we emphasize that this re-

construction does not rely on them and should

thus not discourage readers uncomfortable with

these speciﬁc interpretations.

We shall not explicitly deal with transforma-

tion and measurement (‘eﬀect’) devices as in

GPTs; instead, these will be replaced more gener-

ally by time evolution and questions, respectively.

The s et of all possible (information preserving)

operations that O could perform on S can later

be identiﬁed with the set of all poss ible time evo-

lutions of S. H owever, in analogy to GPTs, we

will assume that O has access to some method of

preparing S in diﬀerent ways such that S’s an-

swers to O’s q uestions depend on the precise way

of preparation. (O could either control himself a

preparation device or have a distinct observer O

′

prepare systems for him.)

When constructing the new landscape L of the-

ories describing O’s acquisition of information

about S within which we shall later, in section

4,

formulate our postulates, we will make a number

of restrictions and assumptions. In order to fa-

cilitate f uture generalizations and improvements

of the present construction of quantum theory –

which constitutes a proof of principle – we shall

attempt to be as clear as possible about the as-

sumptions made throughout this work.

As a starter, we would like to keep Q as sim-

ple as possible, while still having non-trivial ques-

tions. In particular, we do not wish to consider

trivial propositions which are always true or al-

ways false. We shall therefore assume the follow-

ing.

Assumption 1. The set of questions Q which

we shall permit O to ask S only contains binary

questions Q

i

. Any Q

i

∈ Q is a non-trivial ques-

tion such that S’s answer (‘yes’ or ‘no’) is not

independent of i ts preparation. Furthermore, any

Q

i

∈ Q is repeatable such that O, b y asking the

same S the same Q

i

m times in succession will

receive m times the same answer.

The restriction to elementary ‘yes-no’-

questions greatly simpliﬁes the discussion and,

ultimately, will give rise to the quantum theory

of qubit systems. For instance, in quantum

theory, a binary question could be ‘is the spin of

the qubit up in x-direction?’ However, it will not

be too diﬃcult to generalize Q to also consist of

ternary, quaternary, quinary, etc. questions, but

we shall not attempt to do so here. Since trivial

questions are not considered, we already see

that not even all implementable questions will

be taken into account. We shall impose further

restrictions on Q such that, ultimately, it will be

a strict subset of all possible binary questions

which O could, in principle, ask S.

Since this is an operational approach, it is fair

to assume that O can record the answers to his

questions asked to any system (e.g., by writing

them on a piece of paper) and that he can do

statistics over the outcomes (e.g., by counting the

frequency of outcomes). We shall require that

every possible way of preparing S will give rise

to a particular s tatistics over the answers to all

Q

i

∈ Q; O could test these statistics by interro-

gating a large number n of identically prepared

systems

1

S

a

, a = 1, . . . , n, s uﬃciently often with

1

Given the present structure, two systems S

1

and S

2

could only be considered as distinct in nature if either t he

(maximal) set of questions Q which O can ask the sys-

Accepted in Quantum 2017-11-27, click title to verify 8

(at least ideally) all Q

i

∈ Q. In fact, this is pre-

cisely how O will operationally distinguish diﬀer-

ent preparations of systems.

By having interrogated, in this manner, the

n always identically prepared S

a

for all possible

ways of preparation, we shall assume O to have

gained a ‘stable’ knowledge of the set Σ of all pos-

sible answer statistics for S over Q, i.e., the an-

swer statistics for all Q ∈ Q and all possible ways

of preparing a system S.

2

While ideally n → ∞ is

necessary, ‘practically’ n should be large enough

for O to develop a theoretical model of both Q

and Σ up to some accuracy which agrees with

his observations. It is not our ambition to clarify

further what n is nor how precisely O has de-

veloped his theoretical model, instead, we shall

henceforth just assume that O has puzzled out

the pair (Q, Σ). As any experimenter in an ac-

tual laboratory, O shall interpret the outcomes

of his interrogations by means of his model for

(Q, Σ) and he can decide whether a given ques-

tion is contained in the set Q or not. Henceforth,

we assume that O only asks questions from Q.

Our task will be to establish what this model is,

based on the ensuing assumptions and postulates.

3.2.2 From information to probabilities: the state

of S relati v e to O

The previous subsection did not refer to the no-

tion of probabilities. But it deﬁned two prepa-

rations as being identical if they give identical

statistics. This permits us to regard Σ equiva-

tems or the totality of answer statistics for all p ossible

preparations were distinct for S

1

and S

2

(if always the

same two S

1

, S

2

are interrogated and thereafter freshly

prepared again). If O can not distinguish S

1

and S

2

in this

way for suﬃciently many trials (ideally inﬁnitely many),

we shall call them identical. Let S

1

and S

2

be identi-

cal and O prepare both systems with the same procedure

(for instance, the setting of the preparation device is the

same for both systems). If the answer statistics (frequ en-

cies) for S

1

, S

2

for all Q

i

∈ Q become indistinguishable af-

ter suﬃciently many trials (of preparing and interrogating

the same systems with the same procedure), we consider

these systems as ‘identically prepared’. (After completion

of this work, the author was made aware that this notion

is similar to the deﬁnition of ‘operational equivalence’ put

forward in [

63].)

2

We assume the preparation method to be ideal in the

sense that it accounts for any possible answer statistics

which S can admit in O’s world for questions in Q. That

is, there do not exist other methods which can prepare S

in ways th at O’s method does not en compass.

lently as the set of all possible answer statistics

or as the set of all (distinct) preparations.

Based on this notion, we shall now identify

probabilities as degrees of belief. The ‘knowledge’

of what Σ is for a given S will permit O to assign

probabilities to the outcomes of his questions. It

is very natural for O to assign probabilities to

questions because he deals with statistical ﬂuctu-

ations and furthermore, as we shall see later, with

systems about which he always has incomplete in-

formation in the sense that the corresponding Σ

is such that he can never know the answers to all

Q

i

∈ Q at the same time.

More precisely, for a s peciﬁc S and any Q

i

∈ Q

that he may ask the system next, O can assign a

probability y

i

that the answer will be ‘yes’ (or a

probability n

i

that the answer will be ‘no’), ac-

cording to

(i) O’s knowledge of Σ, and

(ii) any prior information that O may have about

the s peciﬁc S.

Given our setup, the only prior information that

O may have about the particular S (apart from

what the associated Σ and Q are) must result ei-

ther from having interrogated some ensemble of

systems identically prepared to S with some sub-

set of Q beforehand

3

and from the corresponding

accumulated statistics of the asked Q

j

∈ Q (e.g.,

that O may have recorded on a piece of paper),

or from any other prior information about the

method of preparation. In particular, this previ-

ous ensemble could have been empty.

For instance, if every time that O asked the

speciﬁc Q

i

to any of the identically prepared sys-

tems gave a ‘yes’ answer before, he will assign

the prior probability y

i

= 1 to Q

i

and to the next

identically prepared S that he will interrogate. If,

on the other hand, the number of ‘yes’ and ‘no’

answers to some other Q

j

was equal for the previ-

ously identically prepared systems, O will assign,

as a best guess, the prior probability y

j

=

1

2

to

this Q

j

and to the next S. Similarly, for any

other answer statistics, O would assign y

i

to the

next S depending on the recorded frequencies of

‘yes’ answers. But, thanks to his knowledge of

Σ and therefore of any possible relations in the

3

If O asks more than one question to any S, the order-

ing of the questions may matter. But then O could ask

the questions for any S always in the same order.

Accepted in Quantum 2017-11-27, click title to verify 9

answer statistics, O can also assign prior proba-

bilities y

k

to questions Q

k

that he did not ask the

previous set of identically prepared systems. For

example, Σ may b e such that whenever S gives

a ‘yes’ answer to Q

i

, it will give a ‘no’ answer to

an immediately following Q

k

. Accordingly, if O

assigns a prior probability y

i

= 1 to Q

i

as above,

he will also assign a prior y

k

= 0 to Q

k

without

previously having asked Q

k

. But other relations

between questions will be permitted too. In par-

ticular, it may be that the information gained

from the questions he previously asked the iden-

tically prepared systems and the structure of Σ

make it equally likely that the answer to Q

k

asked

to the next S will be ‘yes’ or ‘no’. In this case,

O will assign y

k

=

1

2

to Q

k

that he may ask the

next S. This will become more precise along the

way.

We therefore take a broadly

4

Bayesian perspec-

tive on probabilities: O assigns probabilities to

questions according to his ‘degree of belief’ about

S. These probabilities y

i

are thereby relative to

the observer O. A diﬀerent observer O

′

may have

diﬀerent information about S and thereby assign

diﬀerent probabilities to the various outcomes of

questions posed to S (for a discussion, within

quantum theory, of the consistency of diﬀerent

observers having diﬀerent information about a

system, see [

28,29,64,65]). E.g., O

′

could be the

one preparing S. She could ‘know’ the statistics

for the speciﬁc preparation setting (from previous

tests) and then send O the speciﬁcally prepared

S without informing him about her knowledge.

For consistency, we tacitly assume the set Σ of

all possible answer statistics to coincide with the

set of possible ‘beliefs’. That is, to every equiva-

lence class of preparations of S (with ‘identically

prepared’ deﬁning the equivalence relation) there

shall correspond a unique ‘belief’ {y

i

}

Q

i

∈Q

and

vice versa. Since the only way for O to acquire in-

formation about S is by interrogation with ques-

tions in Q, the prior probabilities y

i

that O as-

signs to every Q

i

∈ Q encode the entire informa-

tion that O has about S. Hence, we shall make

the f ollowing identiﬁcation.

Deﬁnition 3.1. (State of S relative to O)

The collection of all probabilities y

i

∀ Q

i

∈ Q is

the state of S relative to O. Accordingly, the set

4

We add the qualiﬁer ‘broadly’ here since we also allow

for the typical laboratory situation of ensembles of systems

(which may or may not contain more than one element).

Σ of all possible answer statistics on Q which S

admits is the state space of S.

Of course, ultimately not all y

i

will be inde-

pendent such that the full collection of probabili-

ties will yield a redundant parametrization of the

state. However, this is not imp ortant for the mo-

ment and we shall come back to this shortly.

This deﬁnition of the state of a system S ex-

plicitly identiﬁes it with the ‘state of informa-

tion’ that O has acquired about S; O assigns

this state to S according to his information about

the Q

i

∈ Q. As such, the state of system S

is epistemic (i.e. a ‘state of knowledge’) and a

priori only meaningful relative to the observer

O. The interpretation of the quantum state as

a ‘state of information’ is certainly not new and

has been proposed in various ways before (see

also, e.g., [

36–38, 40]). However, the above def-

inition is closest in spirit to the ideas underly-

ing Relational Quantum Mechanics [28, 29] and

the Zeilinger-Brukner interpretation [

15, 31–34]

and thereby generalizes them to the landscape of

theories describing O’s acquisition of information

which we are in the process to establish.

While the state of S is thus a priori only mean-

ingful relative to O, we emphasize that both the

set of questions Q which O may ask and the s tate

space Σ are to be intrinsic to the system S. Oth-

erwise, it would be diﬃcult for two observers to

agree on the description of a given S.

3.2.3 ‘Belief’ updating and ‘collapse’ of the state

At this stage it is important to distinguish single

from multiple shot interrogations. In a

single shot interrogation O interrogates a

single s ystem S, in some prior s tate, with

a number of questions from Q without in-

termediate re-preparations of S. The deﬁ-

nite answers to these questions give O def-

inite information about this speciﬁc S after

the interrogation. Furthermore, his knowl-

edge of Σ and any prior knowledge of S

(acquired through previous interrogations of

identically prepared systems) give him statis-

tical information about any questions he did

not ask S. In conjunction, the new answers

and his prior knowledge thus determine the

state of S after the interrogation. This will

constitute a posterior state update rule, and

thereby a ‘belief’ update form a prior to a

Accepted in Quantum 2017-11-27, click title to verify 10

posterior s tate for a single system which we

shall turn to shortly (and which clearly de-

pends on the speciﬁc way O interrogates S).

This posterior state of S will reﬂect O’s deﬁ-

nite information about every asked question

Q

i

by featuring either y

i

= 0 or y

i

= 1 due to

repeatability, depending on whether the an-

swer was ‘no’ or ‘yes’, respectively (assuming

for now, of course, that Σ is such that the

answers to the selection of questions that O

asked can be known simultaneously).

If this posterior state does not coincide with

the prior state that O assigned to S before

the interrogation, based on his prior infor-

mation about S, then S’s s tate has ‘col-

lapsed’ relative to O during the interroga-

tion. Hence, a state ‘collapse’ only occurs

if O’s posterior information about S does

not coincide with his prior information about

S, i.e. if O experienced an information gain

about S via the interrogation. We shall

therefore view a state ‘collapse’ as O’s infor-

mation gain about this speciﬁc S rather than

a ‘disturbance’

5

of S (we refer the reader also

to [

26,60,66,67] for a related discussion).

mult iple shot interrogation O interrogates

an ensemble of identically prepared systems

S

a

, a = 1, . . . , n, where the interrogation

of every S

a

is a single shot interrogation.

O will carry out such a multiple shot

interrogation to do state tomography, i.e.

to estimate the state of the ensemble {S

a

}

for the speciﬁc setting of preparation or, in

other words, the state of any of the systems

prior to being interrogated by O.

This will also be a ‘belief’ updating, how-

ever, a prior (or ensemble) state updating.

After every interrogation of a system in the

ensemble, O will assign probabilities y

i

to the

Q

i

in the manner described above. This will

then deﬁne the prior state of the next sys-

tem in the ensemble to be interrogated. By

interrogating more and more systems, O will

gain more and more information about the

5

A ‘disturbance’ of the system S is only meaningful if

there was an underlying ontic state (i.e. ‘state of reality’)

to which, however, O would have no access. Here we shall

merely speak about the information th at O has access to

and therefore not make any ontic statements, regarding

them as excess baggage for our purposes.

ensemble state such that his ass ignments of

the y

i

will ﬂuctuate less the larger the num-

ber of interrogated systems. This process

gives rise to an ensemble state updating. In-

dependent of this updating, the prior state

that O assigns to any individual system may

experience a ‘collapse’ during the interroga-

tion of that speciﬁc system because his infor-

mation about the speciﬁc system may have

changed. Accordingly, O will have to distin-

guish the ensemble state from the posterior

state of any system in the ensemble. But the

collection of posterior states determines the

ensemble state.

3.2.4 Elementary s tructure on Σ and Q

The present structure is still too rudimentary for

a quantum (re)construction. We therefore need

more.

Firstly, as in GPTs we will need O to be able to

assign a single prior state to any pair of identical

systems whenever he ﬂips a biased coin in order

to decide which of the two systems he will in-

terrogate. (Equivalently, the preparation method

could involve a biased coin toss , the outcome of

which determines the preparation setting.) That

is, O will be permitted to build convex combina-

tions of states.

Assumption 2. The state space Σ of S is a

closed convex set.

Closure of Σ is assumed because points lying in

the boundary of Σ can be arbitrarily well approx-

imated by states in the interior. Perfect prepa-

ration and arbitrarily good preparation are op-

erationally indistinguishable and so adding the

boundary does not change the physical predic-

tions [

16].

Next, we need to deﬁne some elementary struc-

ture on Q in order to meaningfully speak about

relations among questions (and answers). To this

end, we shall establish additional structure on

the pair (Q, Σ). We declared in assumption

1

that there are to be no trivial questions in Q the

answers to which would be independent of S’s

preparation. We also insisted before that O will

distinguish the diﬀerent ways of preparing a sys -

tem S – and thus the diﬀerent states he can assign

– by the particular answer statistics. We shall

now strengthen these requirements, by asserting

Accepted in Quantum 2017-11-27, click title to verify 11

that there exists a distinguished state of ‘no infor-

mation’, corresponding to the situation that O’s

prior knowledge about S makes it equally likely

for him that the answer to any Q ∈ Q is ‘yes’ or

‘no’.

6

We note that this ass umption is a restric-

tion on O’s model for the pair

(Q, Σ).

7

Assumption 3. There exists a special state in

Σ, called the state of no information, which is

given by y

i

=

1

2

, ∀ Q

i

∈ Q.

This state shall be the prior state that O as-

signs to S (or an ensemble thereof) in a belief

updating whenever he has ‘no prior information’.

For example, a distinct observer O

′

may prepare

a system S and send it to O in such a way that

the latter knows only the associated Σ but not

the preparation setting. In this case, as a prior,

O will assign the s tate of no information to S,

i.e. y

i

=

1

2

, ∀ Q

i

∈ Q. Similarly, there will exist

a special preparation setting which is such that a

multiple shot interrogation on an ensemble pre-

pared in this setting will give totally random an-

swers to O such that he will assign the state of no

information to the ensemble. But note that the

state of any individual system after the interroga-

tion of that system will not be the state of no in-

formation because, through the interrogation, O

will have acquired information about that speciﬁc

system (see the discussion of the state ‘collapse’

above).

Given the state of no information, we shall pre-

liminarily quantify the amount of information α

i

that the deﬁnite answer to any Q

i

∈ Q deﬁnes as

one bit. Similarly, whenever y

i

=

1

2

, as in the

state of no information, we shall say that O has

α

i

= 0 bits of information about Q

i

. In general,

under the premise that information can neither

be negative nor complex, O’s information about

6

We emphasize that GPTs are more general by, in prin-

ciple, permitting state spaces which do not contain such

a distinguished state. However, most operationally inter-

esting GPTs do possess such a state.

7

For instance, we note that on account of the exis-

tence of binary POVMs with an inherent bias, such as

(E

0

= 2/3 · 1, E

1

= 1/3 · 1), the pair given by Q =

{binary POVMs} and Σ = {unit trace density matrices}

cannot satisfy this condition because no unit t race

density matrix exists which yields probability 1/2 for

(E

0

, E

1

). This pair will therefore ultimately not be the

solution of this reconstruction. However, a subset of

{binary POVMs} together with the full quantum state

space will be the solution.

Q

i

should satisfy

0 bit ≤ α

i

≤ 1 bit. (1)

We shall not propose an explicit information mea-

sure α

i

(as a function on Σ) here because this

must follow from the rules on information acqui-

sition postulated b elow and we shall indeed derive

it therefrom later in section

6.8. Until then it will

be suﬃcient to work with this implicit notion of

quantify ing O’s information about any Q

i

.

But the questions O can ask, and the informa-

tion that the corresponding answers deﬁne, may

not be independent. However, a priori the no-

tion of independence of questions, in the sense of

stochastic independence, is state dependent. For

example, for a pair of qubits in quantum theory

the questions Q

1

, ‘is the spin of qubit 1 up in x-

direction?’, and Q

2

, ‘is the spin of qubit 2 up in

x-direction?’, are stochastically independent rel-

ative to the completely mixed state, but fully de-

pendent relative to an entangled state (with cor-

relation in x-direction). As a result of the state

dependence, the independence of questions may

also be viewed as observer dependent. For in-

stance, in quantum theory an observer O

′

could

send O an entangled pure state (with correlation

in x-direction) and refuse to tell O which state it

is. Relative to O

′

, Q

1

and Q

2

will be dependent,

but they will be independent relative to O be-

cause the latter will ass ign the completely mixed

state to the pair prior to measurement.

Since the notion of independence of q uestions

is state dependent, we need a distinguished state

in order to unambiguously deﬁne it – this is the

second purpose of assumption

3 and brings us in

contact with a state update rule. Indeed, suppose

O acquires S in the state of no information and

poses the q uestion Q

i

∈ Q. By assumption

1,

any Q

i

∈ Q is repeatable such that, upon receiv-

ing either the answer ‘yes’ or ‘no’, O will assign

y

i

= 1 or y

i

= 0, respectively, as the probability

for S giving the answer ‘yes’ if posing Q

i

again.

The posterior state update rule, which enables O

to update his information about a speciﬁc S in

compliance with the given answers, must respect

this repeatability. It depends on the details of this

update rule what the probabilities y

j

for all other

Q

j

∈ Q is after having asked only Q

i

. Rather

than fully specifying at this stage what this rule

is, we shall simply assume that O employs one

which is consistent. Whatever this posterior state

update rule, we shall refer to Q

i

, Q

j

∈ Q as

Accepted in Quantum 2017-11-27, click title to verify 12

independent if, after having asked Q

i

to S in

the state of no information, the probability

y

j

=

1

2

. That is, if the answer to Q

i

relative

to the state of no information tells O ‘noth-

ing’ about the answer to Q

j

. We shall require

this relation to be symmetric, i.e., Q

i

is in-

dependent of Q

j

if and only if Q

j

is indepen-

dent of Q

i

.

8

This is equivalent to saying that

Q

i

, Q

j

are stochastically independent with

respect to the state of no information, i.e. the

joint probabilities factorize relative to the

latter, p(Q

i

, Q

j

) = y

∗

i

· y

∗

j

=

1

2

·

1

2

=

1

4

. Here

p(Q

i

, Q

j

) = p(Q

j

, Q

i

) denotes the probabil-

ity that Q

i

and Q

j

give ‘yes’ answers if asked

in sequence to the same S arriving in the

state of no information and y

∗

i

, y

∗

j

are the

individual ‘yes’-probabilities in this distin-

guished state.

9

dependent if, after having asked Q

i

to S in

the state of no information, the probability

y

j

= 0, 1. That is, if the answer to Q

i

rel-

ative to the state of no information implies

also the answer to Q

j

. Again, we require this

relation to be symmetric. This is equivalent

to saying that, relative to the state of no in-

formation, Q

i

, Q

j

are stochastically fully de-

pendent as either p(Q

i

, Q

j

) = y

∗

i

= y

∗

j

=

1

2

= p(¬Q

i

, ¬Q

j

) or p(Q

i

, ¬Q

j

) = y

∗

i

= y

∗

j

=

1

2

= p(¬Q

i

, Q

j

), where ¬Q is the negation of

Q.

10

partially dependent if, after hav ing asked Q

i

to S in the state of no information, the prob-

ability y

j

6= 0,

1

2

, 1. That is, if the answer

to Q

i

relative to the state of no information

8

It should be noted that, while this is true for projec-

tive measurements in quantum theory, it does not hold

for generalized measurements. I thank Tobias Fritz for

pointing this out.

9

We emphasize that this is a deﬁnition of maximal ind e-

pendence. For example, for a qubit two linearly indepen-

dent directions ~n

1

, ~n

2

in the Bloch sphere with ~n

1

· ~n

2

6= 0

would d eﬁne two spin observables ~n

1

· ~σ and ~n

2

· ~σ whose

corresponding projectors would not be maximally inde-

pendent according to this deﬁnition. The corresp onding

questions would be partially d ependent (see below) be-

cause whenever the observer knows the answer to one

question, the probability for a ‘yes’ answer to the other

would be distinct from

1

2

.

10

The most trivial example of a pair of dep endent ques-

tions are clearly Q and ¬Q. But there exist less trivial

ones. E.g., see theorem

5.7 below.

gives O partial information about the answer

to Q

j

. Again, this relation is required to be

symmetric. This is equivalent to saying that

the joint probabilities p(Q

i

, Q

j

) relative to

the state of no information do not factorize

and the answer to one question does not fully

imply the answer to the other.

We shall henceforth assume that Σ is such that

two questions Q

i

, Q

j

which are fully or partially

dependent relative to the state of no information

are also fully or partially dependent relative to

any other state in Σ. These deﬁnitions of (in-

)dependence depend a priori on the update rule

through the j oint probabilities. For our purposes

it will turn out not to be necessary to fully specify

this update rule.

Next, we need a notion of compatibility and

complementarity. Q

i

, Q

j

are

maximally compatible if O may know the an-

swers to both Q

i

, Q

j

simultaneously, i.e. if

there exists a state in Σ such that y

i

, y

j

can

be simultaneously 0 or 1.

maximally complementary if maximal infor-

mation about the answer to Q

i

forbids O to

have any information about the answer to Q

j

at the same time (and vice versa). That is,

every state in Σ which features y

i

= 0, 1 will

necessarily have y

j

=

1

2

(and vice versa).

Consequently, complementary questions are in-

dependent, but independent questions are not

necessarily complementary. Finally, Q

i

, Q

j

are

partially compatible (or complementary) if max-

imal information about one precludes maximal,

but permits non-maximal information about the

other.

This permits us to further constrain the up-

date rule. Firstly, maximal complementarity has

an obvious consequence for an update rule. Sec-

ondly, we shall ass ume the following.

Assumption 4. If Q

i

, Q

j

are maximally com-

patible and independent then asking either shall

not change O’s information about the other, re-

gardless of S’s state. That is, asking Q

i

must

leave y

j

invariant – and vice versa for i, j inter-

changed.

This is to prevent O from losing or gaining in-

formation about some question by asking another

Accepted in Quantum 2017-11-27, click title to verify 13

question which is compatible with but indepen-

dent of the ﬁrst. These constraints on the update

rule turn out to be suﬃcient for our reconstruc-

tion.

Assumption

4 leads to a ﬁrst result which we

shall use at times: it implies what sometimes is

referred to as ‘Specker’s principle’ [68] (see also

[

69–72]). We shall call Q

1

, . . . , Q

n

∈ Q mutually

maximally compatible if there ex ists a state of S

where the answers to all of Q

1

, . . . , Q

n

are known

simultaneously to O.

Theorem 3.2. (‘Specker’s principle’) If

Q

1

, . . . , Q

n

∈ Q are pairwise maximally compat-

ible and pairwise independent then they are also

mutually maximally compatible.

Proof. Let Q

1

, . . . , Q

n

∈ Q be pairwise indepen-

dent and pairwise maximally compatible. On ac-

count of repeatability, after asking S the ques-

tion Q

1

, O will assign th e probability y

1

to be

either 0 or 1, depending on the answer. O may

subsequently ask S the question Q

2

upon w hich

the probability y

2

will likewise be either 0 or 1.

According to assumption

4, this will not change

y

1

because, by assumption, both are maximally

compatible and independent. By the same ar-

gument, upon next asking and receiving an an -

swer to Q

3

, O will assign either y

3

= 1 or

y

3

= 0, while both y

1

, y

2

are unchanged. Re-

peating this argument recursively, it is clear that

O can thereby generate a state of S in which all

y

i

, i = 1, . . . , n, are either 0 or 1 which represents

a state of S where O knows the answers to all of

Q

1

, . . . , Q

n

.

Notice that this property is satisﬁed for classi-

cal bit theory and in quantum theory for projec-

tive, however, not for generalized measurements.

3.2.5 Logical compositions of questions and rules

of inference

Given multiple questions, nothing, in principle,

stops O from composing them via logical con-

nectives to all kinds of propositions which them-

selves constitute questions. The issue is, however,

whether he will be able to get an answer from S to

such a question, i.e. whether it is implementable

on S and thus whether it could b e contained in

Q (see section

3.2.1). For example, given any two

questions Q

i

, Q

j

∈ Q, O could consider the cor-

relation question Q

ij

, ‘are the answers to Q

i

, Q

j

the same?’. Do we allow Q

ij

to also be imple-

mentable on S? Clearly, if Q

i

, Q

j

are maximally

compatible, then Q

ij

is implementable because O

can always ﬁnd the answer to the latter by asking

Q

i

, Q

j

. In this case, since Q

i

, Q

j

are simultane-

ously deﬁned relative to O, we can also write

Q

ij

:= Q

i

↔ Q

j

, (2)

where ↔ is the logical biconditional or

(XNOR).

11

Q

ij

will then automatically be maxi-

mally compatible with both Q

i

, Q

j

.

Let us now contrast this with the situation in

which Q

i

, Q

j

are (partially or maximally) comple-

mentary. The structure introduced thus far does

not preclude the ‘correlation question’ Q

ij

to also

be implementable and, ultimately, contained in Q

even if Q

i

, Q

j

are complementary. In fact, it is not

possible to decide without a theoretical model f or

describing and interpreting (Q, Σ) whether Q

ij

is

implementable on S in the case that Q

i

, Q

j

are at

least partially complementary. What, however, is

unambiguously clear is that for all practical pur-

poses Q

ij

will not be implementable in this case.

Namely, Q

ij

would be a statement about the cor-

relation of two complementary questions Q

i

, Q

j

and O could say ‘the answers are the same’, but

he can never directly test them individually and

see that they are actually ‘the same’ at the same

time. Indeed, Q

i

, Q

j

, Q

ij

would need to form

a mutually (partially or maximally) complemen-

tary set. Thus, from a purely operational per-

spective alone, O could never tell, even in princi-

ple, that Q

ij

was implementable; he simply can-

not get an answer to Q

ij

which he could inter-

pret, based on operationally accessible informa-

tion alone, as a correlation of the complementary

Q

i

, Q

j

.

It thus depends on the model for (Q, Σ )

whether Q

ij

is implementable when Q

i

, Q

j

are

complementary. Since it is impossible for O to

settle this question op erationally, it would re-

quire a mo del which employs hidden and opera-

tionally inaccessible ontic information and which

is devoid of complementarity at an ontic level

to conclude that Q

ij

∈ Q in this case. For ex-

ample, Spekkens’ elegant toy model [

40] and the

‘black boxes’ of [

54] employ ontic states, satisfy

the structure established thus far (at least at the

11

That is Q

ij

= ‘yes’ if Q

i

= Q

j

= ‘yes’ or ‘no’ and

Q

ij

= ‘no’ otherwise.

Accepted in Quantum 2017-11-27, click title to verify 14

epistemic level and modulo restrictions on the no-

tion of convexity), and explicitly feature such a

triple of q uestions.

12

However, such a model is

in conﬂict with our premise of following a purely

operational approach which only speaks about in-

formation that O has access to via direct inter-

rogation. The model by means of which O in-

terprets the answers that he gets from S – and,

hence, the information he can acquire about S –

shall be based entirely on operational statements

that O can, in principle, check through interroga-

tion, and not on propositions that require hidden

and inaccessible ontic information. Accordingly,

O’s theoretical model for Q should not contain

(a question which is logically equivalent to) Q

ij

whenever Q

i

, Q

j

are at least partially complemen-

tary.

Of course, the ‘correlation’ (represented by the

XNOR connective ↔) is only one of many possi-

ble logical connectives. More generally, we shall

require that O’s model for Q does not contain any

logical connectives of complementary questions.

He can only logically connect questions which are

maximally compatible – and thus are simultane-

ously deﬁned with respect to him – such that he

could meaningfully write down a truth table for

the questions to be connected and the connective

question. If he cannot connect questions, he can

also not ask for the connective.

Assumption 5. Let Q

i

, Q

j

∈ Q and ∗ be a logi-

cal connective. According to O’s theoretical model

for Q, Q

i

∗ Q

j

is implementable if and only if

Q

i

, Q

j

are maximally compatible.

12

The reader familiar with Spekkens’ toy model [

40] will

recall the simplest (epistemic) 1-bit system which has four

ontic states ‘1’, ‘2’, ‘3’ and ‘4’. An epistemic restriction

forbids an observer to know the ontic state. Instead, the

epistemic states of maximal knowledge correspond to ei-

ther of the following three questions (and their negations)

Q

1

: “1 ∨ 2”

, Q

2

: “2 ∨ 3” ,

Q

3

: “2 ∨ 4”

,

where ∨ is to be read as ‘or’. Q

1

, Q

2

, Q

3

are mutually

complementary and it can be easily checked that Q

3

coin-

cides with the ‘correlation’ Q

12

of Q

1

and Q

2

: it gives ‘yes’

when the (ontic) answers to Q

1

, Q

2

are equal and ‘no’ oth-

erwise. This relation is cyclic: Q

1

is also the ‘correlation’

Q

23

of Q

2

, Q

3

and Q

2

is the ‘correlation’ Q

13

of Q

1

, Q

3

.

Accordingly, there are three complementary questions for

this 1-bit system, in principle the correct number for a

qubit. Later in section

5.2.3, we will develop a diﬀerent

approach, without ontic states, in order to reason for the

three-dimensionality of the Bloch-sphere.

We recall from section

3.2.1 that O’s theoretical

model for Q only contains implementable ques-

tions (but not necessarily all implementable ques-

tions).

Since ∗ is only applied to connect maximally

compatible questions which have also opera-

tionally simultaneous truth values relative to O,

we shall allow ∗ to be any of the 16 binary connec-

tives (or binary Boolean functions) ¬, ∨, ∧, ↔, . . .

of classical Boolean or propositional logic.

We stress that assumption

5 is a restriction on

O’s model for Q; this assumption clearly does

not rule out that there may exist other (namely,

ontic) models which also y ield a consistent de-

scription of O’s experiences with the s ystems he

is interrogating, yet which ascribe Q

i

∗ Q

j

to be

(logically equivalent to a question) contained in

Q regardless of whether Q

i

, Q

j

are compatible

or not. However, we shall not worry about such

models here.

We note that as sumption

5 does not only ap-

ply to logical connectives of two q uestions, but

arbitrarily many. For example, if Q

i

∗ Q

j

is im-

plementable (and contained in Q), according to

O’s model, then (Q

i

∗ Q

j

) ∗ Q

k

is implementable

too iﬀ Q

i

∗ Q

j

and Q

k

∈ Q are compatible, and

so on. It is also important to note that assump-

tion

5 is a statement about which questions can

be directly connected logically. It does not entail

that O, according to this model, can only form

meaningful logical expressions containing exclu-

sively questions which are mutually maximally

compatible. Namely, questions which are com-

patible might be logically further decomposable

into other questions for which diﬀerent compat-

ibility relations hold. For instance, it might be

that Q

k

in the expression (Q

i

∗ Q

j

) ∗ Q

k

is only

compatible with Q

i

∗Q

j

, yet not with Q

i

or Q

j

in-

dividually. In this case, there is no harm in never-

theless forming the expression (Q

i

∗Q

j

)∗Q

k

even

though, say, Q

j

and Q

k

might be complemen-

tary s uch that O cannot write down a truth table

with all three Q

i

, Q

j

, Q

k

separately. However, it

is clear that in this case there is a hierarchy in

which the logical connectives are to be executed.

More speciﬁcally, the connective ∗ in (Q

i

∗Q

j

)∗Q

k

cannot be associative since Q

i

∗ (Q

j

∗ Q

k

) is not

implementable according to O’s model because

Q

j

, Q

k

are, by assumption, complementary and

can therefore not be directly connected. That is

to say, the left ∗ in (Q

i

∗Q

j

)∗Q

k

must be executed

Accepted in Quantum 2017-11-27, click title to verify 15

ﬁrst and Q

k

can only be logically connected with

the result. Thus, assumption

5 does admit logical

expressions containing also mutually complemen-

tary questions as long as the latter are not directly

connected through a logical connective. What as-

sumption

5 entails is that the ordering of the exe-

cution of the various logical connectives in a com-

position has to respect a hierarchy which ensures

that at every step two compatible subexpressions

are connected. This will become important later

in the reconstruction where we shall encounter

concrete such examples.

This brings us to the rules of inference by

means of which O may transform or evaluate

logical expressions of questions and derive logi-

cal identities, e.g. to establish possible logical re-

lations among various question outcomes. The

state update rules need, in particular, respect

these rules of inference. Again, the rules of in-

ference which O may employ depend on his the-

oretical model for (Q, Σ). For instance, in ontic

models all q uestions will typically have simultane-

ous values, in contrast to purely operational ones,

which will require diﬀerent rules of inference.

In line with our operational premise, we shall

require that the rules of inference of O’s model

must not only be consistent with O’s experiences

with the s ystems he is interrogating, but also

testable. In particular, any logical identities de-

rived from these rules must be testable through

interrogations. It is thus appropriate to call them

operational rules of inference.

Classical rules of inference require that the

questions or propositions in a logical expression

have truth values simultaneously. In O’s model

any truth value must be operational such that

only mutually maximally compatible questions

have simultaneous meaning. Nothing stops O

from applying classical rules to such kind of ques-

tions. We shall thus require that it is appropri-

ate for O to employ classical rules of inference,

according to Boolean logic, – and only those –

for any logical expression or subexpression which

only contains questions that are mutually maxi-

mally compatible. In all other cases (which still

must abide by assumption

5), any possible rule of

inference transforming the composition of ques-

tions will directly involve mutually complemen-

tary questions whose truth values, however, have

no simultaneous operational meaning to O. Ac-

cordingly, classical rules will be operationally in-

appropriate in such cases and we shall demand

more precisely the following.

Assumption 6. O’s model for Q allows him to

apply exclusively classical rules of inference, ac-

cording to the rules of Boolean logic, to (trans-

form or evaluate) any logical expression (or

subexpression of a larger expression) which is

composed purely of mutually maximally compati-

ble questions. This holds regardless of whether

the mutually compatible q uestions can be logi-

cally decomposed into other questions which fea-

ture diﬀerent compatibility relations. In all other

cases, no classical rules of inference (and thus no

classical logical identities) are valid.

In consequence, O can take any set of mutually

compatible questions and treat them, with clas-

sical logic, as a Boolean algebra. The ass ump-

tion states what is possible for compositions of

compatible questions and what is not possible f or

comp os itions involving also complementary ques-

tions. This will become useful later in section

5.2.7 where we shall also see what is possible for

comp os itions of complementary questions.

We emphasize that as sumption 6 by itself does

not severely constrain the nature of any conceiv-

able ‘hidden variable model’ which could also con-

sistently describe O’s world. However, in con-

junction with two of the subsequent quantum

principles, it will rule out local ‘hidden variables’

in section 5.2.7.

3.2.6 Parametrization of S’s state and t omogra-

phy

Now that we have a notion of independence on Q

we can say more about the parametrization and

thus representation of S’s state relative to O. Not

all Q

i

∈ Q will be necessary to describe the state;

the pairwise independent questions shall be the

fundamental building blocks of the landscape L

of inference theories.

Supp ose there is a maximal set Q

M

=

{Q

1

, . . . , Q

D

∈ Q} of D pairwise independent

(but not necessarily compatible) questions, such

that no further question Q ∈ Q\Q

M

exists which

is pairwise independent from all members of Q

M

too. Then, every other Q ∈ Q \ Q

M

is either

(i) dependent on exactly one Q

j

∈ Q

M

and in-

dependent of all other Q

M

∋ Q

l

6= Q

j

(if Q

was dependent on Q

j

∈ Q

M

and partially de-

pendent on Q

M

∋ Q

l

6= Q

j

, then Q

j

, Q

l

could

Accepted in Quantum 2017-11-27, click title to verify 16

not be independent), (ii) partially dependent on

some and independent of the other questions in

Q

M

, or (iii) partially dependent on all Q

j

∈ Q

M

.

While O will not be able to infer maximal infor-

mation about the answers to questions of cases

(ii) and (iii) from his information about individ-

ual (or even subsets of) memb ers of Q

M

alone,

the question is whether his information about the

full set Q

M

will be suﬃcient to do so.

Deﬁnition 3.3. (Informational Complete-

ness) A maximal set Q

M

= {Q

1

, . . . , Q

D

∈ Q}

of pairwise independent questions is said to be in-

formationally complete if O’s information about

the questions in Q

M

determines his information

about all other Q ∈ Q \ Q

M

in such a way

that the probabilities y

i

which O assigns to eve ry

Q

i

∈ Q

M

are suﬃcient in order for him to com-

pute the probabilities y

j

∀ Q

j

∈ Q for all prepa-

rations of S. In this case, the set of probabilities

{y

i

}

D

i=1

of the Q

i

∈ Q

M

parametrizes the state

that O assigns to S and thereby yields a com-

plete description of the state space Σ. We shall

call D the dimension of Σ.

We emphasize that the dimension D of Σ will

ultimately not be the Hilbert space dimension in

quantum theory, but the dimension of the set of

density matrices.

If Q

M

was not informationally complete, O

would require further questions that are partially

dependent on at least some of the elements in

Q

M

in order to fully describe the system S and

its state. This situation cannot be precluded,

given the structure we have devis ed so far. How-

ever, we deem it undesirable, given that we would

like to employ pairwise independent questions as

building block s for system descriptions. We shall

therefore require that no more independent infor-

mation about S can be learned from any question

in addition to a maximal set Q

M

.

Assumption 7. Every maximal set Q

M

of pair-

wise independent questions is informationally

complete.

There may exist (even continuously) many such

informationally complete sets of questions on Q

which, at this stage, may still be either discrete or

continuous. However, their dimensions are equal.

Lemma 3.4. The dimensions of all maximal sets

Q

M

on Q are equal (if ﬁnite).

Proof. Consider any Q

M

. On account of pairwise

independen ce, for each Q

i

∈ Q

M

there must exist

a state such that O only knows the answer to

this Q

i

with certainty, but nothing about any

other Q

j6=i

∈ Q

M

, i.e. y

i

= 0 or 1 and all other

y

j6=i

= 1/2. (Namely, O could have asked Q

i

to

S prepared in the state of no information.) But

since also in the state of no information y

j6=i

=

1/2, y

i

cannot in general be computed fr om the

y

j6=i

. Hence, given an inform ationally complete

set Q

M

, all associated probabilities {y

i

}

D

i=1

are

necessary to parametrize the full state space Σ.

Now Σ is a convex set (see assum ption

2).

Given the lack of redundancy in {y

i

}, a given

informationally complete Q

M

with D elements

therefore encodes Σ as a D-dimensional convex

subregion of R

D

. Su ppose there is a second Q

′

M

with D

′

elements. In terms of Q

′

M

, Σ would be

described as a D

′

-dimensional convex s ubregion

of R

D

′

. But clearly, th ese two descriptions of Σ

must be isomorphic which, for D ﬁnite, is only

possible if D ≡ D

′

.

Finiteness of D will later be established in sec-

tion

5 with the help of the principles.

Any such Q

M

establishes a question refe rence

frame on Q (see also [73]) and thereby also a ‘co-

ordinate s ystem’ on Σ. In particular, in order

to do state tomography with a multiple shot in-

terrogation, as outlined in section

3.2.3, it will

be suﬃcient for O to interrogate an ensemble of

identically prepared S with the questions within

a given Q

M

only. Given a speciﬁc Q

M

, there

are now three equivalent ways for O to describe

S’s state: he could represent it by either the D-

dimensional ye s- or no-vector

~y

O→S

:=







y

1

y

2

.

y

D







, ~n

O→S

:=







n

1

n

2

.

n

D







, (3)

of probabilities y

i

and n

i

, i = 1, . . . , D, that the

answers to question Q

i

∈ Q

M

are ‘yes’ and ‘no’,

respectively. That is,

~y

O→S

+ ~n

O→S

= p

~

1, (4)

where

~

1 is a D-dimensional vector with a 1 in each

of its entries. Here we have introduced a new pa-

rameter: p is the probability that S is present

at all. (For example, the method of preparation

Accepted in Quantum 2017-11-27, click title to verify 17

could be such that O cannot always tell with cer-

tainty when a system is spit out by the prepara-

tion device.) This probability will rescale all y

i

and n

i

simultaneously. However, it only becomes

relevant in subsection

3.2.8 and otherwise is usu-

ally taken to be p = 1. Evidently, the assignment

of which answer to Q

i

is ‘yes’ and which is ‘no’ is

arbitrary, but any consistent such assignment is

ﬁne for us.

But he could also represent the state redun-

dantly as a 2D-dimensional vector

~

P

O→S

=

~y

~n

!

(5)

which will turn out to be convenient especially

when p < 1. We shall write a state with a sub-

script O → S to emphasize that it is the state of

O’s information about S.

Lastly, this structure also puts us into the po-

sition to specify O’s total amount of informa-

tion about S. Clearly, the total amount of in-

formation must be a function of the state. Let

Q

M

= {Q

1

, . . . , Q

D

} be an informationally com-

plete set of questions in Q. Given that these ques-

tions carry the entire information O may know

about S, we deﬁne the total information I

O→S

as

the sum of O’s information about the Q

i

∈ Q

M

,

as measured by the α

i

(

1):

I

O→S

(~y

O→S

) :=

D

X

i=1

α

i

. (6)

(We imagine O as an agent who can write down

results on a piece of paper and add these up.) The

speciﬁc relation between α

i

and ~y

O→S

will be de-

rived later in section 6.8; it will not be the Shan-

non entropy which, as discussed in section

6.1,

describes average information gains in repeated

experiments rather than the information content

in the state ~y

O→S

.

3.2.7 Composite systems

For later purpose we need to clarify the notion

of a composite system. Since we are pursuing a

purely operational approach, the notion of a com-

position of systems must be deﬁned in terms of

the information accessible to O through interro-

gation. O should, in principle, be able to tell

a composite system apart into its constituents.

Accordingly, we require that the set of questions

which can be pos ed to the composite system con-

tains all questions about the individual subsys-

tems and that the remaining questions are lit-

erally composed of these individual questions or

iteratively composed of compositions of them.

13

Deﬁnition 3.5. (Composite System) Let

Q

A

, Q

B

denote the full question sets associated

to S

A

, S

B

, respectively. Two systems S

A

, S

B

are

said to form a composite system S

AB

if any

Q

a

∈ Q

A

is maximally compatible with and in-

dependent of any Q

b

∈ Q

B

and if

Q

AB

= Q

A

∪ Q

B

∪

˜

Q

AB

, (7)

where

˜

Q

AB

only contains composite ques-

tions which are i terative compositions, Q

a

∗

1

Q

b

, Q

a

∗

2

(Q

a

′

∗

3

Q

b

), (Q

a

∗

4

Q

b

) ∗

5

Q

b

′

, (Q

a

∗

6

Q

b

) ∗

7

(Q

a

′

∗

8

Q

b

′

), . . ., via some logical con-

nectives ∗

1

, ∗

2

, ∗

3

, · · · , of individual questions

Q

a

, Q

a

′

, . . . ∈ Q

A

about S

A

and Q

b

, Q

b

′

, . . . ∈ Q

B

about S

B

. For an N -partite system, we use this

deﬁnition recursively.

Of course, thanks to assumption

5, O can only

directly compose questions with a logical connec-

tive ∗ if they are compatible.

There are further repercussions: let Q

M

A

, Q

M

B

be informationally complete sets for S

A

, S

B

, re-

spectively. Then an informationally complete set

Q

M

AB

for a composite S

AB

can be formed itera-

tively by joining (in a set union sense) Q

M

A

, Q

M

B

and adding to it a maximal pairwise independent

set of questions which are the logical connectives

of members of Q

M

A

with elements of Q

M

B

and/or

iterative compositions of such compositions and

possibly questions from Q

M

A

, Q

M

B

. In section

5.2.1 below, we shall determine which logical con-

nectives ∗ we may employ to build up Q

M

AB

from

Q

M

A

, Q

M

B

.

3.2.8 Time evolution of S’s state

We shall assume that O has access to a clock and

begin with a deﬁnition:

static states: A state

~

P

O→S

which is constant

in time – corresponding to the situation that

13

For the GPT specialists, we emphasize that this deﬁ-

nition has noth ing to do with the usual requirement of local

tomography [

15–18,20,55,59] in GPTs. To give a concrete

example, we shall see later that two-level systems over real

Hilb ert spaces (rebits) satisfy this deﬁnition while violat-

ing local tomography.

Accepted in Quantum 2017-11-27, click title to verify 18

O will always assign the same probability

to each of his question outcomes – is called

static.

If all the states O would assign to any system

S were static – according to his information –

O’s world would be a rather boring place. In or-

der not to let O die of boredom, we allow the

probability vector

~

P

O→S

to change in time. (But

we require that Q and Σ are time independent.)

We shall now make use of operational reasoning,

to brieﬂy consider how

~

P

O→S

evolves under time

evolution. The following argument bears some

analogy to an argument typically employed in

the GPT framework [

14–18, 55] concerning con-

vex mixtures and measurements, however, is dis-

tinct in nature as we instead consider states and

their mixtures under time evolution.

Let O have access to two identical

14

non-

interacting systems S

1

and S

2

. O’s information

about S

1

and S

2

can be diﬀerent such that they

are allowed to be in distinct states ~y

O→S

1

and

~y

O→S

2

relative to O. N ow let O perform a (bi-

ased) coin ﬂip which yields ‘heads’ with probabil-

ity λ and ‘tails’ with probability (1 − λ).

15

Given

that S

1

and S

2

are identical, O can ask the same

questions to both s ystems. If the coin ﬂip yields

‘heads’, O will interrogate S

1

, if it yields ‘tails’ O

will interrogate S

2

.

PSfrag replacements

O

S

1

S

2

λ 1 − λ

In particular, before toss ing the coin, say at

time t

1

, the probability he assigns to receiving a

‘yes’ answer to question Q

i

(asked to either S

1

or S

2

depending on the outcome of the coin ﬂip)

is simply the convex s um y

12

i

(t

1

) = λ y

1

i

(t

1

) +

(1 − λ) y

2

i

(t

1

). (O is allowed to build this con-

vex combination thanks to assumption

2.) But

this holds at any time t before O tosses the coin

(which O can also choose never to do).

16

As will

14

That is, both S

1

and S

2

carry the same Q and Σ.

15

Equivalently, O m ay use another system to which he

has assigned a stable state vector by repeated interroga-

tions on identically prepared systems. Given an arbitrary

elementary question Q, he can use the probability that the

answer is ‘yes’ as λ and the probability that the outcome

is ‘no’ as (1 − λ) and thus use Q as the ‘coin’.

16

We assume λ to be constant in time.

become clear shortly, it is convenient to consider

the states

~

P

O→S

with 0 ≤ p ≤ 1 for now. We

shall return to the yes-vector later in section

6.

We then have for all t

~

P

O→S

12

(t) = λ

~

P

O→S

1

(t) + (1 − λ)

~

P

O→S

2

(t). (8)

This convex combination is the state of an ‘eﬀec-

tive’ system S

12

(identical to S

1

, S

2

) describing

O’s information about the coin ﬂip scenario. Note

that S

12

is not a composite system according to

deﬁnition

3.5 because Q

12

= Q

1

= Q

2

.

Denote by T

k

the time evolution map of the

state of system S

k

, k = 1, 2, 12, from t = t

1

to

t = t

2

. Under the assumption that

~

P

O→S

1,2

(t)

evolve independently of each other, given that S

1

and S

2

do not interact, equation (

8) implies

T

12

[

~

P

O→S

12

(t

1

)] = T

12

[λ

~

P

O→S

1

(t

1

)

+ (1 − λ)

~

P

O→S

2

(t

1

)]

= λ T

1

[

~

P

O→S

1

(t

1

)] (9)

+ (1 − λ) T

2

[

~

P

O→S

2

(t

1

)].

All three states

~

P

O→S

k

, k = 1, 2, 12, are elements

of the same Σ. As such, we assume the following.

Assumption 8. Every time evolution map can

act on any state in Σ.

Next, we restrict to the situation where O ex-

poses both S

1

, S

2

to evolve under the same time

evolution T := T

1

= T

2

. Thanks to assumption

8, (9) becomes in this case

T [

~

P

O→S

12

(t

1

)] = λ T [

~

P

O→S

1

(t

1

)]

+ (1 − λ) T [

~

P

O→S

2

(t

1

)]

and T is convex linear. This is clear since O may,

in particular, prepare S

1

, S

2

in identical states

~

P

O→S

1

(t

1

) =

~

P

O→S

2

(t

1

) =

~

P

O→S

12

(t

1

). This spe-

cial case, when inserted into (

9), y ields T

12

= T .

This scenario can be easily generalized to ar-

bitrarily many identical systems. Remarkably, it

can be shown that convex linearity of operations

on probability vectors, in fact, implies full lin-

earity of the operation [

14, 55]. Hence, the time

evolution of the state must be linear:

~

P

O→S

(t

2

) = T [

~

P

O→S

(t

1

)]

= A(t

1

, t

2

)

~

P

O→S

(t

1

) +

~

V (t

1

, t

2

),

where A(t

1

, t

2

) is a 2D × 2D matrix and

~

V some

2D-dimensional vector. Given our assumption

above that O’s world is not static, A(t

1

, t

2

) will

Accepted in Quantum 2017-11-27, click title to verify 19

generally be a non-trivial matrix. Requiring that

the last relation als o holds at t

1

= t

2

for all

initial states, it follows that

~

V (t, t) = 0 and

A(t, t) = 1. Demanding further that the sp e-

cial state

~

P

O→S

=

~

0, corresponding to p = 0

and the absence of a system, is invariant un-

der time evolution (such that no system or any

information is created out of ‘nothing’), ﬁnally

yields

~

V ≡

~

0, ∀ t

1

, t

2

.

17

Given that

~

P

O→S

(t) is a

probability vector for all t and (

4) must always

hold, A(t

1

, t

2

) must be a nonnegative (real) ma-

trix which is stochastic in any pair of components

i and i + D of

~

P

O→S

.

Consequently, the elementary and natural as-

sumption

8 rules out non-linear and state de-

pendent time evolution, such as in Weinberg’s

non-linear extension of quantum mechanics [74],

and thereby also eliminates the many operational

problems that arise with it (e.g., superluminal

signaling) [

75–77].

18

For later purpose, we impose a further condi-

tion on the evolution of S’s state.

Assumption 9. (Temporal Translation In-

variance) There exist no distinguished instants

of time in O’s world such that O is free to set

any instant he desires as the instant t = 0. Time

evolution, as perceived by O, is therefore (tempo-

rally) translation invariant A(t

1

, t

2

) = A(t

2

−t

1

).

Since the time evolution matrix A can only de-

pend on the duration elapsed, but not on the par-

ticular instant of time, we can collect the above

results in the simple form:

~

P

O→S

(t

2

) = A(∆t)

~

P

O→S

(t

1

), ∆t = t

2

−t

1

. (10)

We do not yet have suﬃcient evidence to con-

clude that a given time evolution will be described

by a continuous one-parameter matrix group be-

cause we neither know (i) that A is invertible for

17

A similar argument would not work for the evolution

of ~y

O→S

because, in principle, ~n

O→S

6=

~

0 is possible even

if ~y

O→S

=

~

0 (where

~

0 is a D-dimensional zero vector).

This is the reason why here it is more convenient to work

with

~

P

O→S

, which contains the information about both

~y

O→S

, ~n

O→S

, rather than ~y

O→S

alone. In fact, we shall

see later in section 6 that the evolution of the latter does

involve an aﬃne D-dimensional vector

~

V

′

6=

~

0.

18

It should be emphasized that assumption

8 is also tac-

itly made in the GPT framework [

14–18, 55] for any kind

of transformations on states, such that the present setup

is in this regard not less potent.

all ∆t, (ii) that the evolution is actually continu-

ous, nor (iii) that every composition of evolution

matrices is again an evolution matrix. We shall

return to this question in section

6 with the help

of the set of principles for quantum theory and

also defer the discussion of the time evolution of

~y

O→S

until then.

For now we note, however, ﬁrstly, that any

A(∆t) acting on

~

P

O→S

implies a unique map

T

∆t

(~y

O→S

) acting on the yes-vector thanks to (

4,

5). Secondly, a multiplicity of time evolutions

of S is possible, depending on the physical cir-

cumstances (interactions) to which O may sub-

ject S. The set of all p ossible time evolutions

which O shall henceforth be able to implement

will be denoted by T . This set need not contain

all physically possible time evolutions. Indeed,

upon imposing the quantum principles, T will be-

come the set of unitaries (rather than of arbitrary

completely pos itive maps). Clearly, T is part of

O’s model for describing S; his model is thus a

triple (Q, Σ, T ).

3.2.9 The theory landscape L

The landscape L of theories describing O’s acqui-

sition of information about a physical system S

is now the set of all theories which comply with

the structure and assumptions established in this

section.

In the sequel, we shall restrict O’s attention

solely to composite systems of N ∈ N general-

ized bits (gbits), where a single gbit is character-

ized by the fact that O can maximally know the

answer to a single question at once such that it

can carry at most one bit of information. Ev-

ery inference theory speciﬁes for every system of

N gbits a triple (Q

N

, Σ

N

, T

N

). We shall be con-

cerned with the landscape of gbit theories L

gbit

which contains all gbit theories, satisfying as-

sumptions

1–9. To name concrete examples at

this stage, L

gbit

contains, among a continuum of

other theories, classical bit, rebit and qubit the-

ory for all N ∈ N. We shall see more of this later,

but for now we summarize their characteristics

for N = 1,

classical bit theory gives Q

1

= {Q, ¬Q},

Σ

1

≃ [0, 1] (for normalized states) with ex-

tremal points corresponding to Q = ‘yes’ and

‘no’, respectively, and T

1

≃ Z

2

.

19

There are precisely two states of maximal information

Accepted in Quantum 2017-11-27, click title to verify 20

rebit theory (two-level systems on real Hilbert

spaces) yields Q

1

≃ S

1

and every two maxi-

mally complementary questions are informa-

tionally complete. Σ

1

≃ D

2

(for normalized

states) and T

1

≃ SO(2).

qubit theory has Q

1

≃ S

2

and every triple

of mutually maximally complementary ques-

tions are informationally complete. Σ

1

≃ B

3

(for normalized states) and T

1

≃ SO(3).

With all the assumptions made along the way,

the theory landscape devised here is perhaps not

quite as general as the commonly employed land-

scape of generalized probability theories [

14–18,

23, 55, 60, 61]. In particular, GPTs easily handle

arbitrary ﬁnite dimensional systems, while this

remains to be done within the present frame-

work. N evertheless, the new landscape L

gbit

is

large enough and prov ides new tools for a non-

trivial and instructive (re)construction of qubit

quantum theory that approaches the latter from a

conceptually and technically new angle. To facil-

itate future generalizations of this work, we have

also attempted to spotlight the assumptions un-

derlying L

gbit

as clearly as possible in this s ection.

4 Principles for the quantum theory of

qubits as rules on information acquisi-

tion

We shall now use the landscape L

gbit

of gbit theo-

ries and formulate the principles for the quantum

theory of qubit systems on it as rules on an ob-

server’s information acquisition. These principles

constitute a set of ‘coordinates’ of qubit quantum

theory on L

gbit

.

that O can assign to a classical bit

~

P

O→S

=



1

0



,

~

P

O→S

=



0

1



,

corresponding to ‘yes’ and ‘no’ answers to Q. Time evo-

lution is described by the abelian group Z

2

, given by

1 =



1 0

0 1



, P =



0 1

1 0



,

where P swaps the two states. For classical bit theory, the

permitted time evolution is therefore discrete.

4.1 The rules

As we are reconstructing a technically and empir-

ically well-established theory, we are in the fortu-

nate position to avail ourselves of empirical ev-

idence and earlier ideas on characterizing quan-

tum theory in order to motivate a set of basic

postulates. However, the ultimate justiﬁcation

for these postulates will be their success in sin-

gling out qubit quantum theory within the infer-

ence theory landscape L

gbit

which will be com-

pleted in [

1]. As ‘coordinates’ on a theory space

these principles will not be unique and one could

ﬁnd other equivalent sets. (As usual, at least

many roads lead to Rome.) But we shall take

the below ones as a ﬁrst working set which re-

stricts the informational relation between O and

S, where S will be a composite system (c.f. def-

inition

3.5) of N ∈ N gbits. Each principle will

be ﬁrst expressed as an intuitive and colloquial

statement, followed by its mathematical meaning

within L

gbit

.

We take it as an empirical fact that there ex-

ist physical systems about which only a limited

amount of information can be known at any one

moment of time. The standard quantum example

is a spin-

1

2

particle about which an experimenter

may only know its polarization in a given spatial

direction, but nothing independent of that; the

polarization ‘up’ or ‘down’ corresponds to one bit

of information. But there is also a typical classi-

cal example, namely a ball which may be located

in either of two identical boxes, the deﬁnite po-

sition ‘left’ or ‘right’ corresponding to one bit of

information. Informationally, these two examples

incarnate the most elementary of systems, a gbit,

supporting maximally just one proposition at a

time. But, clearly, there are more complicated

systems supporting other limited amounts of in-

formation. We shall take this simple observation

and raise the existence of an information limit to

the level of a principle.

More precisely, in analogy to von Weizsäcker’s

‘ur-theory’ [

78–80], we shall restrict O’s world

to be a world of elementary alternatives which

thereby consists only of systems which can be de-

comp os ed into elementary gbits.

20

All physical

20

However, in contrast to [

78–80], we shall be much

less ambitious here and will not attempt t o deduce the

dimension of space or space-time symmetry groups from

systems of elementary alternatives. Recent developments

[

21, 42, 59, 81], on the other hand, unravel a deep rela-

Accepted in Quantum 2017-11-27, click title to verify 21

quantities in O’s world are to be ﬁnite such that

he can record his information about them on a

ﬁnite register. We wish to characterize the com-

posite systems in O’s world according to the ﬁnite

limit of N bits of information he can maximally

inquire about them.

Rule 1. (Limited Information) “The observer

O can acquire maximally N ∈ N independent

bits of information about the system S at any

moment of time.”

There exists a maximal set Q

i

, i = 1, . . . , N, of

N mutually independent and maximally compat-

ible questions in Q

N

and no su bset in Q

N

can

contain more than N questions with that prop-

erty.

In other words, O can ask maximally N inde-

pendent questions

21

at a time to S. Accordingly,

this rule immediately implies that O can distin-

guish maximally 2

N

states of S in a single shot

interrogation because there will be 2

N

possible

answers to the Q

1

, . . . , Q

N

. Since Q

N

, Σ

N

are in-

trinsic to S, also the maximal amount of N bits

that each S can carry must be intrinsic to it and

thus be observer independent. In fact, this rule

can be regarded as a deﬁning property of all in-

ference theories in the gbit landscape L

gbit

and

can thus clearly not distinguish between classical

bits, rebits, qubits, etc.

We take another empirical f act and elevate

it to a fundamental principle: despite the lim-

ited information accessible to an experimenter

at any moment of time, there always exists ad-

ditional independent information that she may

learn about the observed system at other times.

This is Bohr’s complementarity principle [

82].

Consider, for instance, the prototypical quantum

physics experiment: Young’s double slit exper-

iment. The experimenter can choose whether

to obtain which-way-information or an interfer-

ence pattern, but not both; the complete knowl-

edge of whether the particle went through the

left or right slit is at the expense of total ig-

norance about the information pattern and vice

tion between (a) simple conditions on operations with sys-

tems carrying ﬁn ite information (e.g., communication with

physical systems) and (b) the dimension and symmetry

group of the ambient space or space-time.

21

Obviously, combinations (e.g., ‘correlations’) of the

compatible Q

i

will deﬁne other bits of information which,

however, will be dependent once the Q

1

, . . . , Q

N

are asked

(we shall return to this in detail in section

5).

versa. (For an informational discussion of the

particle-wave duality in Young’s double slit ex-

periment and a Mach-Zehnder interferometer, see

also [

32,35].) Similarly, in a Stern-Gerlach exper-

iment an experimenter may determine the polar-

ization of a spin-

1

2

particle in x-direction, but will

be entirely oblivious about the polarization in y-

and z-direction. A subsequent measurement of

the spin of the same particle in y-direction will

render her previous information about the polar-

ization in x-direction obsolete and keep her ig-

norant about the spin in z-direction and so on.

That is, systems empirically admit many more

independent questions than they are able to an-

swer at a time – thanks to the information limit.

We shall now return to the relation between O

and S and accordingly stipulate that complemen-

tarity exists in O’s world, however, we shall say

nothing more about how much complementary in-

formation may exist.

Rule 2. (Complementarity) “The observer O

can always get up to N new independent bits of

information about the system S. But whenever

O asks S a new question, he experiences no n et

loss in his total information about S.”

There exists another maximal set Q

′

i

, i =

1, . . . , N, of N mutually independent and

maximally compatible questions in Q

N

such

that Q

′

i

, Q

i

are maximally complementary and

Q

′

i

, Q

j6=i

are maximally compatible and indepen-

dent.

22

That is, after asking S a set of N indepen-

22

Alternatively, one could formulate the technical part

of this rule as follows: In the N = 1 case of a single

gbit there exists, for every Q

1

∈ Q

1

, another Q

′

1

∈ Q

1

such that Q

1

, Q

′

1

are maximally complementary. Deﬁni-

tion

3.5 would then immediately imply that a compos-

ite sy stem of N gbits features two sets of questions, Q

i

,

i = 1, . . . , N, and Q

′

j

, j = 1, . . . , N , with the following

properties: Q

i

, Q

′

i

are questions about the elementary gbit

labeled by i, all Q

i

are maximally independent and max-

imally compatible, all Q

′

j

are maximally independent and

maximally compatible and any pair Q

i

, Q

′

j6=i

is maximally

independent and compatible and every pair Q

i

, Q

′

i

corre-

sponding to th e same gbit is maximally complementary.

This is a priori not equivalent to the current formulation

of the technical part of rule

2 in the main text as the lat-

ter does not necessarily refer to questions about individual

subsystems of a composite system. However, the alterna-

tive formulation would also be enough to get the correct

structure of the informationally complete sets for N qub its

and N rebits below. While the alternative formulation is

simpler, we keep the other one as it is also published in

this form in the companion article [

1].

Accepted in Quantum 2017-11-27, click title to verify 22

dent elementary propositions Q

i

, i = 1, . . . , N,

in a single shot interrogation, O can pose a new

(N + 1)th elementary question Q

′

1

to S. Since O

can only know N independent bits of informa-

tion about S at a time and asking a new question

does not lead to a net loss of information, the

single bit of his previous information about Q

1

must have become obsolete upon learning the an-

swer to Q

′

1

, while O’s total information about S

is still N bits. The rule allows O to perform the

same procedure until he replaces his N old by N

new bits of information about S. As O’s infor-

mation about S has changed, the state of S rel-

ative to O will necessarily experience a ‘collapse’

whenever he asks a new complementary question.

As a result, it is the observer who decides which

information (e.g., spin in x- or y-direction) he will

obtain about the system by asking speciﬁc ques-

tions. But clearly O will have no inﬂuence on

what the answer to these questions will be and

any answer will come at the price of total igno-

rance about complementary questions.

A few further explanations concerning this

complementarity rule are in place. First of all,

this rule clearly rules out classical bit theory. Sec-

ondly, the postulate asserts the existence of max-

imally complementary questions in O’s world,

however, makes no statement about whether par-

tially independent or partially complementary

questions may exist too. Thirdly, the p eculiar

requirement that every question of rule

2 is com-

plementary to exactly one from rule

1 is chosen

such that each s ubsystem of a composite system

features complementarity. For example, consider

the N = 1 case, corresponding to a single gbit,

for which rules

1 and 2 entail a complementary

pair Q

1

, Q

′

1

. This complementarity per gbit gen-

eralizes to arbitrary N since every Q

i

in rule

1

and every Q

′

j

in rule

2 may correspond to one of

N gbits. More complicated complementarity re-

lations arising from rules

1 and 2 will be discuss ed

in section

5.

Notice that the complementarity rule implies

the existence of a notion of ‘su perposition’ – even

in states of maximal information. For example,

take N = 1 and let O know the answer to Q

1

with certainty, y

1

= 1, such that his informa-

tion about S saturates the limit of one bit. This

will leave him oblivious about the complementary

Q

′

1

such that he would have to assign y

′

1

=

1

2

.

The state of information he has about S can be

interpreted as being in a ‘superposition’ of the

Q

′

1

alternatives ‘yes’ and ‘no’, but in this case

with ‘equal weight’ because both alternatives are

equally likely according to O’s knowledge.

23

We

emphasize that the necessary presence of super-

positions of elementary alternatives, as perceived

by O, is a consequence of the information limit

and complementarity.

We would like to stress that rules

1 and 2

are conceptually motivated by related propos-

als which have been put forward ﬁrst by Rov-

elli within the context of relational quantum me-

chanics [

28] and later, indep endently, by Brukner

and Zeilinger within attempts to understand the

structure of quantum theory via limited informa-

tion [

31,32,34,35,73]. However, in order to com-

plete these ideas to a full reconstruction of qubit

quantum theory, we have to impose further rules.

Speciﬁcally, rules

1 and 2 say nothing about

what happens in-between interrogations. We re-

quire that O shall not gain or lose information

about an otherwise non-interacting S without

asking questions.

Rule 3. (Information Preservation) “The to-

tal amount of information O has about (an oth-

erwise non-interacting) S is preserved in -between

interrogations.”

I

O→S

is constant in time in-between interroga-

tions for (an otherwise non-interacting) S.

Corresp ondingly, I

O→S

is a ‘conserved charge’

of time evolution; this is a simple observation

which will become extremely us eful later. In fact,

notice that rule

3 could also be viewed as deﬁn-

ing the notion of non-interacting systems (as per-

ceived by O).

Finally, we come to the last rule of this

manuscript which is about time evolution. Em-

pirical evidence suggests, at least to a good ap-

proximation, that the time evolution of an ex-

perimenter’s ‘catalogue of k nowledge’ about an

observed system is continuous – in-between mea-

surements. More precisely, the speciﬁc prob-

23

We shall later see that, even in a state of maximal

information of N bits, O m ay possibly have only partial

or incomplete knowledge about the outcomes of any ques-

tion in an informationally complete set Q

M

N

of pairwise

independent questions (c.f. assumption

7). In this case,

O’s information about all questions in Q

M

N

will be in a

general state of superposition because their correspond-

ing probabilities cannot all be

1

2

, otherwise it would be

the state of no information.

Accepted in Quantum 2017-11-27, click title to verify 23

abilistic statements an experimenter can make

about the outcomes of measurements , i.e. his ac-

tual information about the s ystems, change con-

tinuously in time. We promote this to a further

postulate for O’s world, however, with a further

requirement.

We note that the state space and the time evo-

lutions are interdependent as every legal time evo-

lution must map a legal state to a legal state, i.e.

T

∆t

(~y

O→S

) ∈ Σ

N

, ∀ T

∆t

∈ T

N

and ∀ ~y

O→S

∈ Σ

N

.

Accordingly, what is the set of legal states de-

pends on what is the set of legal time evolutions

– and vice versa. We would like the pair (Σ

N

, T

N

)

to be as ‘big’ as compatibility with the other rules

allows in order to equip the other rules with as

general a validity as possible. But there are mul-

tiple ways of ‘maximizing’ the pair. Namely, the

interdependence implies that the larger the num-

ber of states, the tighter the constraints on the set

of time evolutions – and vice versa. We required

before that O’s world should not be ‘boring’ and

therefore feature a non-trivial time evolution of

states. We shall now sharpen this requirement: it

is more interesting for O to live in a world which

‘maximizes’ the number of possible ways in which

any given state of S can change in time, rather

than the number of states which it can be in rela-

tive to O. We shall thus require that any consis-

tent, continuous time evolution of O’s ‘catalogue

of knowledge’ about S in-between interrogations

is physically realizable. Since diﬀerent ways of

evolving the state of a system correspond to dif-

ferent interactions (e.g., among the members of a

comp os ite system) we thereby maximize the set

of possible interactions among systems. O’s world

is a maximally interactive place!

Rule 4. (Time Evolution) “O’s ‘catalogue of

knowledge’ about S evolves continuously in time

in-between interrogations and every consistent

such evolution is physically realizable.”

T

N

is the maximal set of transformations T

∆t

on

states which is continuous in ∆t and compatible

with rules

1-3 and the structure of L

gbit

.

Of course, during an interrogation, ~y

O→S

may

change discontinuously, i.e. ‘collapse’, on account

of complementarity. As innocent as the require-

ment of continuity of time evolution in rule

4 ap-

pears, it turns out to be absolutely crucial in or-

der to single out a unique information measure

α

i

(~y

O→S

). Notice also that classical bit theory

violates this postulate due to its discrete time

evolution group (see section

3.2.9).

It is now our task to verify what the triples

of (Q

N

, Σ

N

, T

N

) for each N ∈ N are which obey

rules

1–4. Remarkably, it turns out that these

four rules cannot distinguish between real and

complex quantum theory, i.e. between real and

complex Hilbert spaces. But the state spaces and

the orthogonal and unitary groups of time evo-

lutions of rebit and qubit quantum theory are,

in fact, the only ‘solutions’ within L

gbit

to these

rules. In the sequel of this manuscript we shall

employ rules

1–4 in order to develop the necessary

tools, within L

gbit

, for eventually proving the fol-

lowing more precise claim in [1, 2]. In fact, as a

simple example of the newly develop ed tools, we

shall already prove the claim for N = 1 at the

end of this article.

Claim. L

gbit

contains only two solutions for the

pair (Σ

N

, T

N

) which are compatible with rules

1–

4:

1. rebit quantum theory [

2], where Σ

N

co-

incides with the space of 2

N

× 2

N

density

matrices

24

over (R

2

)

⊗N

and T

N

is PSO(2

N

).

2. qubit quantum theory [1], where Σ

N

co-

incides with the space of 2

N

× 2

N

density

matrices over (C

2

)

⊗N

and T

N

is PSU(2

N

).

Furthermore, states evolve unitarily accord-

ing to the von Neumann evolution equ ation.

We remind the reader that the time evolution

groups T

N

for density matrices in rebit and qubit

quantum theory are projective because they cor-

respond to ρ 7→ U ρ U

†

, where for (1) rebits ρ is a

2

N

× 2

N

real symmetric matrix, U ∈ SO(2

N

) and

† denotes matrix transpos e; and (2) for qubits ρ

is a 2

N

× 2

N

hermitian matrix, U ∈ SU(2

N

) and

† denotes hermitian conjugation.

Furthermore, using one additional rule on Q

N

which allows O to ask S any question that “makes

(probabilistic) sense”, we show in [

1,2] that

1. in the rebit case [

2], Q

N

is (isomorphic

to) the set of projective measurements on to

the +1 eigenspaces of Pauli operators

25

over

R

2

N

. This is the set of rank-(2

N−1

)-

projectors.

24

A real density matrix is a symmetric, positive semidef-

inite matrix on (R

2

)

⊗N

.

25

The set of Pauli operators over R

2

N

is the set of trace-

less real symmetric 2

N

×2

N

matrices with eigenvalues ±1.

Accepted in Quantum 2017-11-27, click title to verify 24

2. in the qubit case [1], Q

N

is (isomorphic

to) the set of projective measurements on to

the +1 eigenspaces of Pauli operators

26

over

C

2

N

(i.e., the set of rank-(2

N−1

)-projectors)

and the outcome probability for any Q ∈ Q

N

to be answered with ‘yes’ by S in some state

is given by the Born rule for proj ective mea-

surements.

Since both real and complex quantum theory

come out of rules

1–4, we shall impose a further

rule on O’s information acquisition in [

1] in order

to also eliminate rebit theory. However, we note

that rebit theory is mathematically contained in

qubit quantum theory and that rebits can ac-

tually be produced in laboratories. Hence, one

might also hold rules

1–4 and the rule that O

may ask S any question that “makes sense” as

physically suﬃcient.

Notice that it is necessary to directly recon-

struct the space of density matrices over Hilbert

spaces from the rules rather than the underly-

ing Hilbert spaces themselves. The reason is that

the latter contain physically redundant informa-

tion (norm and global phase), while the rules refer

only to information which is directly accessible to

O.

4.2 Strategy for building the necessary tools

and proving the claim

Our strategy and procedure f or developing tools

and intermediate results before proving the main

claim is best summarized as a diagram (see ﬁgure

3).

PSfrag rep lacements

limited information complementarity information preservation time evolution

time evolution

independen ce, compatibility and cor-

relation structure on Q

N

(in sec.

5)

reversible time evolution (in sec.

6)

and information measure (in sec.

6.8)

Σ

1

is a ball with D = 2, 3, and

T

1

is either SO(2) or SO(3) (in sec.

7)

T

2

is either PSO(4) or PSU(4) ⇒

Σ

2

convex hull of RP

3

or CP

3

(in [1,2])

(Q

N

, Σ

N

, T

N

) for N > 2 (in [1, 2])

Figure 3: Strategy an d steps for the reconstruction.

That is, we shall ﬁrstly ascertain, in section 5,

the independence, compatibility and correlation

structure on Q

N

which is induced by rules 1 and

2. This section will be particularly instructive

and deliver many elementary technical results

26

The set of Pauli operators over C

2

N

is the set of trace-

less hermitian 2

N

× 2

N

matrices with eigenvalues ±1.

which, besides becoming important later, yield

simple explanations f or typical quantum phenom-

ena such as, e.g., entanglement, non-locality and

monogamy relations. This is also where we shall

determine the dimensionality of Σ

1

by a simple

argument and, using this result, derive the di-

mensionalities of all other state spaces too. Next,

in section

6, we shall use rules 3 and 4 in order

Accepted in Quantum 2017-11-27, click title to verify 25

to conclude that a speciﬁc time evolution, is re-

versible and, in fact, described by a continuous

one-parameter matrix group. These results will

then be used in section

6.8, together with rules

3 and 4, to derive an explicit information mea-

sure α

i

(~y

O→S

) which is unique under elementary

consistency conditions. In section 7, the conjunc-

tion of these outcomes will then be employed to

show that for N = 1 one either obtains a two-

or three-dimensional Bloch ball as a state space

Σ

1

and that the group of all possible time evo-

lutions T

1

is accordingly either SO(2) or SO(3).

The claim for N > 1 will require s ubstantial addi-

tional work and will be proven in the companion

paper [

1] for qubits and in [2] for rebits.

5 Question structure and correlations

In this section, we shall solely employ rules

1 and

2 in order to deduce the independence, compati-

bility and correlation structure for the questions

in an informationally complete set Q

M

N

(c.f. as-

sumption

7). To this end it is instructive to look

at the individual cases N = 1, 2, 3 in some de-

tail before considering general N ∈ N. We shall

denote by D

N

the dimension of the state space

Σ

N

.

5.1 A single gbit

Rules

1 and 2 taken together imply that for N = 1

there exists at least one maximally complemen-

tary pair Q

1

, Q

′

1

. Since N = 1 is ﬁxed, it is now

more convenient to count the indep endent ques-

tions by an index, rather than a prime.

27

Let us

therefore slightly change the notation and write

Q

′

1

henceforth as Q

2

, such that the rules imply

the existence of a maximally complementary pair

Q

1

, Q

2

. But, applied to a single gbit only, rules

1 and 2 tell us nothing more about how many

questions maximally complementary to Q

1

exist.

There may arise another Q

3

maximally comple-

mentary to both Q

1

, Q

2

etc. Notice that, for a

single gbit, an informationally complete set of

pairwise independent questions must be given by

a maximal set of mutually maximally comple-

mentary questions (i.e., no further Q ∈ Q

1

can

be added to this set without destroying mutual

27

Rules

1 and 2 count compatible questions, here we

need to count maximally complementary questions.

maximal complementarity)

Q

M

1

= {Q

1

, Q

2

, . . . , Q

D

1

} (11)

because rule

1 prohibits further pairwise indepen-

dent (partially or maximally) compatible ques-

tions. (Recall that maximally complementary

questions are automatically independent.) Rules

1 and 2 imply D

1

≥ 2. It turns out that we need

to consider two gbits in order to upper bound D

1

.

We shall call such questions for a single gbit

individual questions.

5.2 Two gbits

The N = 2 case requires substantially more work.

First of all, since this is a composite system (c.f.

deﬁnition 3.5), the individual questions about the

two single gbits must be contained in an infor-

mationally complete set Q

M

2

. We shall again

slightly change the notation, as compared to sec-

tion

4.1: the maximal complementary set Q

M

1

(

11) for gbit 1 will be denoted by Q

1

, . . . , Q

D

1

,

while the elements of Q

M

1

for gbit 2 will be de-

noted with a prime Q

′

1

, . . . , Q

′

D

1

. It will be conve-

nient to depict the q uestion structure graphically.

Representing individual questions henceforth as

vertices, Q

M

2

certainly contains the following

PSfrag replacements

gbit 1 gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

.

. (12)

This structure abides by rules

1 and 2 and any Q

i

will be independent of and maximally compatible

with any Q

′

j

(cf. deﬁnition

3.5).

5.2.1 Logical connectives of single gbit questions

But a composite system may also admit compos-

ite questions, built with logical connectives of the

subsystem questions. We must now determine

which composite questions are allowed and which

must be added to the individual questions in or-

der to form an informationally complete Q

M

2

of

pairwise independent questions. To this end, we

must ﬁrstly unravel which logical connective ∗ O

can employ at all in order to construct compos-

ite questions which are pairwise independent of

Accepted in Quantum 2017-11-27, click title to verify 26

the individual ones. Recall assumption 5 that

O, according to his theoretical model, may only

comp os e compatible questions with logical con-

nectives ∗ and consider Q

i

, Q

′

j

as an exemplary

compatible pair. In truth tables, we shall hence-

forth symbolize ‘yes’ by ‘1’ and ‘no’ by ‘0’.

Remark. (‘=’ in logical expressions.) To

avoid confusion, we emphasize that whenever we

write an equality sign ‘=’ in a logical expression

in the sequel, we do not interpret it as another

logical connective ∗, but as an actual eq uality of

the values of the propositions on the left and right

hand side. For example, the expression Q

i

= Q

′

j

is not itself a proposition which takes truth val-

ues 0 or 1, but is intended to mean that the truth

value which Q

i

takes is identical to that which Q

′

j

takes.

While permitted binary connectives, it is clear

that, e.g., the AND and OR operations ∧ and ∨,

respectively, cannot be employed alone to build

a new independent q uestion from Q

i

, Q

′

j

because

Q

i

∧Q

′

j

= 1 implies Q

i

= Q

′

j

= 1 and Q

i

∨Q

′

j

= 0

implies Q

i

= Q

′

j

= 0 such that certain answers

to these two connectives imply answers to the in-

dividuals. But if Q

i

∗ Q

′

j

is to be pairwise inde-

pendent from Q

i

, Q

′

j

, an answer to it must not

imply the answer to either of Q

i

, Q

′

j

and, con-

versely, the answer to only Q

i

or only Q

′

j

cannot

imply Q

i

∗ Q

′

j

. As can be easily checked, it must

satisfy the following truth table:

Q

i

Q

′

j

Q

i

∗ Q

′

j

0 1 a

1 0 a

1 1 b

0 0 b

a 6= b a, b ∈ {0, 1}. (13)

The only logical connectives ∗ satisfying this

truth table are the XNOR ↔ for a = 0 and b = 1

and its negation, the XOR ⊕ for a = 1 and b = 0.

As Q

i

⊕ Q

′

j

= ¬(Q

i

↔ Q

′

j

), we may only choos e

one of the two binary connectives to deﬁne new

pairwise independent questions. We henceforth

choose to use the XNOR , already introduced in

(

2), and operationally interpret it as a ‘correlation

question’; Q

ij

:= Q

i

↔ Q

′

j

is to be read as ‘are

the answers to Q

i

and Q

′

j

the same?’. But the

XOR could equivalently be employed.

28

Since, according to assumption 7, O must be

able to build an informationally complete set

for a composite system which, in particular, in-

volves pairwise independent composite questions,

we would like to conclude that Q

i

↔ Q

′

j

are not

only implementable, but also contained in a Q

M

2

together with the individuals. However, this is

subject to a few consistency checks on pairwise

independence.

The XNOR ↔ is a symmetric logical connec-

tive, Q

ij

= Q

i

↔ Q

′

j

= Q

′

j

↔ Q

i

(but note that

Q

ij

6= Q

ji

), and, thanks to its associativity,

Q

i

↔ Q

ij

= Q

i

↔ (Q

i

↔ Q

′

j

)

= (Q

i

↔ Q

i

)

|

{z }

1

↔ Q

′

j

= Q

′

j

,

Q

ij

↔ Q

′

j

= Q

i

(14)

such that ↔ deﬁnes a closed relation on

Q

i

, Q

′

j

, Q

ij

. This has the following ramiﬁcations:

(1) For any compatible pair Q

i

, Q

′

j

, the ‘correla-

tion’ operation ↔ gives rise to precisely one ad-

ditional question Q

ij

, and (2) the set Q

i

, Q

′

j

, Q

ij

will indeed be pairwise independent, according to

the deﬁnition in section

3.2.4, because neither Q

i

nor Q

′

j

alone can determine Q

ij

relative to the

state of no information (not even partially), oth-

erwise, together with the determined Q

ij

, they

would (at least partially) determine each other

via (

14) – in contradiction with the independence

of Q

i

, Q

′

j

. That is, Q

ij

is independent of both Q

i

and Q

′

j

and since we assume independence to be a

symmetric relation, it must also be true the other

way around.

Consequently, the D

2

1

correlation questions

Q

ij

, i, j = 1, . . . , D

1

, are candidates for addi-

tional questions in Q

M

2

besides the 2D

1

indi-

vidual ones of (

12). Graphically, we shall repre-

sent Q

ij

as the edge connecting the vertices cor-

responding to Q

i

and Q

′

j

. For example, the fol-

lowing question graphs

28

As an aside, the XNOR can be expressed in terms of

the basic Boolean op erations as Q

i

↔ Q

′

j

= (¬Q

i

∨ Q

′

j

) ∧

(Q

i

∨ ¬Q

′

j

).

Accepted in Quantum 2017-11-27, click title to verify 27

PSfrag replacements

gbit 1 gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

31

Q

22

Q

23

Q

D

1

D

1

.

PSfrag replacements

gbit 1 gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

31

Q

22

Q

23

Q

D

1

D

1

.

PSfrag replacements

gbit 1 gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

31

Q

22

Q

23

Q

D

1

D

1

.

represent legal sets of questions. Note that due

to assumption

5, there can only be edges between

gbit 1 and gbit 2 but not between the vertices of

a single gbit, e.g., Q

1

and Q

2

, because they are

complementary.

Our next task is to analyse the independence

and compatibility structure of the Q

ij

.

5.2.2 Independence, complementarity and entan-

glement

We begin with a simple, but important observa-

tion which we will frequently make use of.

Lemma 5.1. Let Q

1

, Q

2

, Q

3

∈ Q

N

be such that

Q

1

is maximally compatible with and pairwise in-

dependent of both Q

2

, Q

3

and that Q

3

is maxi-

mally complementary to Q

1

↔ Q

2

. Then Q

2

, Q

3

are maximally complementary.

Proof. Suppose Q

2

, Q

3

were at least partially

compatible. In that case there must exist a state

of S in which O has maximal information α

2

= 1

bit about Q

2

and at least partial information

α

3

> 0 bit about Q

3

(or vice versa). Let O

now ask Q

1

which is maximally compatible with

and pairwise independent of both Q

2

, Q

3

to S

in this state. Hence, according to assumption

4,

by asking Q

1

, O cannot change his in formation

about either Q

2

, Q

3

. Thus, after asking Q

1

, O

will have m aximal information about Q

1

, Q

2

and

at least partial in formation about Q

3

. But max-

imal information about Q

1

, Q

2

implies maximal

information about Q

1

↔ Q

2

which is maximally

complementary to Q

3

. Consequently, Q

2

and Q

3

must be maximally complementary too.

This has immediate useful implications.

Lemma 5.2. Q

i

is maximally compatible with

Q

ij

, ∀ j = 1, . . . , D

1

and maximally complemen-

tary to Q

kj

, ∀ k 6= i and ∀ j = 1, . . . , D

1

. That

is, graphically, an individual question Q

i

is max-

imally compatible with a correlation question Q

ij

if and only if its corresponding vertex is a vertex

of the edge corresponding to Q

ij

. By symmetry,

the analogous result holds for Q

′

j

.

Proof. Q

i

and Q

ij

are maximally compatible by

construction. Consider therefore Q

i

and Q

kj

for

k 6= i and j = 1, . . . , D

1

. Clearly, Q

i

and Q

′

j

are

maximally compatible and pairwise independent

and so are Q

′

j

and Q

kj

. Maximal complemen-

tarity of Q

i

and Q

kj

for k 6= i now follows from

Q

k

= Q

kj

↔ Q

′

j

(see (

14)), which is maximally

complementary to Q

i

, an d lemma

5.1.

For example, Q

1

and Q

12

are maximally com-

patible, while Q

1

and Q

22

are maximally com-

plementary. The intuitive explanation for the

incompatibility of Q

1

and Q

22

is as follows: if

O knew the answers to both simultaneously, he

would know more than one bit of information

about gbit 1 because Q

1

deﬁnes a full bit of in-

formation about it while Q

22

could be regarded

as deﬁning half a bit about each of gbit 1 and

2. But in view of rule

1, O should never know

more than one bit about a single gbit, even in a

comp os ite system. Considering the information

contained in correlation questions to equally cor-

respond to gbit 1 and 2, lemma

5.2 suggests that

O’s information will always be equally distributed

over the two gbits for states of maximal knowl-

edge, i.e. O will know eq ually much about gbit 1

and 2.

29

We shall make this more precise in [1 ].

We saw before that Q

i

, Q

′

j

, Q

ij

are pairwise in-

dependent. Lemma

5.2 implies that also Q

i

and

Q

kj

for i 6= k are independent. We can make use

of this result to show the following.

Lemma 5.3. The Q

ij

, i, j = 1, . . . , D

1

are pair-

wise independent.

Proof. Consider Q

ij

and Q

kl

and suppose i 6=

k. Then Q

i

and Q

ij

are maximally compatible,

29

Clearly, this is not tru e for non-maximal knowledge.

E.g., let O always ask only Q

1

.

Accepted in Quantum 2017-11-27, click title to verify 28

while Q

i

and Q

kl

are maximally complementary

by lemma

5.2. Hence, when knowing Q

i

and Q

ij

O cannot know Q

kl

such that Q

ij

and Q

kl

must

be independent. (Recall from section

3.2.4 that

dependence requires the answer to Q

ij

to always

imply at least partial knowledge about Q

kl

, i.e.,

y

ij

= 0, 1 implies y

kl

6=

1

2

). The analogous argu-

ment holds for when j 6= l.

Recalling from subsection 5.2.1 that the set of

comp os ite questions in Q

M

2

must be non-empty,

lemmas

5.2 and 5.3 have an important corollary.

Corollary 5.4. Q

i

, Q

′

j

, Q

ij

are pairwise indepen-

dent for all i, j = 1, . . . , D

1

and, thanks to as-

sumption

7, will thus be part of an information-

ally complete set Q

M

2

.

Next, we consider the compatibility and com-

plementarity structure of the correlation ques-

tions Q

ij

.

Lemma 5.5. Q

ij

and Q

kl

are maximally com-

patible if and only if i 6= k and j 6= l. That is,

graphically, Q

ij

and Q

kl

are maximally compat-

ible if their corresponding edges do not intersect

in a vertex and maximally complementary if they

intersect in one vertex.

Proof. Suppose Q

ij

and Q

kj

with i 6= k were

at least partially compatible. Then there must

exist a state of S in which the answer to Q

ij

is fully known to O and in which also α

kj

> 0

bit (or vice versa). Let O ask Q

′

j

to S in this

state. Since, by lemma

5.2, this question is max-

imally compatible with both Q

ij

, Q

kj

(and pair-

wise independ ent of both), according to assump-

tion

4, by asking Q

′

j

, O cannot change his infor-

mation about both Q

ij

, Q

kj

. T hat is, after hav-

ing also asked Q

′

j

, O must have maximal informa-

tion about Q

ij

, Q

′

j

and partial inform ation about

Q

kj

. But maximal information about Q

ij

, Q

′

j

im-

plies also maximal information about Q

i

which,

however, is maximally complementary to Q

kj

by

lemma

5.2. Hence, Q

ij

, Q

kj

with i 6= k must be

maximally complementary.

Consider now Q

ij

and Q

kl

with i 6= k and j 6= l.

Let O ask S both Q

i

and Q

′

j

the answers to which

imply the answer to Q

ij

according to (

2). Thanks

to rule

1, this deﬁ nes a state of maximal knowl-

edge of 2 independent bits and O may not know

any further independent information. Next, let

O ask the same S the question Q

kl

. From lemma

5.2 it follows that Q

kl

is maximally complemen-

tary to both Q

i

and Q

′

j

such that the answer to

Q

kl

will give one new independent bit of infor-

mation but renders O’s information about Q

i

, Q

′

j

obsolete. But by rule

2 O cannot experience a net

loss of information by asking a new question and

after asking Q

kl

he must still know 2 bits about

S. Hence, after acquiring the answer to Q

kl

he

must still know the answer to Q

ij

such that both

are m aximally compatible.

To give graphical examples, Q

11

and Q

21

are

maximally complementary due to the intersection

in Q

′

1

, while Q

13

and Q

21

are maximally compati-

ble because their edges do not intersect in vertices

PSfrag replacements

gbit 1

gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

21

Q

22

Q

13

Q

D

1

D

1

.

PSfrag replacements

gbit 1

gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

21

Q

22

Q

13

Q

D

1

D

1

.

This question structure has signiﬁcant conse-

quences: since the correlation questions Q

ij

and

Q

kl

are both independent and maximally com-

patible for i 6= k and j 6= l, O can ask both of

them simultaneously, thereby spending the maxi-

mal amount of N = 2 independent bits of infor-

mation he may acquire according to rule

1 over

comp os ite questions. For example, he may ask S

Q

11

and Q

22

simultaneously

PSfrag replacements

gbit 1

gbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

21

Q

22

Q

13

Q

D

1

D

1

.

(15)

upon which he must be entirely oblivious about

the individual gbit properties represented by

Q

i

, Q

′

j

. (Lemma

5.2 implies that no individual

question is maximally compatible with two non-

intersecting bipartite questions at once.) That is,

O has only composite, but no individual infor-

mation about the two gbits; they are maximally

correlated. But this is precisely entanglement.

Indeed, the question graph (

15) will ultimately

correspond to four Bell states in either of the two

Accepted in Quantum 2017-11-27, click title to verify 29

question bases {Q

1

, Q

′

1

} and {Q

2

, Q

′

2

}, represent-

ing the four p os sible answer conﬁgurations ‘yes-

yes’, ‘yes-no’, ‘no-yes’ and ‘no-no’ to the corre-

lation questions Q

11

, Q

22

(see [

31] for a related

perspective within quantum theory). In fact, if

O now ‘marginalized’ over gbit 2, corresponding

to discarding any information about questions in-

volving gbit 2, he would ﬁnd gbit 1 in the state

of no information relative to him because the

discarded inf ormation also contained everything

he knew about gbit 1. Other compatible edges

will correspond to Bell states in diﬀerent ques-

tion bases.

Inspired by the compelling proposal within

quantum theory in [

33,34], we shall deﬁne s tates

~y

O→S

of a bipartite gbit system in L

gbit

as

entangled if the composite information satisﬁes

P

D

1

i,j=1

α

ij

> 1 bit,

where α

ij

(~y

O→S

) is O’s information about the

correlation question Q

ij

. While there will be

states with

P

D

1

i,j=1

α

ij

≤ 1 bit which can indeed

be considered as classically composed,

30

in con-

trast to [

33,34], we emphasize that the above will

only be a suﬃcient, but not a necessary condi-

tion for entanglement, as in the quantum case

there will exist entangled states violating it.

31

It

30

E.g., consider a state with y

11

= 1 and y

i

=

1

2

for all

other questions in Q

M

2

, corresponding to O knowing with

certainty that Q

11

= 1 and nothing else. The discussion

surrounding (

1) implies already that

P

D

1

i,j=1

α

ij

= 1 bit,

while the individual information is zero. In quantum the-

ory, this state would correspond to the separable state

ρ =

1

4

(1 + σ

x

⊗ σ

x

).

31

For the quantum case, states with

P

D

1

=3

i,j=1

α

ij

≤ 1 bit

where deﬁn ed as classically composed in [

33, 34]. This is,

however, in conﬂict with the usual deﬁnition of entangled

states as those which are non-separable. Namely, consider

the family of quantum states (on the r.h.s. expressed in

z-basis) for −

1

3

≤ a ≤

1

3

ρ

a

:=

1

4



1 +

1

3

(σ

z

⊗ 1 + 1 ⊗ σ

z

− σ

z

⊗ σ

z

)

+2a(σ

x

⊗ σ

x

+ σ

y

⊗ σ

y

)) =







1

3

0 0 0

0

1

3

a 0

0 a

1

3

0

0 0 0 0







.

This clearly is a family of non-separable states for a 6= 0.

In particular, for a =

1

3

this is the reduced state which

one obtains from a W-state upon marginalizing over a

qubit. These states y ield y

z

1

= y

z

2

=

2

3

, y

zz

=

1

3

and

y

xx

= y

yy

= a +

1

2

and all other y

i

=

1

2

, where, e.g., y

z

1

and y

zz

are th e probabilities that the spin of qu bit 1 is

up in z-direction and that th e spins of both qubits are

would be an interesting question to investigate

what the precise suﬃcient and necessary condi-

tions for entanglement are in the quantum case

in this informational formulation. However, we

shall leave this open for now.

Since there are only N = 2 independent bits of

information to be gained, according to rule

1, the

above suﬃcient condition for entanglement thus

means, in particular, that O has more compos-

ite than individual information. In general, two

gbits will be referred to as maximally entangled

relative to O if he has only composite informa-

tion about them which is incompatible with any

individual information. By contrast, two gbits

would be in a ‘product state’ if O has maximal

individual knowledge of 2 bits about them. For

instance, if he knows that Q

1

= Q

′

1

= 1, S would

be in such a product state relative to him. But

notice that even in this case, O would have one

dependent bit of information about the correla-

tions because clearly Q

11

= 1 too.

Note also that O can ‘collapse’ two gbits into

an entangled state: as the most extreme exam-

ple, consider the case that O receives a ‘product

ensemble state’ of two gbits in a multiple shot

interrogation. O may then ask the next pair of

gbits the questions Q

11

, Q

22

. Upon receiving the

answers, the two gbits will have ‘collapsed’ into

a maximally entangled posterior state relative to

O, despite having been in a ‘product state’ prior

to interrogation. Entanglement is thus a property

of O’s information about S.

We emphasize that, for systems with limited in-

formation content, entanglement is a direct con-

sequence of complementarity – at least for states

of maximal information. To illustrate this obser-

vation, consider two classical bits (cbits). Since

the same in z-direction, resp ectively. Using the quadratic

information measure α

ij

= (2 y

ij

− 1)

2

which we derive in

subsection

6.8 and which was also proposed for quantum

theory by the authors of [

33, 34], these states give

I

comp

:=

3

X

i,j=1

α

ij

= (2 y

xx

− 1)

2

+(2 y

yy

− 1)

2

+(2 y

zz

− 1)

2

+0 · · · + 0 =



1

9

+ 8a

2



bit ≤ 1 bit,

and for the individual information I

indiv

:= (2y

z

1

− 1)

2

+

(2y

z

2

− 1)

2

+ 0 · · · + 0 = 2/9 bit. I

comp

can be arbitrar-

ily close to 1/9 bit for a suﬃciently small a 6= 0 so that

even I

comp

< I

indiv

, and yet the corresponding state is en-

tangled in violation of the p roposed condition of classical

composition in [

33, 34].

Accepted in Quantum 2017-11-27, click title to verify 30

there is no complementarity in this case, there are

only three pairwise independent questions: the

individuals Q

1

, Q

′

1

about cbit 1 and cbit 2, re-

spectively, and the correlation Q

11

would form an

informationally complete Q

M

N

, graphically rep-

resented as

PSfrag replacements

cbit 1 cbit 2

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

21

Q

22

Q

13

Q

D

1

D

1

.

Q

1

, Q

′

1

, Q

11

are mutually maximally compatible

and this composite system satisﬁes rule

1 too such

that O may only acquire maximally N = 2 inde-

pendent bits of information about this pair of

cbits. Clearly, also in this classical case it is pos-

sible for O to acquire only composite information

about the pair of cbits, namely by only asking

Q

11

and not bothering about the individuals. O

would be able to ﬁnd out whether identically pre-

pared pairs of cbits in an ensemble are correlated

by alway s asking Q

11

in a multiple shot interro-

gation. There will indeed exist states such that

y

11

= 1 and y

1

= y

′

1

=

1

2

, however, the pair of

cbits will not be in a state of maximal informa-

tion relative to O because he has only spent one of

his two available bits. A classical state of max-

imal information corresponds to O knowing the

answers to two questions of the three Q

1

, Q

′

1

, Q

11

,

but then by (

2, 14) O will also know the answer

to the third. That is, in a state of maximal in-

formation about two cbits, O will always have

maximal individual information about the pair.

For cbits, O cannot spend the second bit also in

comp os ite information. By contrast, it is a conse-

quence of the complementarity rule 2 that in our

case here, O can actually exhaust the entire infor-

mation available to him by composite questions,

thereby giving rise to entanglement.

Thus far, we have only considered individual

and correlation questions. All are part of Q

M

2

.

But can there be more pairwise independent ques-

tions in Q

M

2

? These would have to be obtained

from the individuals and correlations via another

comp os ition with an XNOR. Composing the indi-

viduals Q

i

, Q

′

j

with the correlation questions via

XNOR, e.g. Q

i

↔ Q

il

, will yield nothing new

because questions need to be maximally compat-

ible in order to be logically connected by O and

for compatible pairs it already follows from (

14)

that the individual and correlation questions are

logically closed under ↔. However, what about

correlations of correlation questions, e.g., what

about Q

11

↔ Q

22

?

5.2.3 A logical argument for the dimensionality of

the Bloch-sphere

The answer to the last question, in fact, is inter-

twined with the dimensionality D

1

of the state

space Σ

1

of a single gbit and thus, ultimately,

with the dimensionality of the Bloch-sphere. De-

riving the dimension of the Bloch sphere has also

been a crucial step in the successful reconstruc-

tions of quantum theory via the G PT framework.

While H ardy’s pioneering reconstruction [

14] did

not fully settle this issue (it required an opera-

tionally somewhat obscure ‘simplicity axiom’ to

obtain D

1

= 3), it was later explicitly solved

by impressive group-theoretic arguments in [59]

which show that, under certain information the-

oretic constraints, qubit quantum theory is the

only theory with non-trivial entangling dynam-

ics. This result was then exploited to relate the

dimension of the Bloch-sphere to the dimension

of space via a communication thought experi-

ment [

21]. However, unfortunately, the ingredi-

ents of these arguments neither ﬁt mathemati-

cally nor conceptually fully into our framework

which relies on distinct structures than GPTs.

On the other hand, although not yielding full

quantum theory reconstructions, approaches as-

serting an epistemic restriction (i.e. a restriction

of knowledge) over ontic states admit very sim-

ple and elegant arguments for a 1-bit system to

have a state space spanned by three indep endent

epistemic states [

40, 41, 54]. These arguments

can be carried out by considering a single sy stem

over a 2-bit ontic state which at the epistemic

level, however, is eﬀectively a 1-bit system due

to epistemic restrictions. The arguments involve

binary connectives (at the ontic level) of comple-

mentary questions which, while non-problematic

when dealing with ‘hidden variables’, are, how-

ever, deemed illegal in O’s model of the world ac-

cording to our assumption

5. We do not wish to

make any ontological commitments in our purely

operational approach. Consequently, we must

reason diﬀerently and without ontic states to de-

duce the dimensionality of Σ

1

. In our case, this

requires to consider two gbits rather than just

one and involves entanglement, in analogy to the

group-theoretic GPT derivation in [

21,59] which

likewise requires two gbits and entanglement.

Accepted in Quantum 2017-11-27, click title to verify 31

Theorem 5.6. D

1

= 2 or 3.

Proof. By lemma

5.5, Q

ij

and Q

kl

are indepen-

dent and maximally compatible if i 6= k and

j 6= l. But any maximal set of pairwise maxi-

mally compatible correlation questions then con-

tains precisely D

1

questions. Graphically, this

is easy to see: one can always ﬁnd D

1

non-

intersecting edges between the D

1

vertices of gbit

1 and the D

1

vertices of gbit 2. For example, the

D

1

‘diagonal correlations’ Q

ii

, i = 1, . . . , D

1

PSfrag rep lacements

.

Q

11

Q

22

Q

D

1

D

1

Q

33

are pairwise maximally compatible and pairwise

independent (lemma

5.3) su ch that, by theorem

3.2 (Specker’s principle), they must also be mu-

tually maximally compatible. Hence, O m ay ac-

quire the answers to all D

1

Q

ii

at the same time.

However, rule

1 forbids O’s information about

S to exceed the limit of N = 2 independent

bits. Accordingly, the D

1

correlation questions

Q

ii

cannot be mutually independent if D

1

> 2

and the answers to any pair of them must imply

the answers to all others. For instance, suppose

O has asked Q

11

and Q

22

. The pair must de-

termine the answers to Q

33

, . . . , Q

D

1

D

1

such th at

the latter must be Boolean f unctions of Q

11

, Q

22

.

Hence, Q

jj

= Q

11

∗Q

22

, j 6= 1, 2, for some logical

connective ∗ w hich preserves that Q

11

, Q

22

, Q

jj

are pairwise independ ent. Table (

13) implies that

∗ must either be th e XNOR ↔ or the XOR ⊕

such that either

Q

jj

= Q

11

↔ Q

22

, or

Q

jj

= ¬(Q

11

↔ Q

22

), ∀ j = 3, . . . , D

1

.

But th en clearly Q

jj

, j = 3, . . . , D

1

could not

be pairwise independent if D

1

> 3. The same

argument can be carried out for any set of D

1

pairwise independent and maximally compatible

correlations Q

ij

, i.e. for any qu estion graph with

D

1

non-intersecting edges, and any choice of a

pair of maximally compatible correlations which

O may ask ﬁrst. We must therefore conclude that

D

1

≤ 3, while rule

2 implies that D

1

≥ 2.

We shall refer to D

1

= 2 as the ‘rebit case’ and

to D

1

= 3 as the ‘qubit case’; for the moment

this is just suggestive terminology, however, we

shall see later and in [

2] that the D

1

= 2 case

will, indeed, result in rebit quantum theory (two-

level s ystems over real Hilbert spaces), while the

D

1

= 3 case will give rise to s tandard q uantum

theory of qubit systems [1].

5.2.4 An informationally complete set for N = 2

qubits

The two cases have distinct properties as regards

questions which as k for the correlations of bi-

partite questions and it is necessary to consider

them separately. We begin with the simpler qubit

case D

1

= 3 for which correlations of correlation

questions do not deﬁne new information f or O

about S such that six individual and nine cor-

relation questions constitute an informationally

complete set Q

M

2

. These will ultimately cor-

respond to the propositions ‘the spin is up in

x−, y−, z−direction’ for the individual qubits 1

and 2 and ‘the spins of qubit 1 in i− and qubit 2

in j− direction are the same’ where i, j = x, y, z.

Note that the density matrix for two qubits has

15 parameters.

Theorem 5.7. (Qubits) If D

1

= 3 then the correlation questions Q

ij

are logically closed under ↔

and Q

M

2

= {Q

i

, Q

′

j

, Q

ij

}

i,j=1,2,3

constitutes an informationally complete set for N = 2 with D

2

= 15.

Furthermore, f or any two permutations σ, σ

′

of {1, 2, 3} either

Q

σ(3)σ

′

(3)

= Q

σ(1)σ

′

(1)

↔ Q

σ(2)σ

′

(2)

, or Q

σ(3)σ

′

(3)

= ¬(Q

σ(1)σ

′

(1)

↔ Q

σ(2)σ

′

(2)

) (16)

and either

Q

σ(3)σ

′

(3)

= Q

σ(1)σ

′

(2)

↔ Q

σ(2)σ

′

(1)

, or Q

σ(3)σ

′

(3)

= ¬(Q

σ(1)σ

′

(2)

↔ Q

σ(2)σ

′

(1)

) (17)

Accepted in Quantum 2017-11-27, click title to verify 32

such that either

Q

σ(1)σ

′

(1)

↔ Q

σ(2)σ

′

(2)

= Q

σ(1)σ

′

(2)

↔ Q

σ(2)σ

′

(1)

, or

Q

σ(1)σ

′

(1)

↔ Q

σ(2)σ

′

(2)

= ¬(Q

σ(1)σ

′

(2)

↔ Q

σ(2)σ

′

(1)

). (18)

Proof. Statements (16, 17) are an immediate

consequence of the argument in the proof of the-

orem 5.6 which can be applied to any correla-

tion question graph with D

1

= 3 non-intersecting

edges and thus to any two permutations σ, σ

′

of

{1, 2, 3}. From this it directly follows that the

correlation questions Q

ij

are logically closed u n-

der ↔ such that in the case D

1

= 3 correlations

of correlations such as Q

ij

↔ Q

kl

do not deﬁne

any n ew independent questions. Accordingly,

Q

M

2

= {Q

i

, Q

′

j

, Q

ij

}

i,j=1,2,3

is an informationally complete set of questions

for N = 2 and D

1

= 3 such that D

2

= 15.

For example, if σ, σ

′

are both trivial then the

theorem applied to the following two question

graphs

PSfrag replacements

.

Q

11

Q

22

Q

12

Q

21

Q

33

PSfrag replacements

.

Q

11

Q

22

Q

12

Q

21

Q

33

implies for D

1

= 3 either

Q

33

= Q

11

↔ Q

22

, or Q

33

= ¬(Q

11

↔ Q

22

) (19)

and either

Q

33

= Q

12

↔ Q

21

, or Q

33

= ¬(Q

12

↔ Q

21

) (20)

such that either

Q

11

↔ Q

22

= Q

12

↔ Q

21

, or Q

11

↔ Q

22

= ¬(Q

12

↔ Q

21

) (21)

In sections 5.2.7 and 5.4 we shall determine

whether negations ¬ occur in the expressions (

16–

18). For Q, Q

′

, Q

′′

maximally compatible and re-

lated by an XNOR, we shall henceforth distin-

guish between

even correlation: if Q = Q

′

↔ Q

′′

, and

odd correlation: if Q = ¬(Q

′

↔ Q

′′

).

32

5.2.5 An informationally complete set for N = 2

rebits

Next, let us consider the rebit case D

1

= 2. O can

ask the four individual questions Q

1

, Q

2

, Q

′

1

, Q

′

2

and the four correlations Q

11

, Q

12

, Q

21

, Q

22

. But

O can also deﬁne the two new correlation of cor-

relations questions

Q

33

:= Q

12

↔ Q

21

,

˜

Q

33

:= Q

11

↔ Q

22

(22)

32

We note that this also implies Q

′

= ¬(Q ↔ Q

′′

) and

Q

′′

= ¬(Q ↔ Q

′

).

corresponding to the two correlation questions

graphs

PSfrag replacements

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

21

Q

22

Q

12

Q

33

˜

Q

33

.

PSfrag replacements

Q

1

Q

2

Q

3

Q

D

1

Q

′

1

Q

′

2

Q

′

3

Q

′

D

1

Q

11

Q

21

Q

22

Q

12

Q

33

˜

Q

33

.

(we add the correlation of correlations as a new

edge without vertices to the graph). The dif-

ference to the qubit case is that Q

33

,

˜

Q

33

can

now not b e written as correlations of individ-

ual questions Q

3

, Q

′

3

since the latter do not ex-

ist. Clearly, Q

12

, Q

21

, Q

33

and Q

11

, Q

22

,

˜

Q

33

are

two pairwise independent sets. The question is

whether Q

33

,

˜

Q

33

are independent of each other

and of the individuals. The following lemma even

asserts complementarity to the latter. (Note that

this lemma holds trivially in the case D

1

= 3.)

Accepted in Quantum 2017-11-27, click title to verify 33

Lemma 5.8. The questions asking for the cor-

relation of the bipartite correlations, Q

12

↔ Q

21

and Q

11

↔ Q

22

, are maximally complementary

to and independent of any Q

1

, Q

2

, Q

′

1

, Q

′

2

.

Proof. We ﬁrs tly demonstrate independence of

Q

1

and Q

11

↔ Q

22

. This f ollows from not-

ing that Q

22

is maximally complementary to

Q

1

(lemma

5.2) and maximally compatible with

Q

11

↔ Q

22

and arguments completely analogous

to those in the proof of lemma 5.3. Next, we es-

tablish complementarity of Q

11

↔ Q

22

and Q

1

.

Suppose the contrary, namely, that Q

11

↔ Q

22

and Q

1

were at least partially compatible such

that there must exist a state with α

Q

11

↔Q

22

= 1

bit and α

1

> 0 bit. Clearly, both are maxi-

mally compatible with and independent of Q

11

.

According to assumption

4, asking Q

11

to this

state does not change O’s information about

Q

11

↔ Q

22

and Q

1

. However, maximal infor-

mation about Q

11

↔ Q

22

and Q

11

also implies

maximal information about Q

22

which is max-

imally complementary to Q

1

. Hence, Q

1

and

Q

11

↔ Q

22

must be maximally complementary.

The arguments works analogously for the other

cases.

Finally, we show that (19–21) hold analogously

for the rebit case D

1

= 2. From this it follows

that the four individual questions Q

i

, Q

′

j

, the four

correlations Q

ij

, i, j = 1, 2, and the correlation

of correlations Q

33

form an informationally com-

plete set Q

M

2

for rebits. That is, in contrast

to the qubit case, there is now one non-trivial

correlation of correlations question which is pair-

wise independent from the individuals and corre-

lations.

Theorem 5.9. (Rebits) If D

1

= 2 then Q

M

2

=

{Q

i

, Q

′

j

, Q

ij

, Q

33

}

i,j=1,2

constitutes an informa-

tionally complete set for N = 2 with D

2

= 9 and

either

Q

11

↔ Q

22

= Q

12

↔ Q

21

, or

Q

11

↔ Q

22

= ¬(Q

12

↔ Q

21

).

Proof. O can begin by asking S the maximally

compatible Q

11

and Q

22

upon which he also

knows the answer to

˜

Q

33

. O then possesses the

maximal amount of N = 2 independent bits

of information about S. Next, O can ask Q

12

(or Q

21

) which, according to lemma

5.5, is max-

imally complementary to Q

11

, Q

22

. But, by rule

2, O is not allowed to experience a net loss of

information. Hence, Q

12

(or Q

21

) must be max-

imally compatible with

˜

Q

33

(they are also in-

dependent). That is, Q

12

, Q

21

,

˜

Q

33

are mutu-

ally maximally compatible according to theorem

3.2 (Specker’s principle) and O may also ask all

three at the same time. But then the same ar-

gument as in the proof of theorem 5.6 applies

such that either

˜

Q

33

= Q

12

↔ Q

21

or

˜

Q

33

=

¬(Q

12

↔ Q

21

). This implies either Q

33

=

˜

Q

33

or

Q

33

= ¬

˜

Q

33

. Accordingly, there is only one in-

dependent correlation of correlations Q

33

. Since

Q

11

, Q

12

, Q

21

, Q

22

and Q

33

are then, by construc-

tion, logically closed under the XNOR ↔ and Q

33

is maximally complementary to all individuals

Q

1

, Q

2

, Q

′

1

, Q

′

2

, no fu rther pairwise independent

question can be built from this set such that it is

informationally complete. Hence, D

2

= 9.

5.2.6 Entanglement and non-local tomography for

rebits

The rebit case D

1

= 2 thus has a very spe-

cial question and correlation structure: Q

33

=

Q

12

↔ Q

21

is the only composite question which

is maximally complementary to all individuals,

but maximally compatible with all correlations

Q

ij

. By contrast, e.g., Q

11

is maximally comple-

mentary to Q

2

, Q

′

2

, Q

12

, Q

21

and maximally com-

patible with Q

1

, Q

′

1

, Q

22

and maximally compati-

ble with the correlation of correlations Q

33

. Con-

sequently, Q

33

assumes a special role in the en-

tanglement structure: once O knows the answer

to this question he may no longer have any fur-

ther information about the outcomes of individ-

ual questions, but may have information about

correlation questions. That is, two rebits can be

in a state of non-maximal information of 1 bit

relative to O, corresponding to the latter only

having maximal k nowledge about the answer to

Q

33

and no information otherwise, and still be

maximally entangled because any individual in-

formation is forbidden in that s ituation. Notice

that this is not true for pairs of qubits because

even if everything O knew about the pair was the

answer to Q

33

, he could still acquire information

about the individuals Q

3

, Q

′

3

such that one could

not consider such a state as maximally entangled.

Even stronger, O will always know the answer

to Q

33

if the two rebits are maximally entangled

and he has maximal information about S. This

follows from (

6): for a maximally entangled state

Accepted in Quantum 2017-11-27, click title to verify 34

of maximal information (no information about in-

dividuals), the following must hold

I

O→S

=

2

X

i=1

(α

i

+ α

′

i

)

|

{z }

=0

+

2

X

i,j=1

α

ij

+ α

33

= α

11

+ α

12

+ α

21

+ α

22

+ α

33

= 3 bits.

The last equality follows from the fact that once

O knows the answers to two maximally compat-

ible questions, he will also k now the answer to

their correlation and since this correlation is in

the pairwise independent question set the total

information deﬁned in (

6) will always yield 3 bits

for N = 2 independent bit systems in states of

maximal knowledge.

33

If we now require that O

cannot k now more than a single bit about max-

imally complementary questions then, in a state

of maximal information of N = 2 independent

bits, we must have

α

11

+ α

12

= α

21

+ α

22

= 1 bit

⇒ α

33

= 1 bit

such that O must have maximal information

about Q

33

. Therefore, the correlation of corre-

lations Q

33

can be v iewed as the litmus test f or

entanglement of two rebits.

We note that the individuals Q

1

, Q

2

, Q

′

1

, Q

′

2

will ultimately correspond to projections onto the

+1 eigenspaces of σ

x

, σ

z

, while the Q

ij

corre-

spond to projections onto the +1 eigenspaces of

σ

i

⊗ σ

j

, i, j = x, z, and Q

33

corresponds to the

projection onto the + 1 eigenspace of σ

y

⊗ σ

y

,

where σ

x

, σ

y

, σ

z

are the Pauli matrices [

1,2]. Ob-

servables and density matrices on a real Hilbert

space correspond to real symmetric matrices.

This is the reason why σ

y

is not an observable

on R

2

(it corresponds to the ‘missing’ Q

3

), but

σ

y

⊗ σ

y

is a real symmetric matrix and thus an

observable on R

2

⊗ R

2

.

This gives a novel and simple explanation for

the discovery that σ

y

⊗ σ

y

determines the entan-

glement of rebits [

83]: a two-rebit density matrix

ρ is separable if and only if Tr(ρ σ

y

⊗ σ

y

) = 0,

i.e. if the state has no σ

y

⊗ σ

y

comp onent. This

statement means in our language that a state is

separable if and only if α

33

= 0 and is consistent

with our observation above because any informa-

tion about the individuals is incompatible with

information about Q

33

due to complementarity.

33

We are implicitly using here also rules

3 and 4.

This question structure also has severe reper-

cussions for rebit state tomography: it must ul-

timately be non-local. For rebits, D

1

= 2, the

probability that Q

33

= 1 could be written as

y

33

= p(Q

12

= 1, Q

21

= 1) + p(Q

12

= 0, Q

21

= 0)

where p(Q

i

, Q

j

) denotes here the joint probabil-

ity distribution over Q

i

, Q

j

. That is, in a multiple

shot interrogation, O could ask both Q

12

, Q

21

to

the identically prepared rebit couples and from

the statistics over the answers, he could also de-

termine y

33

. But this probability cannot be de-

comp os ed into joint probabilities over the individ-

ual q uestions according to Q

33

= “(Q

1

↔ Q

′

2

) ↔

(Q

2

↔ Q

′

1

)” because Q

1

, Q

2

and Q

′

1

, Q

′

2

are max-

imally complementary. Therefore, O would not

be able to determine y

33

by only asking individ-

ual questions to the two rebits and the statis-

tics over these answers.

34

That is, for rebits,

state tomography would always require correla-

tion questions and in this sense be non-local.

Note that this stands in stark contrast to qubit

pairs where Q

33

= Q

3

↔ Q

′

3

can be written

in terms of individual questions such that also

y

33

= p(Q

3

= 1, Q

′

3

= 1) + p(Q

3

= 0, Q

′

3

= 0) can

be determined by the statistics over the answers

to Q

3

, Q

′

3

only. Qubit systems (and quantum the-

ory in general) are thus tomographically local.

The requirement of tomographic locality, ac-

cording to which the state of a composite system

can be determined by doing statistics over mea-

surements on its subsystems, is a standard con-

dition in the GPT framework [

15–18, 20, 55, 59]

and thus directly rules out rebit theory. How-

ever, in contrast to derivations within the GPT

landscape, we shall not implement local tomog-

raphy here because it will be interesting to see

the diﬀerences between real and complex quan-

tum theory from the perspective of information

inference and we shall thus carry out the recon-

struction of both here and in [

1,2].

34

For example, in a multiple shot interrogation O could

ﬁrst ask Q

1

, Q

′

2

on a set of identically prepared rebit cou-

ples to ﬁnd out whether the answers are correlated. On a

second identically prepared set, O could then ask Q

2

, Q

′

1

which are maximally complementary to the ﬁrst questions

he asked. From the statistics over the answers, O would be

able to determine also the probabilities for Q

12

and Q

21

.

But since he needed two separate interrogation runs to

determine the statistics for Q

12

and Q

21

, he would not be

able to infer from this any information whatsoever about

the statistics of answers to Q

33

.

Accepted in Quantum 2017-11-27, click title to verify 35

Local tomography is usually taken as the ori-

gin of the tensor product structure for composite

systems in quantum theory [16, 17,55,59]. How-

ever, one has to be careful with this statement

because there exist two distinct tensor products:

there is (a) the tensor product of Hilbert spaces,

e.g., (C

2

)

⊗N

for qubits and (R

2

)

⊗N

for rebits, and

(b) the tensor product of unnormalized probabil-

ity vectors or density matrices. A tensor prod-

uct of type (b) deﬁnes a suﬃcient support only

for composite qubit systems, but not for compos-

ite rebit systems; the space of hermitian matrices

over (C

2

)

⊗N

is the N-fold tensor product of her-

mitian matrices over C

2

, but the space of sym-

metric matrices over (R

2

)

⊗N

is not the N-fold

tensor product of symmetric matrices over R

2

(see

also [

83]). What local tomography implies is the

tensor product (b), but as the example of rebits

shows, the tensor product of type (a) also exists

without it.

5.2.7 A Bell scenario with questions: ruling out

local hidden variables

We shall now settle the issue of the relative nega-

tion ¬, i.e. whether

Q

11

↔ Q

22

= Q

12

↔ Q

21

, or

Q

11

↔ Q

22

= ¬(Q

12

↔ Q

21

),

for both rebits and q ubits; one of these equations

must be true (see theorems 5.7 and 5.9). It is

instructive to illustrate these equations by means

of a question conﬁguration analogous to a Bell

scenario.

(A) Suppose Q

11

↔ Q

22

= Q

12

↔ Q

21

was

true. Before we determine whether it is consis-

tent with our background assumptions and pos-

tulates, we note that this is the case of classi-

cal (or realist) logic: O can consistently interpret

any such conﬁguration by means of a local ‘hid-

den variable’ model. For example, consider the

case Q

11

= Q

22

= 1 (which is compatible with

the rules) such that Q

11

↔ Q

22

= 1 and, conse-

quently, also Q

12

↔ Q

21

= 1. We represent the

last two equations by the following two graphs

PSfrag replacements

Q

12

Q

21

Q

11

Q

22

PSfrag replacements

Q

12

Q

21

Q

11

Q

22

where both edges solid and of equal colour means

that the corresponding questions are correlated.

O could consistently read this conﬁguration as

follows: “Since “ Q

1

= Q

′

1

” and “Q

2

= Q

′

2

”, I

would precisely have to conclude that “Q

12

=

Q

21

”, i.e. Q

12

↔ Q

21

= 1, if the individuals

Q

1

, Q

2

, Q

′

1

, Q

′

2

had deﬁnite values which, how-

ever, I do not know.” For instance, Q

11

= Q

22

=

Q

12

↔ Q

21

= 1 would be consistent with the four

ontic states:

PSfrag replacements

1

0

PSfrag replacements

11

0 0

PSfrag replacements

1

0

00

0

PSfrag replacements

1 1

00

.

Therefore, O could join the two graphs consis-

tently (without knowing the truth values of the

individuals)

(23)

i.e., draw all four edges simultaneously which cor-

responds to all four edges having deﬁnite (al-

beit unknown) values at the same time. Sim-

ilarly, O can interpret any other conﬁguration

with Q

11

↔ Q

22

= Q

12

↔ Q

21

in terms of a

local ‘hidden variable’ model which assigns deﬁ-

nite values to the individuals and, conversely, as

can be easily checked, any assignment of deﬁnite

(or ontic) values to the individuals Q

1

, Q

2

, Q

′

1

, Q

′

2

leads necessarily to Q

11

↔ Q

22

= Q

12

↔ Q

21

.

We now preclude this possibility. We note that

Q

11

↔ Q

22

= Q

12

↔ Q

21

can be rewritten in

terms of the individuals as

(Q

1

↔ Q

′

1

) ↔ (Q

2

↔ Q

′

2

) = (Q

1

↔ Q

′

2

) ↔ (Q

2

↔ Q

′

1

) (24)

Accepted in Quantum 2017-11-27, click title to verify 36

which is a classical logical identity that is al-

ways true for classical Boolean logic. It s tates

that the brackets can be equivalently regroup ed,

i.e. that the order in which the XNOR ↔

is executed in the logical ex pression can be

switched. In classical Boolean logic, this iden-

tity follows from the asso ciativity and symme-

try of the XNOR. Namely, suppose for a moment

that Q

1

, Q

2

, Q

′

1

, Q

′

2

had simultaneous truth val-

ues. Then Boolean logic would tell us that

(Q

1

↔ Q

′

1

) ↔ (Q

2

↔ Q

′

2

) =



(Q

1

↔ Q

′

1

) ↔ Q

2



↔ Q

′

2

=



Q

1

↔ (Q

2

↔ Q

′

1

)



↔ Q

′

2

= Q

′

2

↔



Q

1

↔ (Q

2

↔ Q

′

1

)



= (Q

1

↔ Q

′

2

) ↔ (Q

2

↔ Q

′

1

). (25)

Of course, relative to O, Q

1

, Q

2

, Q

′

1

, Q

′

2

do not

have simultaneous truth values. Instead, assump-

tion

6 implies that, for a classical logical iden-

tity to hold in O’s model, either (a) all ques-

tions in the involved logical expressions are mu-

tually maximally compatible or (b) the identity

follows from applying classical rules of inference

to (sub-)expressions which are entirely written in

terms of mutually maximally compatible ques-

tions and subsequently decomposing the result

logically into other questions (with possibly dis-

tinct compatibility relations). However, since

Q

1

, Q

2

and Q

′

1

, Q

′

2

form maximally complemen-

tary pairs, neither is the case here. In particu-

lar, the right hand side in (

24) does not follow

from applying any rules of Boolean logic to the

total expression Q

11

↔ Q

22

or those subexpres-

sions Q

1

↔ Q

′

1

and Q

2

↔ Q

′

2

on the left hand

side which are fully composed of maximally com-

patible q uestions.

35

Hence, (24) violates assump-

tion

6 such that we must rule out the possibility

Q

11

↔ Q

22

= Q

12

↔ Q

21

.

(B) We are thus already forced to the conclu-

sion that

Q

11

↔ Q

22

= ¬(Q

12

↔ Q

21

). (26)

must hold. Indeed, its equivalent representation

(Q

1

↔ Q

′

1

) ↔ (Q

2

↔ Q

′

2

)

= ¬(Q

1

↔ Q

′

2

) ↔ (Q

2

↔ Q

′

1

)

does not constitute a classical logical identity

and is thus consistent with ass umption

6. It

35

The only relevant rules in this case, as in (

25), follow

from the properties of the XNOR, namely its symmetry

and associativity in Boolean logic. However, it is clear

that, th anks to the occurring complementarity, the argu-

ments of (

25) no longer apply. For instance, the ﬁrst step

in (

25) is illegal, according to assumption 5, because Q

2

and Q

1

↔ Q

′

1

are complementary such that they may not

be d irectly connected.

is also consistent with as sumption

5 since only

maximally compatible questions are directly con-

nected.

It is impossible for O to interpret this situation

in terms of local ‘hidden variables’ which assign

deﬁnite values to the individuals simultaneously.

Namely, consider, again, the case Q

11

= Q

22

=

1 such that Q

11

↔ Q

22

= 1 and, consequently,

now Q

12

↔ Q

21

= 0. This conﬁguration may be

graphically represented as

PSfrag replacements

Q

12

Q

21

Q

11

Q

22

PSfrag replacements

Q

12

Q

21

Q

11

Q

22

where one edge solid the other dashed, but same

colour, means that the corresponding answers are

anti-correlated. One can easily check that, in con-

trast to (

23), it is impossible to consistently join

the two diagrams by drawing all four edges si-

multaneously (which would correspond to all in-

dividuals and thus all edges having deﬁnite truth

values), as one would obtain a frustrated graph

PSfrag replacements

<

PSfrag replacements

. (27)

O must conclude that neither the four individu-

als nor the four edges can have deﬁnite (but un-

known) values all at the same time. The same

verdict holds for any other conﬁguration with

Q

11

↔ Q

22

= ¬(Q

12

↔ Q

21

).

Since either (A ) or (B) must be true by theo-

rems

5.7 and 5.9, and (A) violates assumption 6,

while (B) is consistent with both assumptions

5

and 6, we conclude that (26) is the correct rela-

tion.

Accepted in Quantum 2017-11-27, click title to verify 37

Note that this argument holds for both rebits

and qubits and, for qubits, also for any other pairs

Q

i

, Q

j6=i

and Q

′

k

, Q

′

l6=k

of pairs of maximally com-

plementary individuals (see theorem

5.7). In fact,

even more generally, the same argument can be

made for any f our q uestions (i.e., not necessar-

ily individuals) Q, Q

′

, Q

′′

, Q

′′′

which are such that

Q, Q

′

and Q

′′

, Q

′′′

are maximally complementary

pairs, while Q and Q

′

are each maximally com-

patible with both Q

′′

, Q

′′′

; also in this case one

would have to conclude that

(Q ↔ Q

′′

) ↔ (Q

′

↔ Q

′′′

) = ¬



(Q ↔ Q

′′′

) ↔ (Q

′

↔ Q

′′

)



. (28)

This will become relevant later on. For now we

observe that exchanging the positions of the com-

plementary Q

′′

, Q

′′′

from the left to the right hand

side introduces a negation ¬.

This relative negation ¬ in (

28) precludes a

classical reasoning for the distribution of truth

values over O’s questions. In fact, as just seen,

it rules out local ‘hidden variable models’, analo-

gously to the Bell arguments. We also recall that

assumption

6 was a statement about which rules

of inference and logical identities are not appli-

cable to logical comp ositions involving mutually

complementary questions. By contrast, (

28) is

now a ﬁrst non-classical logical identity showing

one possible non-classical way of reasoning in the

presence of complementarity.

The correlation structure for rebits is now clear

from these results; (26) implies Q

33

= ¬

˜

Q

33

for

the correlations of correlations deﬁned in (

22).

This settles the fate of all possible relative nega-

tions ¬ for rebits. However, these results do

not fully determine the odd and even correla-

tion structure for qubits: we still have to clarify

whether there is an overall negation ¬ relative to

Q

33

in (

19, 20) and more generally in theorem 5.7

for other permutations of non-intersecting edges.

This diﬀerence between rebits and qubits results

again from the fact that Q

33

is deﬁned as the

correlation of the individuals Q

3

, Q

′

3

for qubits,

while it is a correlation of correlations for rebits.

This has a remarkable consequence: rebit theory

is its own ‘logical mirror image’, while qubit the-

ory’s ‘logical mirror image’ is distinct from qubit

theory. This topic will be deferred to section

5.4

because we ﬁrstly need to understand the ques-

tion structure for the N = 3 case in order to

discuss the odd and even correlation structure of

two qubits further.

5.3 Three gbits

It will be both useful and instructive to explicitly

consider the N = 3 case for rebits and qubits.

As a composite system, we can view three gbits,

labeled by A, B, C, either as three individual sys-

tems, as three combinations of one individual and

a bipartite composite system or as a tripartite

system:

PSfrag replacements

A

B

C

PSfrag replacements

A

B

C

PSfrag replacements

A

B

C

PSfrag replacements

A

B

C

PSfrag replacements

A

B

C

(29)

According to deﬁnition 3.5 of a composite system,

Q

3

must then contain the individual questions of

all three gbits, any bipartite correlation questions

and any permissible logical connectives thereof.

This results in diﬀerent structures for rebits and

qubits.

Accepted in Quantum 2017-11-27, click title to verify 38

5.3.1 Three qubits

We shall begin with the qubit case D

1

= 3.

Clearly, according to deﬁnition

3.5, all 3 × 3 =

9 individual questions, henceforth denoted as

Q

i

A

, Q

j

B

, Q

k

C

, and all 3 × 9 = 27 bipartite

correlation questions, from now on written as

Q

i

A

j

B

, Q

i

A

,k

C

, Q

j

B

k

C

, i, j, k = 1, 2, 3, will be part

of an informationally complete set Q

M

3

. As be-

fore, we represent individuals and correlations

graphically as vertices and edges, respectively,

e.g.

PSfrag replacements

A B C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

A

1

C

Q

1

A

3

B

Q

1

B

3

C

Q

2

A

2

B

Q

2

B

2

C

Q

111

Q

333

Q

322

PSfrag replacements

A B C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

A

1

C

Q

1

A

3

B

Q

1

B

3

C

Q

2

A

2

B

Q

2

B

2

C

Q

111

Q

333

Q

322

depict valid question graphs.

But in order to complete the individuals and

bipartite correlations to Q

M

3

we now have to

consider logical connectives of these questions

which are pairwise independent. This will nec-

essarily involve tripartite questions because the

bipartite structure is already exhausted with in-

dividuals and bipartite correlations. Clearly, we

cannot add a question

˜

Q

111

, representing the

proposition “the answers to Q

1

A

, Q

1

B

, Q

1

C

are the

same" to the individuals and bipartite correla-

tions because, e.g.,

˜

Q

111

= 1 would alway s imply

Q

1

A

1

B

= Q

1

A

1

C

= Q

1

B

1

C

= 1. Also, Q

1

A

1

B

= 0

would imply

˜

Q

111

= 0 such that the bipartite

correlations and

˜

Q

111

would not be pairwise in-

dependent.

From (

13) we already know that the logi-

cal connective yielding new pairwise independent

questions must either be the X NOR or the XOR.

For consistency with the bipartite structure, we

continue to employ the XNOR . There is an obvi-

ous candidate for an independent tripartite ques-

tion, namely

Q

ijk

= Q

i

A

↔ Q

j

B

↔ Q

k

C

(30)

which thanks to the associativity and symmetry

of ↔ can also equivalently be written as

Q

ijk

= Q

i

A

↔ Q

j

B

k

C

= Q

i

A

j

B

↔ Q

k

C

= Q

i

A

k

C

↔ Q

j

B

. (31)

(Since the notation for this tripartite ques-

tion is unambiguous from the ordering of

i, j, k we drop the subscripts A, B, C in Q

ijk

.)

This structure is also natural from the diﬀer-

ent compositions in (

29). Q

ijk

thus deﬁned

is by construction maximally compatible with

Q

i

A

, Q

j

B

, Q

k

C

, Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

and, for sim-

ilar reasons to the independence of Q

i

A

j

B

from

Q

i

A

, Q

j

B

(see the discussion below (14 )), also

pairwise independence of the latter. Note that

this question does not stand in one-to-one cor-

respondence with the proposition “the answers

to Q

i

A

, Q

j

B

, Q

k

C

are the same"; e.g., Q

i

A

= 1

and Q

j

B

= Q

k

C

= 0 also gives Q

ijk

= 1. It is

easier to interpret this question via (

31) as ei-

ther of the three questions “are the answers to

Q

i

A

, Q

j

B

k

C

/Q

i

A

j

B

, Q

k

C

/Q

i

A

k

C

, Q

j

B

the same?".

There are 3 × 3 × 3 = 27 such tripartite

questions Q

ijk

, i, j, k = 1, 2, 3. We shall repre-

sent them graphically as triangles. For example,

Q

111

, Q

322

, Q

333

are depicted as follows:

PSfrag replacements

A B C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

A

1

C

Q

1

A

3

B

Q

1

B

3

C

Q

2

A

2

B

Q

2

B

2

C

Q

111

Q

333

Q

322

Accepted in Quantum 2017-11-27, click title to verify 39

5.3.2 Independence and compatibility for three

qubits

We have to delve into some technical details on

the independence and compatibility structure to

explain and understand monogamy and the en-

tanglement structure of three qubits. These re-

sults will also be used to prove the analogous re-

sults by induction in section

5.5.1 for N qubits.

Individual questions from qubit A are maxi-

mally compatible with the individual questions

from qubits B and C, etc. But what about the

compatibility of bipartite and tripartite correla-

tions? The compatibility structure of bipartite

correlations of a ﬁxed qubit pair is clear from

lemma

5.5, but we have to investigate compat-

ibility of bipartite correlation questions involving

all three qubits.

Lemma 5.10. Q

i

A

j

B

and Q

l

B

k

C

are maximally

complementary if j 6= l. On the other hand,

Q

i

A

j

B

and Q

j

B

k

C

are maximally compatible and

it holds

Q

i

A

j

B

↔ Q

j

B

k

C

= Q

i

A

k

C

.

The analogous statements hold for any permuta-

tion of A, B, C. That is, graphically, two bipar-

tite correlations involving three qubits are max-

imally compatible if the corresponding edges in-

tersect in a vertex and maximally complementary

otherwise.

Proof. Maximal complementarity of Q

i

A

j

B

and

Q

l

B

k

C

for j 6= l is proven by noting that both

are maximally compatible with and independent

of Q

i

A

, the relation Q

j

B

= Q

i

A

↔ Q

i

A

j

B

and

lemma

5.1.

Q

i

A

j

B

and Q

j

B

k

C

are evidently maximally com-

patible since Q

i

A

, Q

j

B

, Q

k

C

are maximally com-

patible. Moreover,

Q

i

A

j

B

↔ Q

j

B

k

C

= Q

i

A

↔ (Q

j

B

↔ Q

j

B

)

|

{z }

=1

↔ Q

k

C

= Q

i

A

k

C

thanks to the associativity of ↔.

For example, Q

2

A

2

B

and Q

2

B

2

C

intersect in

Q

2

B

and are thus maximally compatible, while

Q

2

A

2

B

and Q

1

B

1

C

do not share a vertex and are

therefore maximally complementary:

PSfrag replacements

A

B

C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

B

1

C

Q

1

A

3

B

Q

1

B

3

C

Q

2

A

2

B

Q

2

B

2

C

Q

111

Q

333

Q

322

PSfrag replacements

A

B

C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

B

1

C

Q

1

A

3

B

Q

1

B

3

C

Q

2

A

2

B

Q

2

B

2

C

Q

111

Q

333

Q

322

We continue with the tripartite questions.

Lemma 5.11. Q

ijk

is maximally compatible with

Q

i

A

, Q

j

B

, Q

k

C

and maximally complementary to

Q

l

A

6=i

A

, Q

m

B

6=j

B

, Q

n

C

6=k

C

. That is, g raphically,

Q

ijk

is maximally compatible with an individ-

ual Q

i

A,B,C

if the corresponding vertex is one of

the ve rtices of the triangle representing Q

ijk

and

maximally complementary otherwise.

Proof. Q

ijk

is by construction maximally com-

patible with Q

i

A

, Q

j

B

, Q

k

C

. On the other hand,

complementarity of Q

ijk

and Q

l

A

6=i

A

is shown by

noting that both are maximally compatible with

and independent of Q

j

B

k

C

and lemma 5.1. One

argues analogously for the individuals of qubits

B and C.

For instance, Q

111

is maximally compatible

with Q

1

C

and maximally complementary to Q

2

C

:

PSfrag replacements

A

B

C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

B

1

C

Q

1

A

3

B

Q

1

B

3

C

Q

2

A

2

B

Q

2

B

2

C

Q

111

Q

333

Q

322

This lemma also directly implies that any in-

dividual and any tripartite correlation question

are pairwise independent because (1) maximally

complementary questions are in particular inde-

pendent, and (2) for maximally compatible ques-

tion pairs such as Q

i

A

, Q

ijk

independence argu-

ments analogous to those surrounding (

14) apply.

Accepted in Quantum 2017-11-27, click title to verify 40

Next we consider bipartite and tripartite cor-

relation questions.

Lemma 5.12. Q

ijk

is maximally compatible

with Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

and, furthermore, with

Q

m

B

n

C

, Q

l

A

n

C

and Q

l

A

m

B

for l 6= i, m 6= j

and k 6= n. On the other hand, Q

ijk

is max-

imally complementary to Q

i

A

m

B

, Q

i

A

n

C

, Q

l

A

j

B

,

Q

j

B

n

C

, Q

l

A

k

C

, Q

m

B

k

C

for l 6= i, m 6= j and

k 6= n. That is, graphically, Q

ijk

is maximally

compatible with a b i partite correlation if the edge

of the latter is either an edge of the triangle cor-

responding to Q

ijk

or if the edge and triangle do

not intersect. Q

ijk

is maximally complementary

to a bipartite correlation question if the edge of

the latter and the triangle corresponding to Q

ijk

share one common vertex.

Proof. Q

ijk

is by construction maximally com-

patible with Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

. Q

ijk

= Q

i

A

↔

Q

j

B

k

C

and Q

m

B

n

C

are also maximally compat-

ible for j 6= m and k 6= n because Q

m

B

n

C

is

maximally compatible with Q

i

A

and thanks to

lemma

5.5 also with Q

j

B

k

C

. Complementarity of

Q

ijk

and Q

i

A

m

B

for j 6= m follows from noting

that both are maximally compatible with and in-

dependent of Q

k

C

and lemma

5.1. The reasoning

for all other cases is analogous.

To give a graphical example, Q

111

is maximally

compatible with Q

1

B

1

C

and Q

3

A

3

C

and maxi-

mally complementary to Q

1

A

2

B

:

PSfrag replacements

A

B

C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

B

1

C

Q

1

A

3

B

Q

1

A

2

B

Q

3

A

3

C

Q

2

B

2

C

Q

111

Q

333

Q

322

PSfrag replacements

A

B

C

Q

1

A

Q

2

A

Q

3

A

Q

1

B

Q

2

B

Q

3

B

Q

1

C

Q

2

C

Q

3

C

Q

1

B

1

C

Q

1

A

3

B

Q

1

A

2

B

Q

3

A

3

C

Q

2

B

2

C

Q

111

Q

333

Q

322

We still have to check pairwise independence of

the bipartite and tripartite correlation questions.

Lemma 5.13. Any bipartite

Q

m

B

n

C

, Q

l

A

n

C

, Q

l

A

m

B

and any tripartite corre-

lation question Q

ijk

are independent from one

another.

Proof. Lemma

5.12 implies that we only

have to check pairwise independence of

Q

m

B

n

C

, Q

l

A

n

C

, Q

l

A

m

B

from Q

ijk

for l 6= i,

m 6= j and k 6= n because maximally comple-

mentary questions are by deﬁnition independent

and Q

ijk

and Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

are pairwise

independent. Consider therefore Q

ijk

and

Q

m

B

n

C

for j 6= m and k 6= n. By lemma

5.11,

Q

k

C

is maximally compatible with Q

ijk

and by

lemma 5.2 m aximally complementary to Q

m

B

n

C

.

This implies, using the arguments from the proof

of lemma 5.3, independence of Q

ijk

, Q

m

B

n

C

.

The other cases follow similarly.

Lemma 5.14. The tripartite correlation ques-

tions Q

ijk

, i, j, k = 1, 2, 3 are pairwise indepen-

dent.

Proof. Consider Q

ijk

and Q

lmn

for i 6= l. By

lemma

5.11, Q

i

A

is maximally compatible with

Q

ijk

and maximally complementary to Q

lmn

. Us-

ing the analogous arguments from the proof of

lemma

5.3, this implies that Q

ijk

, Q

lmn

are inde-

pendent. The same reasoning holds when j 6= m

and k 6= n.

This has an immediate consequence:

Corollary 5.15. The individuals Q

i

A

, Q

j

B

, Q

k

C

,

the bipartite Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

and the tripar-

tite Q

ijk

, i, j, k = 1, 2, 3 are pairwise independent

and thus, thanks to assumption

7, contained in

an informationally complete set Q

M

3

.

Lastly, we consider the complementarity and

compatibility structure of the tripartite correla-

tions.

Lemma 5.16. Q

ijk

and Q

lmn

are maximally

compatible if {i, j, k} and {l, m, n} overlap in one

or three indices and maximally complementary if

{i, j, k} and {l, m, n} overlap in zero or two in-

dices. That is, graphically, Q

ijk

and Q

lmn

are

maximally compatible if their corresponding tri-

angles intersect in one vertex (or coincide) and

maximally complementary i f the triangles share

an edge or do not intersect.

Proof. Compatibility for an overlap in all three

indices is trivial. But also Q

ijk

= Q

i

A

↔ Q

j

B

k

C

and Q

imn

= Q

i

A

↔ Q

m

B

n

C

are clearly maxi-

mally compatible for j 6= m and k 6= n because

by lemma

5.5 Q

j

B

k

C

, Q

m

B

n

C

are maximally com-

patible in this case. Compatibility for th e other

cases of an overlap of Q

ijk

and Q

lmn

in one index

follows by permutation.

Accepted in Quantum 2017-11-27, click title to verify 41

The proof of the complementarity of Q

ijk

and

Q

lmn

for i 6= l, j 6= m and k 6= n follows from

lemma 5.1. One m ay use the fact that, by lemma

5.12, both questions are maximally compatible

with and independent of Q

j

B

k

C

and that, by

lemma

5.11, Q

i

A

6=l

A

is maximally complementary

to Q

lmn

.

Similarly, one proves complementarity of

Q

ijk

= Q

i

A

↔ Q

j

B

k

C

and Q

ljk

= Q

l

A

↔ Q

j

B

k

C

for i 6= l by using that both are maximally com-

patible with and independent of Q

j

B

k

C

. Com-

plementarity of tripartite correlations for other

overlaps in precisely two indices follows by per-

mutation.

For example, Q

111

and Q

212

intersect in the

vertex Q

1

B

and are thus maximally compatible.

By contrast, Q

111

shares the edge Q

1

A

1

B

with

Q

113

and does not intersect at all with Q

333

such

that Q

111

is maximally complementary to both.

Q

113

and Q

333

intersect in the vertex Q

3

C

and are

therefore maximally compatible:

PSfrag replacements

Q

111

Q

333

Q

113

Q

212

PSfrag replacements

Q

111

Q

333

Q

113

Q

212

5.3.3 An informationally complete set for three

qubits

We now show that the 9 individuals, the 27 bi-

partite and the 27 tripartite correlation questions

form an informationally complete set Q

M

3

for

N = 3 qubits.

Theorem 5.17. (Qubits) The individuals

Q

i

A

, Q

j

B

, Q

k

C

, the bipartite Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

and the tripartite Q

ijk

, i, j, k = 1, 2, 3 are logi-

cally closed under ↔ such that they form an in-

formationally complete set Q

M

3

with D

3

= 63 for

D

1

= 3.

Proof. For individuals and bipartite correlations

of any pair of qubits the logical closure under the

XNOR was already shown in section

5.2. Cor-

relation of individuals with a bipartite correla-

tion question from another pair of qu bits yields

the tr ipartite correlation questions. Lemma

5.10

shows that also the b ipartite correlation ques-

tions involving distinct pairs of qubits are logi-

cally closed und er ↔. We thus only have to check

logical closure of XNOR combinations involving

tripartite correlation questions.

Lemma

5.11 shows that tripartite correlations

and individuals are only maximally compatible if

the individual is a vertex of the tripartite trian-

gle. But then a combination such as

Q

ijk

↔ Q

i

A

= (Q

i

A

↔ Q

i

A

) ↔ Q

j

B

k

C

= Q

j

B

k

C

produces another bipartite correlation, thanks to

the associativity of ↔. Similarly, lemma

5.12 as-

serts that tripartite and bipartite correlations are

only maximally compatible if the edge of the b i-

partite correlation is either contained in the tri-

partite triangle or if the edge and triangle do n ot

intersect. However, combinations of such maxi-

mally compatible pairs also do not yield any new

questions because, e.g.,

Q

ijk

↔ Q

i

A

j

B

= Q

k

C

and, us ing again the associativity of XNOR and

theorem

5.7,

Q

σ

A

(1)σ

B

(1)k

↔ Q

σ

A

(2)σ

B

(2)

= (Q

σ

A

(1)σ

B

(1)

↔ Q

σ

A

(2)σ

B

(2)

)

| {z }

= Q

σ

A

(3)σ

B

(3)

or ¬Q

σ

A

(3)σ

B

(3)

↔ Q

k

C

= Q

σ

A

(3)σ

B

(3)k

or ¬Q

σ

A

(3)σ

B

(3)k

,

where σ

A

, σ

B

are permutations of {1, 2, 3}. Fi-

nally, lemma 5.16 entails that tripartite correla-

tions are only maximally compatible if they inter-

sect in one vertex. But, using theorem

5.7 once

more,

Accepted in Quantum 2017-11-27, click title to verify 42

Q

σ

A

(1)σ

B

(1)k

↔ Q

σ

A

(2)σ

B

(2)k

= Q

σ

A

(1)σ

B

(1)

↔ Q

σ

A

(2)σ

B

(2)

= Q

σ

A

(3)σ

B

(3)

or ¬Q

σ

A

(3)σ

B

(3)

.

Permu ting these examples implies that all in-

dividuals, bipartite and tripartite correlations

are logically closed under ↔ which by (

13) is

the only independent logical connective possibly

yielding new independent questions. These ques-

tions therefore form an informationally complete

set Q

M

3

with D

3

= 63.

5.3.4 Entanglement of t hree qubits and

monogamy

Let us now put all these detailed results to good

use and reward ourselves with the observation

that they naturally explain monogamy of entan-

glement for the q ubit case. This is b est illus-

trated with an example. Let O have asked the

questions Q

1

A

1

B

and Q

2

A

2

B

to qubits A, B such

that the two are in a state of maximal informa-

tion of N = 2 independent bits and maximal

entanglement relative to O.

PSfrag replacements

Q

1

A

1

B

Q

2

A

2

B

Intuitively, this already suggests monogamy of

entanglement because O has spent the maximally

attainable amount of information over the pair

A, B s uch that even if now a third qubit C enters

the game he should no longer be able to ask any

question to the triple which gives him any fur-

ther simultaneous independent information about

the pair A, B. But any correlation of A or B

with C would constitute additional information

about A or B which would violate the informa-

tion bound for the subsystem of the composite

system of three qubits.

We shall now make this more precise. The

question is, which bipartite or tripartite correla-

tions involving qubit C are maximally compatible

with Q

1

A

1

B

, Q

2

A

2

B

. Firstly, lemma

5.10 states

that bipartite correlations of two distinct qubit

pairs are only maximally compatible if their cor-

responding edges intersect in a vertex. Clearly,

there can then not exist any edge which is max-

imally compatible with both Q

1

A

1

B

, Q

2

A

2

B

. Sec-

ondly, lemma

5.12 asserts that a bipartite correla-

tion is only maximally compatible with a tripar-

tite correlation if the corresponding edge is either

contained in the tripartite triangle or does not

intersect the triangle. This means that the only

tripartite correlations maximally compatible with

Q

1

A

1

B

, Q

2

A

2

B

are (a) the correlations of the lat-

ter with the individuals Q

1

C

, Q

2

C

, Q

3

C

of qubit

C and (b) Q

33k

, k = 1, 2, 3.

For example, in the ﬁrst of the following two

question graphs Q

2

B

2

C

is maximally compati-

ble with Q

2

A

2

B

but maximally complementary to

Q

1

A

1

B

, while Q

3

B

3

C

is maximally complementary

to both. But O could ask Q

222

together with

Q

1

A

1

B

, Q

2

A

2

B

, as depicted in the second graph:

PSfrag replacements

Q

1

A

1

B

Q

2

A

2

B

Q

222

Q

2

C

Q

3

B

3

C

Q

2

B

2

C

PSfrag replacements

Q

1

A

1

B

Q

2

A

2

B

Q

222

Q

2

C

Q

3

B

3

C

Q

2

B

2

C

However, asking Q

1

A

1

B

, Q

2

A

2

B

and Q

222

simul-

taneously is equivalent to asking Q

1

A

1

B

, Q

2

A

2

B

and Q

2

C

because Q

2

A

2

B

↔ Q

222

= Q

2

C

.

The analogous conclusion holds for any other

tripartite question maximally compatible with

Q

1

A

1

B

, Q

2

A

2

B

. Therefore, once O has asked

the last to bipartite correlations about qubits

A, B, he can only acquire individual informa-

tion corresponding to Q

1

C

, Q

2

C

, Q

3

C

about qubit

C. Clearly, the same state of aﬀairs is true for

any permutation of the three qubits. This is

monogamy of entanglement in its most extreme

form: two qubits which are maximally entangled

can not be correlated whatsoever with any other

system.

Accepted in Quantum 2017-11-27, click title to verify 43

The non-extremal form of monogamy, namely

the case that a qubit pair is not maximally en-

tangled and can thus share a bit of entanglement

with a third q ubit, can also be explained. To this

end, we recall from quantum information theory

that monogamy is generally described with so-

called monogamy inequalities, e.g., the Coﬀman-

Kundu-Wootters inequality [

84]

τ

A|BC

≥ τ

AB

+ τ

AC

, (32)

where 0 ≤ τ

A|BC

= 2(1 − Tr ρ

2

A

) ≤ 1 measures

the entanglement between qubit A and the qubit

pair B, C and ρ

A

is the marginal s tate of q ubit A

obtained after tracing qubit B and C out of the

(not necessarily pure) tripartite state. τ

AB

, τ

AC

,

on the other hand, measure the bipartite entan-

glement of the pairs A, B and A, C and are s imi-

larly obtained by tracing either C or B out of the

tripartite qubit state. The inequality (32) formal-

izes the intuition that the correlation between A

and the pair B, C is at least as strong as the cor-

relation of A with B and C individually. This

is the general form of monogamy. One usually

also deﬁnes the s o-called three-tangle [

84] as the

diﬀerence between left and right hand side

τ

ABC

= τ

A|BC

− τ

AB

− τ

AC

∈ [0, 1]

to measure the genuine tripartite entanglement

shared among all three qubits. This three-tangle

turns out to be permutation invariant τ

ABC

=

τ

BCA

= τ

CAB

.

In our current case, a measure of entanglement

sharing must be informational. At this stage,

in line with our previous deﬁnition of entangle-

ment, we simply deﬁne the analogous entangle-

ment measures to be the sum of O’s information

about the various independent bipartite or tripar-

tite correlation questions

˜τ

A|BC

:=

3

X

i,j,k=1

α

ijk

+

3

X

i,j=1

α

i

A

j

B

+

3

X

i,k=1

α

i

A

k

C

˜τ

AB

:=

3

X

i,j=1

α

i

A

j

B

, ˜τ

AC

:=

3

X

i,k=1

α

i

A

k

C

. (33)

With this deﬁnition, an informational inequality

analogous to the inequality (

32) is trivial

˜τ

A|BC

≥ ˜τ

AB

+ ˜τ

AC

as is the fact that the informational three-tangle

˜τ

ABC

:= ˜τ

A|BC

− ˜τ

AB

− ˜τ

AC

=

3

X

i,j,k=1

α

ijk

is permutation invariant and measures only the

information contained in the tripartite questions

and thus genuine tripartite entanglement. This

is how one can describe the general form of

monogamy of entanglement in our language. Of

course, at this stage the information measure α

i

about the various questions Q

i

is only implicit,

but we shall derive its explicit form in section

6.8, upon which the above statements become

truly quantitative. These informational versions

of monogamy inequalities and tangles naturally

suggest themselves for simple generalizations to

arbitrarily many qubits, thereby complementing

current eﬀorts in the quantum information liter-

ature (e.g., see [

85]).

Before we move on, we brieﬂy emphasize that

monogamy is a consequence of complementarity

– as is entanglement. For example, three classi-

cal bits also s atisfy the limited information rule

1, but due to the absence of complementarity,

O could ask all correlations Q

AB

, Q

AC

, Q

BC

and

Q

ABC

at once

PSfrag replacements

A B

C

which, however, is eq uivalent to asking the three

individuals Q

A

, Q

B

, Q

C

.

5.3.5 Maximal entanglement for three qubits

Let us elucidate maximal tripartite entanglement

for three qubits, corresponding to O asking three

mutually maximally compatible and independent

tripartite correlation questions such that he ex-

hausts the information bound of N = 3 indepen-

dent bits with tripartite information. Lemma

5.16 asserts that tripartite questions are maxi-

mally compatible if and only if they intersect in

exactly one vertex. For example, O could ask

Q

211

, Q

121

, Q

112

simultaneously. These are mu-

tually independent because by (19, 20, 26)

Q

211

↔ Q

121

= Q

3

A

3

B

or ¬Q

3

A

3

B

Q

211

↔ Q

112

= Q

3

A

3

C

or ¬Q

3

A

3

C

Q

121

↔ Q

112

= Q

3

B

3

C

or ¬Q

3

B

3

C

, (34)

i.e., their binary connectives with an XNOR do

not imply each other and the bipartite correla-

tions are pairwise independent from the tripar-

tite ones. Accordingly, asking Q

211

, Q

121

, Q

112

Accepted in Quantum 2017-11-27, click title to verify 44

will provide O with three independent bits of

information about the qubit triple. We note that

lemma 5.11 implies that once O has posed the

questions Q

211

, Q

121

, Q

112

, every individual q ues-

tion Q

i

A

, Q

j

B

, Q

k

C

is maximally complementary

to at least one of the three tripartite correlations.

That is, O cannot acquire any individual inf or-

mation about the three qubits and will only have

comp os ite information. This is a necessary con-

dition for maximal entanglement.

In addition to the three implied binary corre-

lations (

34), O would clearly also know the tri-

partite correlation of all three Q

211

, Q

121

, Q

112

,

which amounts to

Q

211

↔ Q

121

↔ Q

112

= Q

222

or ¬Q

222

.

This exhausts the list of questions about

which O would have information by asking

Q

211

, Q

121

, Q

112

. The lis t can be represented by

the f ollowing question graph:

(35)

This graph will ultimately correspond to

the eight possible Greenberger-Horne-Zeilinger

(GHZ) states in either of the three question

bases {Q

2

A

, Q

1

B

, Q

1

C

}, {Q

1

A

, Q

2

B

, Q

1

C

} and

{Q

1

A

, Q

1

A

, Q

2

C

}, corresponding to the eight

answer conﬁgurations ‘yes-yes-yes’, ‘yes-yes-no’,

‘yes-no-no’, ‘yes-no-yes’, ‘no-yes-yes’, ‘no-yes-no’,

‘no-no-yes’ and ‘no-no-no’ to the three tripar-

tite correlations Q

211

, Q

121

, Q

112

. Similarly, any

other three maximally compatible tripartite cor-

relations will correspond to GHZ states in other

question bases.

36

The correlation information in the graph (35) is

clearly democratically distributed over the three

qubits A, B, C. (Intuitively, one could even

36

As an aside, we note that these propositions solve the

little riddle in Zeilinger’s festschrift for D. Greenberger

[

31].

view the bit stemming from, say, the answer to

Q

2

A

1

B

1

C

as being carried to one-third by each

of A, B, C.) Furthermore, if O now wanted to

‘marginalize’ over qubit C, i.e. discard any infor-

mation involving qubit C, all that would be left of

his knowledge about the qubit triple would be the

answer to Q

3

A

3

B

. But, as argued in section 5.2.6,

this resulting ‘marginal s tate’ of qubits A, B rel-

ative to O cannot be considered entangled. The

analogous results hold for ‘marginalization’ over

either A or B. As a result, the question graph

(

35) corresponds to genuine maximal tripartite

entanglement.

5.3.6 Three rebits

We shall now repeat the same exercise for three

rebits but can beneﬁt from the results of the N =

3 qubit case. We shall thus be briefer here. The

reader exclusively interested in qubits may jump

to section

5.4.

According to deﬁnition

3.5, O’s questions to

the three rebit system must contain the 6 indi-

viduals Q

i

A

, Q

j

B

, Q

k

C

, 12 bipartite correlations

Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

, i, j, k = 1, 2, and 3 bipartite

correlations of correlations Q

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

.

To render this set informationally complete, we

have to top it up with tripartite questions. There

are now 8 tripartite correlations Q

ijk

, i, j, k =

1, 2, of the kind (

30, 31) and, moreover, 6 tri-

partite correlations of a rebit individual question

with the question Q

33

asking for the correlation

of bipartite correlations of the other rebit pair

Q

i33

:= Q

i

A

↔ Q

3

B

3

C

,

Q

3j3

:= Q

j

B

↔ Q

3

A

3

C

,

Q

33k

:= Q

3

A

3

B

↔ Q

k

C

, i, j, k = 1, 2.

Given that the individuals Q

3

A

, Q

3

B

, Q

3

C

do not

exist for rebits there is now also no tripartite cor-

relation Q

333

.

5.3.7 Independence and compatibility for three

rebits

The independence, compatibility and comple-

mentarity structure for the questions not involv-

ing an index i, j, k = 3 directly follows from the

qubit discussion as lemmas

5.10–5.16 also hold in

the present case for i, j, k = 1, 2. But we now

have to clarify the question structure once two

indices are equal to 3 (an odd number of indices

Accepted in Quantum 2017-11-27, click title to verify 45

cannot be 3). The status of any such purely bi-

partite relations was clariﬁed in section

5.2 such

that here we have to consider the case that all

three rebits are involved.

Lemma 5.18. Q

3

B

3

C

is maximally complemen-

tary to Q

i

A

j

B

, i, j = 1, 2, and maximally compat-

ible with Q

3

A

3

B

. Furthermore,

Q

3

A

3

B

↔ Q

3

B

3

C

= Q

3

A

3

C

. (36)

The same holds for any permutations of A, B, C.

Proof. One proves complementarity of Q

i

A

j

B

and

Q

3

B

3

C

= Q

1

B

2

C

↔ Q

2

B

1

C

by employin g that

both are maximally compatible with and inde-

pendent of Q

i

A

and lemma

5.1. T his also requires

invoking that Q

j

B

and Q

3

B

3

C

are maximally com-

plementary by lemma

5.8.

Compatibility of Q

3

A

3

B

and Q

3

B

3

C

fol-

lows ind ir ectly. Namely, Q

1

A

2

B

, Q

2

B

1

C

and

Q

2

A

1

B

, Q

1

B

2

C

are two maximally compatible

pairs by lemma 5.10. We can then apply the

reasoning around (

28) to ﬁnd

Q

3

A

3

B

↔ Q

3

B

3

C

= (Q

1

A

2

B

↔ Q

2

A

1

B

) ↔ (Q

1

B

2

C

↔ Q

2

B

1

C

)

= ¬((Q

1

A

2

B

↔ Q

2

B

1

C

)

|

{z }

= Q

1

A

1

C

↔ (Q

1

B

2

C

↔ Q

2

A

1

B

)

|

{z }

= Q

2

A

2

C

)

= ¬(Q

1

A

1

C

↔ Q

2

A

2

C

) = Q

3

A

3

C

.

The last equality holds thanks to theorem 5.9.

The same reasoning applies to any permuta-

tion of A, B, C.

Next, we discuss the tripartite correlations of

individuals with the questions Q

33

asking for the

correlation of bipartite correlations of rebit pairs.

Lemma 5.19. Q

i33

is maximally compatible with

Q

i

A

and maximally complementary to Q

j

B

, Q

k

C

.

The same holds for any permutation of A, B, C.

Proof. Compatibility of Q

i33

with Q

i

A

is true

by construction. Complementarity of Q

j

B

and

Q

i33

= Q

i

A

↔ (Q

1

B

2

C

↔ Q

2

B

1

C

) follows from

the observation that Q

i33

and Q

j

B

are both max-

imally compatible with and independent of Q

i

A

and a similar reasoning to the previous proof.

Lemma 5.20. Q

i33

is maximally compatible with

Q

3

B

3

C

, Q

j

B

k

C

, j, k = 1, 2, and Q

l

A

k

C

, Q

l

A

j

B

for

i 6= l. On the other hand, Q

i33

is maximally com-

plementary to Q

i

A

j

B

, Q

i

A

k

C

and Q

3

A

3

B

, Q

3

A

3

C

.

Furthermore, Q

ijk

is maximally compatible with

Q

3

A

3

B

. The same holds for any permutation of

A, B, C.

Proof. Compatibility of Q

i33

with Q

j

B

k

C

and

Q

3

B

3

C

, as well as compatibility of Q

ijk

with

Q

3

A

3

B

is obvious. Compatibility of Q

i33

with

Q

l

A

k

C

for i 6= l follows indirectly by noting that

Q

i

A

, Q

l

A

and Q

3

B

3

C

, Q

k

C

are two pairs of max-

imally complementary questions, but that the

questions in one pair are maximally compatible

with both questions of the other. In this case

the reasoning of (

28) applies and entails the cor-

relation of Q

i

A

with Q

3

B

3

C

must be maximally

compatible with the correlation of Q

l

A

with Q

k

C

.

Compatibility of Q

i33

with Q

l

A

j

B

follows simi-

larly.

Complementarity of Q

i33

and Q

i

A

j

B

follows

from the fact that both are maximally compati-

ble with and independent of Q

i

A

and using argu-

ments as in previous proofs. Likewise, Q

i33

and

Q

3

A

3

B

follows similarly by noting that both are

maximally compatible with Q

3

B

3

C

.

Lemma 5.21. Any of Q

i33

, Q

3j3

, Q

33k

is

independent from any bipartite correla-

tion question Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

and

any bipartite correlation of correlations

questionQ

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

. Q

ijk

is also

pairwise independent from the latter. Further-

more, the Q

ijk

and Q

i33

, Q

3j3

, Q

33k

are pairwise

independent.

Proof. The proof is completely analogous to the

proofs of lemmas

5.3, 5.13 and 5.14.

As before this has an important consequence.

Corollary 5.22. The individuals Q

i

A

, Q

j

B

, Q

k

C

,

the bipartite correlations Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

,

the bipartite correlations of correlations

Q

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

, the tripartite correla-

tions Q

ijk

and the tripartite Q

i33

, Q

3j3

, Q

33k

,

Accepted in Quantum 2017-11-27, click title to verify 46

i, j, k = 1, 2, are pairwise independent and

thus, thanks to assumption

7, part of an

informationally complete set Q

M

3

.

The compatibility and complementarity struc-

ture of questions involving the ‘correlation of cor-

relations’ Q

33

is analogous to lemma

5.16.

Lemma 5.23. Q

ijk

is maximally compatible with

Q

i33

, Q

3j3

, Q

33k

and maximally complementary

to Q

l33

, Q

3m3

, Q

33n

for i 6= l, j 6= m and k 6= n.

Furthermore, Q

i33

is maximally compatible with

Q

3j3

, Q

33k

, but Q

133

and Q

233

are maximally

complementary. The analogous result holds for

all permutations of A, B, C.

Proof. Compatibility of Q

ijk

with

Q

i33

, Q

3j3

, Q

33k

follows from the fact that

the constituents of the latter Q

i

A

, Q

3

B

3

C

, . . .

are maximally compatible with Q

ijk

. Comple-

mentarity of, e.g., Q

ijk

and Q

l33

for i 6= l can

be shown by noting that both are maximally

compatible with and independent of Q

3

B

3

C

and

lemma

5.1. Compatibility of, say, Q

i33

and

Q

3j3

can be demonstrated by using (

28) and

noting that Q

i

A

, Q

3

A

3

C

and Q

j

B

, Q

3

B

3

C

are two

pairs of maximally complementary questions

which are such that each question in one pair

is maximally compatible with both questions

of the other pair. Finally, Q

133

and Q

233

are

maximally complementary because both are

maximally compatible with Q

3

B

3

C

and Q

1

A

, Q

2

A

are m aximally complementary.

This ﬁnishes our considerations of the inde-

pendence and complementarity structure of three

rebits.

5.3.8 An informationally complete set for three

rebits

The rebit questions considered thus far comprise

an informationally complete set of 35 elements.

Theorem 5.24. (Rebits) The individu-

als Q

i

A

, Q

j

B

, Q

k

C

, the bipartite correlations

Q

i

A

j

B

, Q

i

A

k

C

, Q

j

B

k

C

, the bipartite correlations

of correlations Q

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

, the tri-

partite correlations Q

ijk

and the tripartite

Q

i33

, Q

3j3

, Q

33k

, i, j, k = 1, 2, are logically closed

under ↔ and thus form an informationally

complete set Q

M

3

with D

3

= 35 for D

1

= 2.

Proof. Logical closure under the XNOR for any

pair of rebits follows f rom section

5.2. Combin-

ing individuals with bipartite questions of an-

other rebit pair prod uces the tripartite qu estions.

Lemma

5.10 and 5.18 assert logical closure of

bipartite correlation questions an d the bipartite

questions Q

33

asking for the correlation of bipar-

tite correlations among all thr ee rebits. We must

therefore only check logical closure of combina-

tions involving tripartite questions which involve

indices taking the value 3 (the other cases are

covered by theorem

5.17 for i, j, k = 1, 2).

Using theorem

5.9 and lemmas 5.18–5.23, the

proof is mostly analogous to the proof of theorem

5.17 such that here we will only show two non-

trivial cases which are treated diﬀerently. For

example, by lemma 5.20 Q

133

and Q

2

A

2

B

are

maximally compatible. The conjunction with ↔

yields

Q

133

↔ Q

2

A

2

B

=

(

22,26)

(Q

1

A

↔ Q

3

B

3

C

) ↔ ¬(Q

1

A

1

B

↔ Q

3

A

3

B

)

=

(

28,36)

Q

1

B

↔ Q

3

A

3

C

= Q

313

.

Similarly, by lemma 5.23, Q

i33

, Q

3j3

are max- imally compatible. Their XNOR conjunction

gives

Q

i33

↔ Q

3j3

= (Q

i

A

↔ Q

3

B

3

C

) ↔ (Q

3

A

3

C

↔ Q

j

B

) =

(

28)

¬(Q

3

A

3

B

↔ Q

i

A

j

B

)

which again coincides with some Q

l

A

m

B

with i 6= l and j 6= m (or the negation thereof). In

Accepted in Quantum 2017-11-27, click title to verify 47

complete analogy to these explicit examples and

to the proof of theorem

5.17, one shows the log-

ical closure under the XNOR of all other cases.

This gives the desired result.

5.3.9 Monogamy and maximal entanglement for

three rebits

Rebits are considered as non-monogamous in the

literature [86, 87]. However, this conclusion de-

pends somewhat on one’s notion of monogamy

and can be clariﬁed within our language. Firstly,

rebits, just as qubits, are clearly monogamous in

the following sense: if two rebits A, B are maxi-

mally entangled in a state of maximal information

relative to O, then they cannot share any entan-

glement whatsoever with a third rebit C. This

can be seen by repeating the argument of section

5.3.4 which holds analogously for three rebits.

The situation changes slightly for entangled

states of non-maximal information involving ei-

ther of Q

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

and for tripartite

maximally entangled states. As an example for

the former case, O could ask Q

3

A

3

B

, Q

3

B

3

C

simul-

taneously which gives him two independent bits

of information about the rebit triple and implies

Q

3

A

3

C

by (

36) as well such that his information

could be s ummarized as

PSfrag replacements

Q

3

A

3

B

Q

3

A

3

C

Q

3

B

3

C

(we depict the Q

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

without ver-

tices to emphasize the absence of the individu-

als Q

3

A

, Q

3

B

, Q

3

C

for rebits). This graph cor-

responds to non-monogamously entangled rebit

states: by lemma

5.8 all individuals are maxi-

mally complementary to two of the three known

questions such that O cannot acquire any indi-

vidual information about the three rebits at the

same time. The three rebits are pairwise maxi-

mally entangled, albeit not in a state of maximal

information (see also section

5.2.6).

However, the three rebits can be similarly

non-monogamous in a tripartite maximally en-

tangled state of maximal information. As in

the qubit case (

35) in section 5.3.5, one can

write down a question graph corresponding to the

rebit analogues of GHZ states, representing the

eight possible answers to the tripartite q uestions

Q

211

, Q

121

, Q

112

PSfrag replacements

Q

3

A

3

B

Q

3

A

3

C

Q

3

B

3

C

(to justify this graph one has to employ the-

orem

5.24). Thanks to the information about

the answers to Q

3

A

3

B

, Q

3

A

3

C

, Q

3

B

3

C

, such a state

would contain a non-monogamous distribution of

entanglement over A, B, C. In particular, if O

‘marginalized’ over rebit C, by discarding all in-

formation involving C, he would be left with the

answer to Q

3

A

3

B

which deﬁnes a maximally en-

tangled two-rebit state of non-maximal informa-

tion (see section 5.2.6) – in contrast to the qubit

case of section

5.3.5.

5.4 Correlation structure for N = 2 gbits

While we were able to settle the relative negation

¬ between the correlations of bipartite correla-

tion questions for both qubits and rebits in (

28)

(e.g., Q

11

↔ Q

22

= ¬(Q

12

↔ Q

21

)), we s till have

to clarify the odd and even correlation structure

for q ubits in (

19, 20) and more generally in theo-

rem

5.7. For example, we have to clarify whether

Q

33

= Q

11

↔ Q

22

or Q

33

= ¬(Q

11

↔ Q

22

), etc.

This will involve the notion of the ‘logical mir-

ror’ of an inference theory and require the tools

from the previous section, covering the N = 3

case. We recall that the odd and even correla-

tion structure for rebits has already been settled

in section

5.2.7 through (28). This, as we shall

see shortly, is a consequence of rebit theory being

its own mirror image.

Accepted in Quantum 2017-11-27, click title to verify 48

5.4.1 The logical mirror image of an inference the-

ory

Consider a single qubit, described by O via an in-

formationally complete set Q

1

, Q

2

, Q

3

which can

be viewed as a question basis on Q

1

. As such, it

deﬁnes a ‘logical handedness’ in terms of which

outcomes to Q

1

, Q

2

, Q

3

O calls ‘yes’ and which

‘no’.

37

Clearly, this is a convention made by

O and he can easily change the handedness by

simply swapping the assignment ‘yes’↔’no’ of

one question, say, Q

1

, which is tantamount to

Q

1

7→ ¬Q

1

. This corresponds to taking the log-

ical mirror image of a single qubit. For a single

qubit this will not have any severe consequences

and O is free to choose whichever handedness he

desires.

However, the handedness of a single qubit ques-

tion basis does become important when consider-

ing composite systems. Consider, e.g., two ob-

servers O, O

′

each having one qubit and describ-

ing it with a certain handed basis. They can also

consider correlations Q

ij

of their qubit pair. If O

now decided to change the handedness of his local

question basis by Q

1

7→ ¬Q

1

this would result in

Q

11

, Q

12

, Q

13

7→ ¬Q

11

, ¬Q

12

, ¬Q

13

,

Q

ij

7→ Q

ij

, i 6= 1

and therefore

Q

11

↔ Q

22

7→ ¬(Q

11

↔ Q

22

), and

Q

12

↔ Q

21

7→ ¬(Q

12

↔ Q

21

).

Since Q

33

would remain unaﬀected by the change

of local handedness (Q

3

, Q

′

3

remain invariant),

O’s change of local handedness would have re-

sulted in a switch between even and odd correla-

tion structure f or (

19, 20). Since the handedness

of the local question basis is just a convention

by O or O

′

, we shall allow either. Consequently,

whether three correlation questions are related by

even or odd correlation (see theorem

5.7) depends

on the local conventions by O, O

′

and both are, in

fact, consistent. H owever, (28) will always hold.

Usually, of course, one would favour the situation

that both O, O

′

make the same conventions such

that their local question bases are equally handed

(or that one observer O describes all qubits with

the same handedness). Nevertheless, physically

all local conventions are fully equivalent, yet will

lead to diﬀerent representations of composite sys -

tems in the inference theory.

But there are important consistency condi-

tions on the distribution of odd and even cor-

relations. This becomes obvious when examining

three qubits A, B, C. To this end, consider the

trivial conjunction

Q

3

A

3

B

↔ Q

3

A

3

C

↔ Q

3

B

3

C

=

lemma

5.10

1. (37)

From (

19) we know that the correlation is either

even or odd, respectively,

Q

3

A

3

B

= Q

1

A

1

B

↔ Q

2

A

2

B

, or

Q

3

A

3

B

= ¬(Q

1

A

1

B

↔ Q

2

A

2

B

) (38)

and analogously for A, C and B, C. Suppose now

that all three bipartite correlations in the con-

junction (37) were even. Then we immediately

get a contradiction because, using lemma

5.10,

Q

3

A

3

B

↔ Q

3

A

3

C

↔ Q

3

B

3

C

= (Q

1

A

1

B

↔ Q

2

A

2

B

) ↔ (Q

1

A

1

C

↔ Q

2

A

2

C

) ↔ (Q

1

B

1

C

↔ Q

2

B

2

C

)

=

(

28)

¬(Q

1

A

1

B

↔ Q

1

A

1

C

)

|

{z }

= Q

1

B

1

C

↔ (Q

2

A

2

B

↔ Q

2

A

2

C

)

|

{z }

= Q

2

B

2

C

↔ (Q

1

B

1

C

↔ Q

2

B

2

C

)

= 0.

But this violates the identity (37) and results

from the relative negation between the left and

right hand side in (28). One would get the same

37

Ultimately, these three questions will deﬁne an or-

thonormal Bloch vector basis in the Bloch sphere (see sec-

tion

7). The handedness or orientation of this basis will

depend on the lab eling of question outcomes.

contradiction if one of the correlations in (

37)

was even and two were odd because then the two

negations from the odd correlation would cancel

each other and one would still be left with the

negation coming from (

28).

On the other hand, everything is consistent if

either all bipartite correlations in (

38) are odd or

one is odd and two are even because then the odd

Accepted in Quantum 2017-11-27, click title to verify 49

number of negations from the odd correlations cancels the negation coming f rom (28):

Q

3

A

3

B

↔ Q

3

A

3

C

↔ Q

3

B

3

C

= ¬(Q

1

A

1

B

↔ Q

2

A

2

B

) ↔ ¬(Q

1

A

1

C

↔ Q

2

A

2

C

) ↔ ¬(Q

1

B

1

C

↔ Q

2

B

2

C

)

=

(

28)

¬(Q

1

A

1

B

↔ Q

1

A

1

C

)

|

{z }

= Q

1

B

1

C

↔ (Q

2

A

2

B

↔ Q

2

A

2

C

)

|

{z }

= Q

2

B

2

C

↔ ¬(Q

1

B

1

C

↔ Q

2

B

2

C

)

= 1.

We can also quickly check that this is consistent

with (

20)

Q

3

A

3

B

= Q

1

A

2

B

↔ Q

2

A

1

B

, or

Q

3

A

3

B

= ¬(Q

1

A

2

B

↔ Q

2

A

1

B

) (39)

and (

26) (and analogously for A, C and B, C)

Q

1

A

2

B

↔ Q

2

A

1

B

= ¬(Q

1

A

1

B

↔ Q

2

A

2

B

).

That is, if all correlations in (38) are odd, then

all correlations in (

39) must be even. Indeed,

Q

3

A

3

B

↔ Q

3

A

3

C

↔ Q

3

B

3

C

= (Q

1

A

2

B

↔ Q

2

A

1

B

) ↔ (Q

1

A

2

C

↔ Q

2

A

1

C

) ↔ (Q

1

B

2

C

↔ Q

2

B

1

C

)

=

(

28)

¬(Q

1

A

2

B

↔ Q

1

A

2

C

)

|

{z }

= Q

2

B

2

C

↔ (Q

2

A

1

B

↔ Q

2

A

1

C

)

|

{z }

= Q

1

B

1

C

↔ (Q

1

B

2

C

↔ Q

2

B

1

C

)

= Q

3

B

3

C

↔ Q

3

B

3

C

= 1

is consistent.

In conclusion, if O wants to treat the bipar-

tite relations among all three A, B, C i dentically,

then the following distribution of odd and even

correlations

Q

3

A

3

B

= Q

1

A

2

B

↔ Q

2

A

1

B

= ¬(Q

1

A

1

B

↔ Q

2

A

2

B

), (40)

and analogously for A, C and B, C, is the only

consistent solution. We shall henceforth make the

convention that the bipartite correlation structure

among any pair of q ubits be the same such that

(

40) holds. This turns out to be the case of qubit

quantum theory.

The tacit assumption, underlying the stan-

dard representation of qubit quantum theory, is

that the handedness of each local qubit ques-

tion basis for A, B, C is the same, e.g., all ‘left’

or ‘right’ handed. But we emphasize, that it

would be equally consistent to choose one basis as

‘left’ (‘right’) and the other two as ‘right’ (‘left’)

handed. A qubit pair with equally handed bases

will be described by the odd and even correla-

tion distribution as in (

40), while a qubit pair

with oppositely handed bases will be described

by the opposite distribution of odd and even cor-

relations. In terms of whether Q

33

= Q

11

↔ Q

22

is even or odd Q

33

= ¬(Q

11

↔ Q

22

), the three

qubit relations yield only four consistent graphs

for the distribution of ‘left’ and ‘right’ handedness

PSfrag replacements

odd odd

odd

even

‘left’

‘left’ ‘left’

‘right’

PSfrag replacements

odd

odd odd

even

‘left’

‘right’

‘right’‘right’

PSfrag replacements

odd

even

‘left’

‘left’‘right’

PSfrag replacements

odd

even

‘left’ ‘right’

‘right’

(41)

This gives a simple graphical explanation for

the consistency observations above. The ﬁrst

two graphs correspond to quantum theory. The

framework with the oppos ite correlation structure

of quantum theory, corresponding to the last two

graphs, is sometimes referred to as mirror quan-

Accepted in Quantum 2017-11-27, click title to verify 50

tum theory [15]. While mirror quantum theory

was considered inconsistent in [

15], we see here

that inconsistencies would only arise if the bi-

partite correlation structure of mirror quantum

theory (and in particular the even correlation

Q

33

= Q

11

↔ Q

22

) was used for all three pairs of

qubits. However, the proper formulation of mir-

ror quantum theory for three qubits corresponds

to the last two graphs in (

41) and gives a per-

fectly consistent framework.

38

It could be easily

generalized in an obvious manner to arbitrarily

many q ubits.

39

Obviously, the s ame argument can be carried

out f or any other triple of maximally compatible

bipartite correlations appearing in theorem 5.7.

In conclusion, the two distinct consistent distri-

butions of odd and even correlations, correspond-

ing to quantum theory and mirror quantum the-

ory,

(a) result from diﬀerent conventions of local ba-

sis handedness, and

(b) are in one-to-one correspondence through a

local relabeling ‘yes’↔‘no’ of one individual

question.

As such, the two distinct correlation structures

ultimately give rise to two distinct representa-

tions of the same physics and are thus fully equiv-

alent. The transformation (b) between the two

representations – being a translation between two

conventions/descriptions – is a passive one and

38

In fact, one could even produce the correlation struc-

ture of mirror quantum theory in the lab by using oppo-

sitely handed bases for two qubits in an entangled p air.

The resulting state would be represented by the partial

transpose of an entangled qubit state. This state would

not be positive if represented in terms of the standard

Pauli matrices and thus not correspond to a legal quan-

tum state [

88]. However, the point is that mirror quantum

theory would not be represented in the standard Pauli

matrix basis but in the partially transposed basis which

corresponds to replacing

σ

y

=



0 −i

i 0



by σ

T

y

=



0 i

−i 0



,

for one of t he two qubits (this is a switch of the y-axis

orientation). In this basis, the state would be positive.

39

Notice, however, that for more than three qubits one

would get more than two diﬀerent consistent distributions

of odd and even correlations. For ex ample, for four qubits

there will be three cases: (1) all four have equally handed

bases, (2) three have equally hand ed bases, (3) two have

equally handed bases.

can be carried out on a piece of paper; it is always

allowed. However, clearly, there cannot exist any

actual physical transformation in the laboratory

which maps states from one convention into the

other.

Within the formalism of quantum theory the

transformation (b) Q

1

7→ ¬Q

1

corresponds to

the partial transpose (e.g., see [

15]). It is well

known that the partial transpose deﬁnes a sepa-

rability criterion for quantum states which is both

necessary and suﬃcient for a pair of qubits [

88]:

a two-qubit density matrix ρ is separable if and

only if its partial transpose is positive relative to

a basis of standard Pauli matrix products (i.e.,

represents a legal q uantum state). This criterion

holds analogously in our language here: as seen at

the beginning of this section, the transformation

Q

1

7→ ¬Q

1

changes between odd and even corre-

lations of bipartite correlations Q

ij

. This would,

in fact, be unproblematic if O only had individ-

ual information about the two qubits; it would

map a classically composed state even of maxi-

mal information, say, Q

1

= 1 and Q

′

1

= 1 and

thus Q

11

= 1, to another legal classically com-

posed state Q

1

= 0, Q

′

1

= 1 and thus Q

11

= 0.

Both states exist within both conventions. How-

ever, applying this transformation to a maxi-

mally entangled state with odd correlation, say,

Q

11

= Q

22

= 1 and thus, by (

40), Q

33

= 0 yields

an even correlation Q

11

= 0 = Q

33

and Q

22

= 1.

The former state only exists in the quantum the-

ory representation, while the latter exists only in

the mirror image. For other entangled states one

would similarly ﬁnd that Q

1

7→ ¬Q

1

necessar-

ily maps from one representation into the other.

The same conclusion also holds for a transfor-

mation {Q

1

, Q

2

, Q

3

} 7→ {¬Q

1

, ¬Q

2

, ¬Q

3

}, which

one might call total inversion, because the odd

number of negations involved in the transforma-

tion would likewise lead to a swap of odd and

even correlations.

Lastly, we note that the situation is very diﬀer-

ent for rebit theory because it is its own mirror

image, i.e. rebit theory and mirror rebit theory

are identical representations. O will describ e a

single rebit by a question basis Q

1

, Q

2

. Suppose

O decided to swap the ‘yes’ and ‘no’ assignments

to the outcomes of Q

1

, such that equivalently

Q

1

7→ ¬Q

1

. For a pair of rebits, this would have

Accepted in Quantum 2017-11-27, click title to verify 51

the f ollowing ramiﬁcation

Q

11

, Q

12

7→ ¬Q

11

, ¬Q

12

,

Q

21

, Q

22

7→ Q

21

, Q

22

,

and therefore

Q

12

↔ Q

21

7→ ¬(Q

12

↔ Q

21

)

⇒ Q

33

7→ ¬Q

33

.

In contrast to the qubit case, Q

33

is deﬁned as

a correlation of correlations Q

33

:= Q

12

↔ Q

21

(

22) and can not be written in terms of local

questions Q

3

, Q

′

3

. Hence, Q

33

also changes un-

der this transformation by construction. Accord-

ingly, Q

1

7→ ¬Q

1

does not lead to a swap of odd

and even correlations for the rebit case. This

‘partial transpose’ therefore always maps states

to other states within the same representation.

For example, even a maximally entangled state

of maximal information and even correlation, say,

Q

12

= Q

21

= 1 and Q

33

= 1, would be mapped to

another evenly correlated state Q

12

= 0, Q

21

= 1

and Q

33

= 0. As a consequence, the Peres sep-

arability criterion [

88] which is valid for qubits

does not hold in analogous fashion for rebits. For

completely equivalent reasons, the total inver-

sion, corresponding to {Q

1

, Q

2

} 7→ {¬Q

1

, ¬Q

2

},

is also a transformation which preserves the rep-

resentation.

5.4.2 Collecting the results: odd and even corre-

lation s tructure for N = 2

After the many technical details it is useful to

collect all the results concerning the compatibil-

ity, complementarity and correlation structure for

two qubits, derived in lemmas

5.2 and 5.5, theo-

rem

5.7, equation (28) and in the previous section

5.4.1 in a graph to facilitate a visualization. We

shall henceforth abide by the convention that all

bipartite relations for arbitrarily many qubits be

treated eq ually such that (

40) must hold. For

the other relations of theorem

5.7 one ﬁnds the

analogous results. As can be easily veriﬁed, the

ensuing question s tructure has the lattice pattern

of ﬁgure

4, where the triangles

PSfrag replacements

−

+

Q Q

′

Q

′′

⇔ Q = ¬(Q

′

↔ Q

′′

),

PSfrag replacements

−

+

Q Q

′

Q

′′

⇔ Q = Q

′

↔ Q

′′

(42)

denote odd and even correlation, respectively.

We recall that (26, 28) imply alternating odd

and even correlation triangles for the bipartite

correlation questions. However, we emphasize

that the Bell scenario argument of section

5.2.7

does not require the correlation triangles involv-

ing individual questions to also admit such an

alternating odd and even pattern. For instance,

the f ollowing relations of Q

33

Q

33

= Q

3

↔ Q

′

3

= Q

12

↔ Q

21

= ¬(Q

11

↔ Q

22

)

are consistent with assumptions

5 and 6, despite

the absence of a negation in Q

3

↔ Q

′

3

= Q

12

↔

Q

21

because the latter is not a classical logical

identity. The graph corresponding to the last re-

lation,

PSfrag replacements

.

Q

11

Q

22

Q

12

Q

21

Q

33

Q

3

Q

′

3

is also diﬀerent from the graphs (

23) as it in-

volves all six individual questions and the argu-

ment leading to (28) does not apply. Clearly,

given the deﬁnition Q

ij

:= Q

i

↔ Q

′

j

, all such

triangles must be even.

The lattice structure in ﬁgure

4 contains 15 tri-

angles and 15 × 3 = 45 distinct edges correspond-

ing to compatibility relations. There are 60 ‘miss-

ing edges’ corresponding to complementarity re-

lations. Every question resides in three compat-

ibility triangles and is therefore maximally com-

patible with the six other questions in these three

triangles and maximally complementary to the

eight remaining q uestions not contained in those

three triangles. This means that once one ques-

tion is fully known, all other information available

to O must be distributed over the three adjacent

Accepted in Quantum 2017-11-27, click title to verify 52

PSfrag rep lacements

++

+

−

Q

1

Q

2

Q

3

Q

′

1

Q

′

2

Q

′

3

Q

11

Q

22

Q

12

Q

33

Q

13

Q

13

Q

21

Q

23

Q

23

Q

31

Q

32

identify identify

PSfrag rep lacements

+

−

Q

1

Q

2

Q

3

Q

′

1

Q

′

1

Q

′

2

Q

′

2

Q

′

3

Q

′

3

Q

11

Q

22

Q

12

Q

33

Q

13

Q

21

Q

23

Q

31

Q

32

identify

Figure 4: A lattice representation of the compl e t e compati bility, complementarity and correlation s t ructure of the

informationally comple t e s e t Q

M

2

for two qubits. Every vertex corresponds to one of the 15 pairwise independent

questions. If two questions are conn e c ted by an edge, they are maximally compatible. If two questions are not

connected by an edg e , they are maximally complementary. Thanks to the logical structure of the XNOR ↔, deﬁning

a question as a correlation of two other questions, the compatibility structure results in a lattice o f triangles. As

clariﬁed in (

42), red triangles deno te odd, while green triangl e s denote even c orrelation. Note that the two lattices

represented here are c onnected through the nine correlation questions Q

ij

and form a s ingle c losed l attice (which,

however, is easier to represent in this disconnecte d manner). Every question resides in exactly three triangles and is

thereby maximally compatible with six and maximally complementary to eight other qu e stions.

triangles. In particular, if O knows the answers

to two of the questions in the lattice, he will also

know the answer to the third sharing the same

triangle such that every triangle corresponds to

a speciﬁc set of states of maximal information.

(Although, as we shall see shortly in section

6.8

and, in more detail, in [1], the time evolution rule

4 implies that s tates of maximal information will

likewise exist such that two independent bits can

be distributed diﬀerently over the lattice.)

Notice that every triangle is connected by an

edge (adjacent to one of its three vertices) to ev-

ery other vertex in the lattice. This embodies

the statement ‘whenever O asks S a new ques-

tion, he experience no net loss of information’ of

the complementarity rule

2. For instance, if O

has maximal knowledge about two questions in

the lattice, corresponding to maximal informa-

tion about one triangle, it is impossible for him

to loose information by asking another question

from the lattice because it will be connected to

one of the questions from the previous triangle.

As a concrete example, suppose O knew the an-

swers to Q

21

, Q

13

, Q

32

. H e could ask Q

22

next.

Since Q

22

is connected by an edge to Q

13

, he

would k now the answers to both upon asking Q

22

and thereby then also the answer to Q

31

– no net

loss of information occurs.

Accepted in Quantum 2017-11-27, click title to verify 53

It is straightforward to check, e.g., using the

ansatz

|ψi = α |z

−

z

−

i + β |z

+

z

+

i + γ |z

−

z

+

i + δ|z

+

z

−

i

for a two qubit pure state and translating it into

the various basis combinations xx, xy, yx, yy, . . .,

that the lattice structure of ﬁgure

4 is pre-

cisely the compatibility and correlation struc-

ture of qubit q uantum theory. For instance,

Q

11

, Q

22

, Q

33

correspond to projectors onto the

+1-eigenspaces of σ

x

⊗ σ

x

, σ

y

⊗ σ

y

and σ

z

⊗ σ

z

.

The three questions sharing a red triangle means,

e.g., that Q

11

= Q

22

= 1 and Q

33

= 0 is an al-

lowed state while Q

11

= Q

22

= Q

33

= 1 is illegal.

Indeed, ignoring normalization, in quantum the-

ory one ﬁnds

|x

+

x

+

i − |x

−

x

−

i = −i|y

+

y

+

i + i|y

−

y

−

i

= |z

+

z

−

i + |z

−

z

+

i

for α = β = 0 and γ = δ = 1, corresponding to

the prop os itions Q

11

= 1: “the spins are corre-

lated in x-direction"; Q

22

= 1: “the spins are

correlated in y-direction"; and Q

33

= 0: “the

spins are anti-correlated in z-direction".

40

But

no quantum state exists such that the spins are

also correlated in z-direction if they are correlated

in x- and y-direction (this would be mirror quan-

tum theory). Every other triangle in the lattice

corresponds similarly to four pure quantum states

(representing the answer conﬁgurations ‘yes-yes’,

‘yes-no’, ‘no-yes’, ‘no-no’ to the two independent

questions per triangle).

Finally, we also collect the results on the com-

patibility, complementarity and correlation struc-

ture of two rebits, derived in lemmas

5.2, 5.5 and

5.8, theorem 5.9 and equation (26), in a lattice

structure in ﬁgure

5. There are six triangles and

6 × 3 = 18 edges representing compatibility rela-

tions. Every question resides in exactly two tri-

angles and is thereby maximally compatible with

four and maximally complementary to four other

questions. As in the qubit case in ﬁgure

4, every

triangle is connected by an edge to every other

vertex in the lattice, in conformity with rule 2

40

Similarly, |ψi = α |z

−

z

−

i + β |z

+

z

+

i =

α+β

2

(|x

+

x

+

i +

|x

−

x

−

i) +

β−α

2

(|x

+

x

−

i + |x

−

x

+

i) =

α+β

2

(|y

+

y

−

i +

|y

−

y

+

i)+

β−α

2

(|y

+

y

+

i+ |y

−

y

−

i). But |x

+

x

+

i+ |x

−

x

−

i =

|y

+

y

−

i+ |y

−

y

+

i and |y

+

y

+

i+ |y

−

y

−

i = |x

+

x

−

i+ |x

−

x

+

i.

Hence, once Q

33

= 1, one indeed gets Q

11

↔ Q

22

= 0

even though α, β are unspeciﬁed such that Q

11

, Q

22

may

be u nknown.

PSfrag rep lacements

++

+

−

Q

1

Q

2

Q

3

Q

′

1

Q

′

2

Q

′

3

Q

22

Q

11

Q

12

Q

33

Q

′

1

Q

′

1

Q

21

Q

′

2

Q

′

2

Q

1

Q

2

identify identify

Figure 5: A latti ce representation of the complete com-

patibility, complementarity and correlation structure of

the informationally complete set Q

M

2

for two rebits.

The explana tions of the caption of ﬁgure

4 also apply

here.

asserting that O shall not experience a net loss of

information by asking further questions.

5.5 The general case of N > 3 gbits

We are now prepared to investigate the indepen-

dence, compatibility and correlation structure en-

suing from rules

1 and 2 on a general Q

N

, i.e. of

O’s possible questions to a system S composed of

N > 3 gbits. In particular, we shall exhibit an in-

formationally complete set Q

M

N

for both qubits

and rebits. O can view the N gbits as a composite

system in many diﬀerent ways: as N individual

gbits, as one individual gbit and a composite sys-

tem of N − 1 gbits, as a system composed of 2

gbits and a system composed of N − 2 gibts, and

so on. All these diﬀerent compositions yield, of

course, the same question structure. It is simplest

to interpret the N gbits recursively as being com-

posed of a composite system of N − 1 gbits and a

new individual gbit. Deﬁnition

3.5 of a composite

system then implies that Q

N

must contain (1) the

questions of Q

N−1

for N − 1 gbits, (2) the set Q

1

of the new gbit, and (3) all logical connectives of

the maximally compatible questions of those two

sets. This entails slightly diﬀerent repercussions

for qubits and rebits.

5.5.1 An informationally complete set and entan-

glement for N > 3 qubits

A natural candidate f or an informationally com-

plete question set is given by the set of all possible

XNOR conjunctions of the individual questions of

Accepted in Quantum 2017-11-27, click title to verify 54

the N gbits

Q

µ

1

µ

2

···µ

N

:= Q

µ

1

↔ Q

µ

2

↔ · · · ↔ Q

µ

N

(43)

(we recall from (

13) that the logical connective

yielding independent q uestions is either ↔ or ⊕).

Here we have introduced a new index notation:

µ

a

is the question index for gbit a ∈ {1, . . . , N }

and can take the values 0, 1, 2, 3. As before the

index values i = 1, 2, 3 correspond to the indi-

viduals Q

1

a

, Q

2

a

, Q

3

a

. On the other hand, the

index value µ

a

= 0 implies that none of the three

individual questions of gbit a appears in the con-

junction (

43), i.e. always Q

0

a

≡ 1. For instance,

Q

100000···000

:= Q

1

, Q

0003020···0

:= Q

3

4

↔ Q

2

6

,

etc. Note that Q

000···000

corresponds to no ques-

tion. The set (

43) thus contains all individual,

bipartite, tripartite, and up to N-partite cor-

relation questions for N gbits. We emphasize

that, due to the special multipartite structure

(and associativity) of the XNOR, a question such

as Q

111···111

does not incarnate the question ‘are

the answers to Q

1

, Q

1

2

, · · · , Q

1

N

all the same?’.

For ex ample, for N = 4, Q

1

= Q

1

2

= 0 and

Q

1

3

= Q

1

4

= 1 also yield Q

1

2

1

3

1

4

= 1. This is

important for the entanglement structure.

We begin with an important result.

Lemma 5.25. The 4

N

− 1 questions

41

Q

µ

1

···µ

N

,

µ = 0, 1, 2, 3, are pairwise independent.

Proof. Consider Q

µ

1

···µ

N

and Q

ν

1

···ν

N

. The two

questions must disagree in at least one index oth-

erwise they would coincide. Let the questions

diﬀer on the ind ex of gbit a, i.e. µ

a

6= ν

a

. Sup-

pose µ

a

, ν

a

6= 0. Then Q

µ

a

is maximally compat-

ible with Q

µ

1

···µ

a

···µ

N

= Q

µ

1

↔ · · · ↔ Q

µ

a

↔

· · · ↔ Q

µ

N

and maximally complementary to

Q

ν

1

···ν

a

···ν

N

= Q

µ

1

↔ · · · ↔ Q

ν

a

↔ · · · ↔

Q

ν

N

since Q

µ

a

, Q

ν

a

are maximally complemen-

tary. Using the same argument as in the proof of

lemma

5.3 this implies independence of Q

µ

1

···µ

N

and Q

ν

1

···ν

N

. Lastly, suppose now µ

a

= 0 and

ν

a

6= 0 (as we have µ

a

6= ν

a

not both can be 0).

Then Q

i

a

6=ν

a

with i 6= 0 is maximally compatible

with Q

µ

1

···µ

N

and maximally complementary to

Q

ν

1

···ν

N

. By the same argument, this again im-

plies independence of Q

µ

1

···µ

N

and Q

ν

1

···ν

N

.

Consequently, the set (43) will be part of an

informationally complete set. We note that a

hermitian matrix of trace equal to 1 on a 2

N

-

dimensional complex Hilbert space (i.e. qubit

density matrix) is described by 4

N

− 1 param-

eters.

Next, we must elucidate the compatibility and

complementarity structure of this set.

Lemma 5.26. Q

µ

1

···µ

N

and Q

ν

1

···ν

N

are

maximally compatible if the index sets

{µ

1

, . . . , µ

N

} and {ν

1

, . . . , ν

N

} diﬀer by an

even numbe r (incl. 0) of non-zero indices,

maximally complementary if the index sets

{µ

1

, . . . , µ

N

} and {ν

1

, . . . , ν

N

} diﬀer by an

odd number of non-zero indices.

For example, for N = 2, Q

11

and Q

22

diﬀer

by two non-zero indices and are thus maximally

compatible. By contrast, Q

10

= Q

1

and Q

22

dif-

fer by an odd number of non-zero indices and are

thereby maximally complementary.

Proof. Let Q

µ

1

···µ

N

and Q

ν

1

···ν

N

disagree in an

odd number, call it 2n + 1, of non-zero indices.

We can always reshuﬄe the index labeling of the

N qubits such that now a = 1, . . . , 2n + 1 cor-

responds to the q ubits on which Q

µ

1

···µ

N

and

Q

ν

1

···ν

N

diﬀer by non-zero indices, i.e. µ

a

6= ν

a

and µ

a

, ν

a

6= 0. The remaining qubits labeled

by b = 2n + 2, . . . , N are then such that Q

µ

1

···µ

N

and Q

ν

1

···ν

N

either agree on the non-zero in dex,

µ

b

= ν

b

6= 0 or at least one of µ

b

, ν

b

is 0. That

is, after the reshuﬄing the index labeling, we can

write the questions as

Q

µ

1

···µ

N

= (Q

µ

1

↔ · · · ↔ Q

µ

2n+1

)

| {z }

disagreement

↔ (Q

µ

2n+2

↔ · · · ↔ Q

µ

N

)

| {z }

maximally compatible

Q

ν

1

···ν

N

=

z }| {

(Q

ν

1

↔ · · · ↔ Q

ν

2n+1

) ↔

z }| {

(Q

ν

2n+2

↔ · · · ↔ Q

ν

N

) . (44)

41

We deduct the trivial question Q

000···000

. Obviously,

one arrives at the same number by counting the distri-

bution of individuals, bipartite,...and N -partite correla-

tions over N qubits as a binomial series

P

N

k=1



N

k



3

k

=

(3 + 1)

N

− 1.

Accepted in Quantum 2017-11-27, click title to verify 55

The parts of the questions where the index

sets either agree or feature zeros coincide with

Q

µ

2n+2

···µ

N

and Q

ν

2n+1

···ν

N

and are clearly maxi-

mally compatible (drop ping here zero indices).

We can now proceed by induction. Lemmas

5.2, 5.5, 5.10–5.16 imply that the statement of

this lemma is correct for n = 0, 1. Let the state-

ment therefore be true for n and consider n + 1.

Then, (

44) reads

Q

µ

1

···µ

N

=(Q

µ

1

···µ

2n+3

)

| {z }

disagreement

↔ Q

µ

2n+4

···µ

N

| {z }

maximally compatible

= (Q

µ

1

↔ · · · ↔ Q

µ

2n+1

)

| {z }

maximally complementary

↔ (Q

µ

2n+2

µ

2n+3

)

| {z }

disagreement

↔ Q

µ

2n+4

···µ

N

| {z }

max. compat.

Q

ν

1

···ν

N

=

z

}| {

(Q

ν

1

···ν

2n+3

) ↔

z }| {

Q

ν

2n+4

···ν

N

=

z }| {

(Q

ν

1

↔ · · · ↔ Q

ν

2n+1

) ↔

z }| {

(Q

ν

2n+2

ν

2n+3

) ↔

z }| {

Q

ν

2n+4

···ν

N

.

But, by lemma 5.5, Q

µ

2n+2

µ

2n+3

and Q

ν

2n+2

ν

2n+3

are maximally compatible with each other and

therefore also with Q

µ

1

···µ

N

and Q

ν

1

···ν

N

. This

implies that both Q

µ

1

···µ

N

and Q

ν

1

···ν

N

are max-

imally compatible with and, thanks to lemma

5.25, independent of, e.g., Q

µ

2n+2

···µ

N

. Com-

plementarity of Q

µ

1

···µ

N

and Q

ν

1

···ν

N

now fol-

lows from lemma 5.1 and noting th at Q

µ

1

···µ

2n+1

and Q

ν

1

···ν

N

are maximally complementary be-

cause they disagree in 2n + 1 non-zero ind ices

(for which, by assumption, the s tatement of the

lemma holds).

Finally, let Q

α

1

···α

N

and Q

β

1

···β

N

disagree in

an even number 2n of non-zero indices. Using

an analogous reshuﬄing of the index labeling as

in the odd case above, one can rewrite the two

questions as

Q

α

1

···α

N

= Q

α

1

α

2

|

{z }

diﬀer

↔ Q

α

3

α

4

|

{z }

diﬀer

↔ · · · ↔ Q

α

2n−1

α

2n

|

{z }

diﬀer

↔ Q

α

2n+1

···α

N

| {z }

maximally compatible

Q

β

1

···β

N

=

z

}| {

Q

β

1

β

2

↔

z }| {

Q

β

3

β

4

↔ · · · ↔

z }| {

Q

β

2n−1

β

2n

↔

z }| {

Q

β

2n+1

···β

N

. (45)

That is, one can decompose the disagreeing parts

of the questions into bipartite correlations. But

thanks to lemma

5.5 two bipartite correlations of

the same qubit pair are maximally compatible if

and only if they diﬀer in both indices. Conse-

quently, all the pairings of question components

of the up per and lower line, as w ritten in (

45),

are maximally compatible and, hence, so must be

Q

α

1

···α

N

and Q

β

1

···β

N

.

Fortunately, it turns out that the 4

N

− 1 ques-

tions (

43) are logically closed under the XNOR.

Theorem 5.27. (Qubits) The 4

N

−1 questions

Q

µ

1

···µ

N

, µ = 0, 1, 2, 3, are logically closed under

↔ and thus form an informationally complete set

Q

M

N

with D

N

= 4

N

− 1 for the case D

1

= 3.

Proof. We shall prove the statement by induc-

tion. The statement is trivially true for N = 1

and, by theorems

5.7 and 5.17, holds also for

N = 2, 3. Let the statement th erefore be true

for N − 1 and consider a composite system of N

qubits. Since O can treat the N qubit system

in many diﬀerent ways as a composite system

and th e statement holds for N − 1, all XNOR

conjunctions of questions involving at least one

zero ind ex will be contained in (

43). We thus

only need to show th at all ↔ conjunctions in -

volving at least one N-partite correlation Q

i

1

···i

N

,

i

a

6= 0 ∀ a = 1, . . . , N, produce questions already

included in (43).

Consider Q

i

1

···i

N

and Q

ν

1

···ν

N

. Lemma

5.26 im-

plies that these two questions are maximally com-

patible – and can thus be connected by ↔ – if and

only if {i

1

, . . . , i

N

} and {ν

1

, . . . , ν

N

} disagree in

an even number of non-zero indices. There are

now two cases that we must consider:

(a) Suppose Q

i

1

···i

N

and Q

ν

1

···ν

N

disagree in

an even number of non-zero indices and, further-

more, agree on at least one index i

a

= ν

a

. Thanks

to Q

i

a

↔ Q

i

a

= 1, the conjunction then yields

Accepted in Quantum 2017-11-27, click title to verify 56

Q

i

1

···i

a

···i

N

↔ Q

ν

1

···i

a

···ν

N

= Q

i

1

··· 0

position a

···i

N

↔ Q

ν

1

··· 0

position a

···ν

N

.

Hence, the two questions on the right hand side

both contain less than N non-zero indices such

that the result must lie in the set (

43) because

the statement is true up to N − 1 by assumption.

(b) Suppose Q

i

1

···i

N

and Q

ν

1

···ν

N

disagree in an

even number 2n of non-zero in dices and do not

agree on any n on -zero index. Reshuﬄing the in-

dex labelings as in the proof of lemma

5.26, one

can th en write

Q

i

1

···i

N

= Q

i

1

i

2

|

{z}

diﬀer

↔ Q

i

3

i

4

|

{z}

diﬀer

↔ · · · ↔ Q

i

2n−1

i

2n

|

{z }

diﬀer

↔ Q

i

2n+1

···i

N

Q

ν

1

···ν

N

=

z

}|{

Q

ν

1

ν

2

↔

z}|{

Q

ν

3

ν

4

↔ · · · ↔

z }| {

Q

ν

2n−1

ν

2n

↔ Q

0

2n+1

···0

N

.

(We obtain here Q

0

2n+1

···0

N

because Q

i

1

···i

N

and

Q

ν

1

···ν

N

do not agree on any common in dex and

Q

i

1

···i

N

does not feature zero-indices.) By lemma

5.5, the pairs of bip artite correlations d iﬀering in

two in dices, e.g., Q

i

1

i

2

and Q

ν

1

ν

2

, are maximally

compatible and by theorem

5.7 their XNOR con-

junction will yield another bipartite correlation of

the same qubit pair. For example, Q

i

1

i

2

↔ Q

ν

1

ν

2

equals either Q

j

1

j

2

or ¬Q

j

1

j

2

for j

1

6= i

1

, ν

1

and

j

2

6= i

2

, ν

2

. Accordingly, up to negation, one ﬁnds

(j

a

6= i

a

, ν

a

, a = 1, . . . , 2n)

Q

i

1

···i

N

↔ Q

ν

1

···ν

N

= Q

j

1

j

2

↔ Q

j

3

j

4

↔ · · · ↔ Q

j

2n−1

j

2n

↔ Q

i

2n+1

···i

N

= Q

j

1

···j

2n

i

2n+1

···i

N

which is another N-partite correlation contained

in the set (

43).

As in the cases N ≤ 3, we could represent

the compatibility and complementarity relations

for N > 3 qubits geometrically by a simplicial

question graph where a D-partite question corre-

sponds to a (D − 1)-dimensional simplex within

the graph (D ≤ N). The compatibility and com-

plementarity relations of lemma 5.26 then trans-

late into abstract geometric relations according

to whether a D-simplex and a D

′

-simplex share

or disagree on subsimplices. In particular, the

criterion that two distinct questions in Q

M

N

are

maximally compatible if and only if they disagree

on an even number of non-zero indices means ge-

ometrically that the two questions are maximally

compatible if and only if the two subsimplices in

the two question simplices which correspond to

the qubits they share in common either

(a) do not overlap and are odd-dimensional, or

(b) coincide.

For example, for N = 2, Q

11

, Q

22

correspond to

two non-intersecting edges, i.e. 1-simplices; they

are maximally compatible because they disagree

on their 1-simplices, both involving qubit 1 and 2.

On the other hand, for N = 3, Q

1

A

1

B

, Q

2

B

2

C

also

correspond to two non-intersecting 1-simplices

but are maximally complementary because the

two edges involve diﬀerent qubits – A, B for the

ﬁrst and B, C for the second edge.

This also helps us to understand entanglement

for arbitrarily many qubits. Speciﬁcally, maxi-

mal entanglement will correspond to O spending

the N independent bits he is allowed to acquire

about the system S of N qubits on N-partite cor-

relation questions. Lemma

5.26 guarantees that

for every N there will exist N maximally com-

patible N-partite questions. For instance, there

are



N

2



ways of having N − 2 of the indices take

value 1 and 2 indices take the value 2. Any two

of the



N

2



corresponding questions will disagree

in two or four non-zero indices (and agree on the

rest) and will thus be maximally compatible (even

mutually according to theorem

3.2). These will

Accepted in Quantum 2017-11-27, click title to verify 57

correspond to a set of



N

2



maximally compati-

ble (N − 1)-simplices in the question graph such

that any two of them either disagree on an edge

or a tetrahedron. (There will exist even more

maximally compatible N -partite questions.) For

N ≥ 3 it also holds that



N

2



≥ N.

It is deﬁnitely possible to choose N such maxi-

mally compatible N-partite correlation questions

out of the



N

2



many such that these N questions

do not all agree on a s ingle index. For similar rea-

sons to the N = 2, 3 cases, this choice will con-

stitute a mutually independent set such that ev-

ery individual question Q

i

1

, . . . , Q

i

N

will be max-

imally complementary to at least one of these N

N-partite questions. (As a consequence of rule

1,

once the answers to these N N-partite questions

are known, they will also imply the answers to the



N

2



− N remaining ones by the same reasoning

as in s ection

5.2.3.) Accordingly, O can exhaust

the information limit with these N-partite corre-

lation questions, while not being able to have any

information whatsoever about the individuals – a

necessary condition for maximal entanglement.

There will exist many diﬀerent ways of having

such multipartite entanglement for arbitrary N.

One could describe such diﬀerent ways of entan-

glement by generalizing the correlation measures

(

33) and informational monogamy inequalities re-

sulting therefrom. These monogamy inequalities

could also be considered as simplicial relations:

they restrict the way in which the available (inde-

pendent and dependent) information can be dis-

tributed over the various simplices and subsim-

plices in the question graph. However, we abstain

from analyzing such relations here further.

5.5.2 An informationally complete set and entan-

glement for N > 3 rebits

We brieﬂy repeat the same procedure for N

rebits. In analogy to (

43), the natural candidate

set for an informationally complete Q

M

N

will con-

tain

Q

µ

1

µ

2

···µ

N

:= Q

µ

1

↔ Q

µ

2

↔ · · · ↔ Q

µ

N

,

µ

a

= 0, 1, 2, a = 1, . . . , N, (46)

where the notation should be clear from sec-

tion

5.5.1. However, being a composite sys-

tem, by deﬁnition

3.5 we must permit the cor-

relation of correlations Q

3

a

3

b

(

22) for all a, b ∈

{1, . . . , N } because clearly O is allowed to ask

Q

1

a

, Q

2

a

, Q

1

b

, Q

2

b

. Furthermore, thanks to (

36)

we have, e.g.,

Q

3

1

3

2

3

4

= Q

3

1

3

2

↔ Q

3

4

= Q

3

1

3

↔ Q

3

2

3

4

= Q

3

1

3

4

↔ Q

3

2

3

such that no confusion can arise about the mean-

ing of Q

3

1

3

2

3

4

although there are no individu-

als Q

3

a

into which the question could be decom-

posed. The same holds similarly for any other

even number of indices taking the value 3. Con-

sequently, the candidate set for Q

M

N

can be writ-

ten as

˜

Q

M

N

:=



Q

µ

1

···µ

N

, µ

a

= 0, 1, 2, 3, a = 1, . . . , N



only even number of indices taking value 3



with an evident meaning of each such question.

Let us count the number of elements within

˜

Q

M

N

.

Lemma 5.28.

˜

Q

M

N

contains 2

N−1

(2

N

+ 1) − 1

non-trivial questions.

Proof. If arbitrary distributions of the values

µ

a

= 0, 1, 2, 3 over the a = 1, . . . , N were per-

mitted, we would obtain 4

N

− 1 non-trivial ques-

tions upon subtracting Q

0

1

0

2

···0

N

as in the qub it

case. In order to obtain the number of questions

within

˜

Q

M

N

we thus still have to s ubtract all the

possible ways of d istributing an odd number of

3’s over the N indices. There are precisely

O

N

:=

N

1

!

3

N−1

+

N

3

!

3

N−3

+ · · · +

N

2n + 1

!

3

N−(2n+1)

such ways, where 2n+1 is the largest odd number

smaller or equal to N. Similarly, the number of

ways an even number of 3’s can be distributed

over N indices is given by

E

N

:= 3

N

+

N

2

!

3

N−2

+ · · · +

N

2m

!

3

N−2m

,

Accepted in Quantum 2017-11-27, click title to verify 58

where 2m is the largest even number smaller or

equal to N. We then have

E

N

+ O

N

= (3 + 1)

N

= 4

N

,

E

N

− O

N

= (3 − 1)

N

= 2

N

and thus

O

N

=

1

2

(4

N

− 2

N

)

which yields

4

N

− 1 − O

N

= 2

N−1

(2

N

+ 1) − 1

non-trivial questions in

˜

Q

M

N

.

We note that the number of parameters in a

symmetric matrix with trace equal to 1 on a 2

N

-

dimensional real Hilbert space (i.e. rebit density

matrix) is precisely

1

2

N

(2

N

+ 1) − 1.

Next, we assert pairwise independence as re-

quired f or an informationally complete set.

Lemma 5.29. The 2

N−1

(2

N

+ 1) − 1 non-trivial

questions in

˜

Q

M

N

are pairwise independent.

Proof. The proof is entirely analogous to the

proofs of lemmas

5.3, 5.13, 5.14 and 5.25.

Likewise, the complementarity and compatibil-

ity structure of

˜

Q

M

N

is analogous to the qubit

case.

Lemma 5.30. Q

µ

1

···µ

N

, Q

ν

1

···ν

N

∈

˜

Q

M

N

are

maximally compatible if the index sets

{µ

1

, . . . , µ

N

} and {ν

1

, . . . , ν

N

} diﬀer by an

even number (incl. 0) of non-zero indices,

and

maximally complementary if the index sets

{µ

1

, . . . , µ

N

} and {ν

1

, . . . , ν

N

} diﬀer by an

odd number of non-zero indices.

Proof. Thanks to lemmas

5.8, 5.18–5.20 and

5.23, the proof of lemma 5.26 also applies to

the rebit case with the sole diﬀerence that only

an even number of indices in the questions can

take the value 3 and that correlations of correla-

tions Q

3

a

3

b

cannot be decomposed into individu-

als Q

3

a

, Q

3

b

.

Finally,

˜

Q

M

N

is indeed logically closed.

Theorem 5.31. (Rebits)

˜

Q

M

N

is logically

closed under ↔ and is thus an informationally

complete set

˜

Q

M

N

= Q

M

N

with D

N

= 2

N−1

(2

N

+

1) − 1 for the case D

1

= 2.

Proof. Thanks to theorems 5.9, 5.24 and lemma

5.30, the proof of theorem 5.27 also applies here,

except that only an even number of indices can

take the value 3 and Q

3

a

3

b

cannot be decomposed

into individuals Q

3

a

, Q

3

b

.

We close with the observation that simi-

larly to the N qubit case in section

5.5.1, one

could represent the compatibility and comple-

mentarity structure v ia a simplicial question

graph. Maximally entangled states (of maximal

information) will correspond to O spending

all N available bits over a mutually inde-

pendent set of N N -partite questions which

is maximally complementary to every indi-

vidual question. In fact, the prescription for

constructing a maximally N -partite entangled

qubit state provided at the end of section

5.5.1

also applies to N rebits since only indices with

values 1, 2 were employed. Furthermore, for

rebits one can similarly generate maximally

entangled states of non-maximal information;

e.g., O could ask only the N − 1 questions

Q

3

1

3

2

000···

, Q

03

2

3

00···

, Q

003

3

4

0···

, . . . , Q

0···03

N−1

3

N

.

If O has maximal information about each such

question, every rebit pair will be maximally

entangled, while O has still not reached the

information limit (see also section

5.2.6).

6 Information measure, time evolution

and state space

Thus far we have only applied rules

1 and 2 to

derive the basic question structure on Q

N

. We

shall now slightly switch topic and return to the

problems of time evolution, begun in subsection

3.2.8, and of explicitly quantifying the informa-

tion which O has acquired about S by means of

interrogations with questions. The information

measure and time evolution of S’s state in be-

tween interrogations are intertwined through rule

3 of information preservation and rule 4 of max-

imality of time evolution such that we have to

discuss these topics together.

In (

6) we have implicitly deﬁned the total

amount of O’s information about S – once the

latter is in a s tate ~y

O→S

– as

I

O→S

(~y

O→S

) =

D

N

X

i=1

α

i

(~y

O→S

),

Accepted in Quantum 2017-11-27, click title to verify 59

where α

i

quantiﬁes O’s information about the

outcome of question Q

i

∈ Q

M

N

and satisﬁes the

bounds (1). We shall begin in subsection 6.1 by

clarifying that the information measure we seek

to derive here is conceptually distinct from the

standard Shannon entropy, before imposing el-

ementary consistency conditions on the relation

α

i

(~y

O→S

) in subsection

6.2. Subsequently, we use

these conditions and implement rules

3 and 4 to

establish that the set of possible time evolutions

deﬁnes a group and, ﬁnally, to derive the explicit

functional f orm of I

O→S

in subsection

6.8. This

discussion will also unravel properties of state

spaces and lead to the notion of pure states.

6.1 Why the Shannon entropy does not apply

here

Consider a random variable experiment with dis-

tinct outcomes each of which is described by a

particular probability. Outcomes with a smaller

likelihood are more informative (relative to the

prior information) than those with a larger like-

lihood. The Shannon information quantiﬁes the

information an observer gains, on average, via a

speciﬁc outcome if she repeated the experiment

many times. (Equivalently, it quantiﬁes the un-

certainty of the observer before an outcome of

the experiment occurs.) The probability distri-

bution over the diﬀerent outcomes is assumed to

be known. In particular, none of the outcomes in

any run of the experiment will lead to an update

of the probability distribution.

Here, by contrast, we are not interested in how

informative one speciﬁc question outcome is rel-

ative to another and how much information O

would gain, on average, with a speciﬁc answer if

he repeated the interrogation with a ﬁxed q ues-

tion many times on identically prepared systems.

Instead, we wish to quantify O’s prior informa-

tion (e.g., gained from previous interrogations)

about the possible question outcomes on only

the next system to be interrogated. The role of

the probability distribution over the various out-

comes (to all questions) is here assumed by the

state which O ascrib es to S. This state is not as-

sumed to be ‘known’ absolutely, instead, as dis-

cussed in section

3.2, it is deﬁned only relative

to the observer and represents O’s ‘catalogue of

knowledge’ or ‘degree of belief’ about S. In par-

ticular, this state is updated after any interro-

gation if the outcome has led to an information

gain. This update takes the form of a ‘collapse’ of

the prior into a posterior state of s single system

in a single shot interrogation, while it yields an

update of the ensemble state in a multiple shot

interrogation (see subsection

3.2.3). The proba-

bility distribution represented by the state is thus

a prior distribution for the next interrogation and

only valid until the next answer – unless the lat-

ter yields no information gain. We thus seek to

quantify the information content in the state of

S relative to O. This is not eq uivalent to the sum

of the average information gains, relative to that

same state used as a ﬁxed prior probability dis-

tribution, which are provided by speciﬁc answers

to the questions in an informationally complete

set over repeated interrogations.

The reader should thus not be surprised to ﬁnd

that the end result of the below derivation will

not yield the Shannon entropy. We emphasize,

however, that this clearly does not invalidate the

use of the Shannon entropy (in the more general

form of the von Neumann entropy) in quantum

theory as long as one employs it what it is de-

signed f or: to describe average information gains

in repeated experiments on identically prepared

systems – relative to a prior state which is as-

signed before the repeated experiments are car-

ried out and which represents the ‘known’ prob-

ability distribution.

6.2 Elementary conditions on the measure

There are a few natural requirements on the re-

lation between α

i

and ~y

O→S

:

(i) The Q

i

∈ Q

M

N

are pairwise independent.

Accordingly, α

i

should not depend on the

‘yes’-probabilities y

j6=i

of other questions

Q

j6=i

such that α

i

= α

i

(y

i

).

(ii) All Q

i

∈ Q

M

N

are informationally of equiv-

alent status. The functional relation be-

tween y

i

and α

i

should be the same for all i:

α

i

= α(y

i

), i = 1, . . . , D

N

.

(iii) If O has no information about the outcome

of Q

i

, i.e. y

i

= 1/2, then α

i

= 0 bit.

(iv) If O has maximal information about the out-

come of Q

i

, i.e. y

i

= 1 or y

i

= 0, then α

i

= 1

bit; both possible answers give 1 bit of in-

formation.

Accepted in Quantum 2017-11-27, click title to verify 60

(v) The assignment of which answer to Q

i

is

‘yes’ and which is ‘no’ is arbitrary and the

functional relation between α

i

and y

i

(or n

i

)

should not depend on this choice. Hence,

α(y

i

) = α(n

i

) must be symmetric around

y

i

= 1/2 (see also (iv)); α

i

quantiﬁes the

amount of information about Q

i

, but does

not encode what the answer to Q

i

is.

(vi) On the interval y

i

∈ (1/2, 1], the relation

between α and y

i

should be monotonically

increasing such that O’s information about

the answer to Q

i

is quantiﬁed as higher, the

higher the assigned probability for a ‘yes’

outcome. Likewise, on [0, 1/2), α(y

i

) must

be monotonically decreasing. In particular,

α shall be continuous and strictly convex;

i.e., for all y

1

i

, y

2

i

∈ [0, 1] and every λ ∈ [0, 1]

α(λ y

1

i

+ (1 − λ) y

2

i

) ≤ λ α(y

1

i

) + (1 − λ) α(y

2

i

)

and the inequality shall be strict whenever

y

1

i

6= y

2

i

and 0 < λ < 1. H ence, y

i

=

1

2

is the

global minimum of α on [0, 1].

Since now

I

O→S

(~y

O→S

) =

D

N

X

i=1

α(y

i

) (47)

and each summand is strictly convex, we also

have that I

O→S

is a strictly convex function on

Σ

N

. Accordingly, in the coin ﬂip scenario of sub-

section

3.2.8, O’s information about the mixed

state ~y

O→S

12

= λ ~y

O→S

1

+(1−λ) ~y

O→S

2

is smaller

than his maximal inf ormation about any of its

constituents ~y

O→S

1

, ~y

O→S

2

– unless the outcome

of the coin ﬂip was certain, or S

1

, S

2

are in the

same state. This is consistent with the fact that

O’s information about the outcome of any ques-

tion (asked to either S

1

or S

2

, depending on the

outcome of the coin ﬂip) is clouded by the random

coin ﬂip outcome.

6.3 The state of no information is an interior

state

In order to discuss the action of time evolution

on the state space Σ

N

, we have to derive a few

further properties of the latter, some of which

depend on the properties of I

O→S

.

Firstly, since D

N

is ﬁnite for ﬁnite N, Σ

N

is ﬁnite dimensional in this case. In fact, Σ

N

contains a basis of R

D

N

42

so that it is a D

N

-

dimensional closed convex subset of R

D

N

. Fur-

thermore, since y

i

∈ [0, 1] for all y

i

in ~y

O→S

, Σ

N

is clearly bounded and thus, by the Heine-Borel-

theorem, compact. As a compact convex set with

non-empty interior, Σ

N

is thus a convex b ody

and, in particular, has volume [

89].

Next, we focus on the state of no information

~y

O→S

=

1

2

~

1. Given strict convexity of I

O→S

on

Σ

N

, the conditions of the previous subsection en-

tail that the state of no information is the global

minimum of I

O→S

with I

O→S

(

1

2

~

1) = 0 bits –

in agreement with its uniqueness. It is also an

interior state.

Lemma 6.1. The state of no information lies in

the interior of Σ

N

.

Proof. The Q

i

∈ Q

M

N

are pairwise indepen-

dent so for each such Q

i

there exist two states

~y

y

i

O→S

= (

1

2

, · · · ,

1

2

, 1,

1

2

, · · · ,

1

2

) and ~y

n

i

O→S

=

(

1

2

, · · · ,

1

2

, 0,

1

2

, · · · ,

1

2

) corresponding to O having

only asked Q

i

to S in the state of no information

and having received the answers Q

i

= ‘yes’ and

Q

i

= ‘no’, respectively. Since Σ

N

is convex, it

also contains all convex mixtures of these special

states. The D

N

~y

y

i

O→S

, as well as the D

N

~y

n

i

O→S

each deﬁn e a basis of R

D

N

such that the set of all

their convex mixtures deﬁne a D

N

-dimensional

convex polytope contained in Σ

N

⊂ R

D

N

. In

particular, given that

1

2

~y

y

i

O→S

+

1

2

~y

n

i

O→S

=

1

2

~

1

yields the state of no information for each i, it

is clear that it lies in the interior of the convex

polytope.

Given that time evolution preserves the total

information by rule

3, we shall also be interested

in the level sets of I

O→S

.

Lemma 6.2. Let I

O→S

fulﬁll conditions (i)–(vi)

of sec tion

6.2. For suﬃciently small ε > 0, the

level set

L

ε

:= {~y

O→S

∈ Σ

N

|I

O→S

(~y

O→S

) = ε} (48)

lies in the interior of Σ

N

and is homeomorphic

to S

D

N

−1

. In this case, L

ε

constitutes the full

boundary of the fat level set L

I≤ε

:= {~y

O→S

∈

Σ

N

|I

O→S

(~y

O→S

) ≤ ε} which contains the state

42

Each of the following D

N

-dimensional vectors

(1,

1

2

,

1

2

, · · · ,

1

2

), (

1

2

, 1,

1

2

, · · · ,

1

2

), · · · , (

1

2

, · · · ,

1

2

, 1) rep-

resents a legal state ~y

O→S

, corresponding to O only

knowing the answer to precisely one question from Q

M

N

with certainty and ‘nothing else’.

Accepted in Quantum 2017-11-27, click title to verify 61

of no information and is homeomorphic to the

closed unit ball in R

D

N

.

Proof. The state of no information is the global

minimum of I

O→S

with I

O→S

(

1

2

~

1) = 0 and,

thanks to lemma

6.1, lies in the interior of Σ

N

.

Furthermore, condition (vi) of subsection

6.2 re-

quires that α(y

i

) increases monotonically away

from y

i

=

1

2

for each i so that I

O→S

in (

47) must

likewise increase monotonically in each direction

away from ~y

O→S

=

1

2

~

1 in R

D

N

. Owing to the

continuity of I

O→S

, we can therefore ﬁnd a suﬃ-

ciently small ε > 0 su ch that any state attained

by moving away from the state of no inform ation

in any direction until reaching I

O→S

= ε still lies

in the interior of Σ

N

. Accordingly, for such ε > 0,

L

ε

must lie in the interior of Σ

N

and so must L

I≤ε

with the state of no information in its own inte-

rior. Given that I

O→S

: Σ

N

→ R is continuous

and Σ

N

is closed, I

O→S

is a closed convex func-

tion. Closed convex functions have closed convex

fat level sets [

89] so that L

I≤ε

is a closed convex

subset in the interior of Σ

N

. Clearly, L

I≤ε

is D

N

-

dimensional since we moved away in all directions

from ~y

O→S

=

1

2

~

1 to reach L

ε

and the states with

I

O→S

= ε deﬁne the limit points of L

I≤ε

in each

direction so that L

ε

constitutes its f ull bound-

ary. Any D

N

-dimensional closed convex subset

of R

D

N

with non-empty interior is homeomorphic

to th e closed unit ball and its boundary is home-

omorphic to S

D

N

−1

.

These results will become important for estab-

lishing that the set of time evolutions is a group.

6.4 Time evolution of the ‘Bloch vector’

In subsection 3.2.8, we have only considered the

time evolution of the redundantly parametrized

2D

N

-dimensional state

~

P

O→S

. According to (10),

this time evolution is linear, described by a ma-

trix A(∆t), and only depends on the interval

∆t = t

2

− t

1

, where t

1

, t

2

are arbitrary instants of

time in between O’s interrogations on a given sys-

tem. However, because of (

4), it clearly suﬃces

to consider the yes-vector ~y

O→S

(or no-vector

~n

O→S

) alone to describe the state of S relative

to O. Let us therefore determine, how ~y

O→S

and

~n

O→S

evolve under time evolution. To this end,

we decompose A(∆t) and

~

P

O→S

in (

10),

~y

O→S

(∆t)

~n

O→S

(∆t)

!

=

a(∆t) b(∆t)

c(∆t) d(∆t)

!

~y

O→S

(0)

~n

O→S

(0)

!

,

where a(∆t), b(∆t), c(∆t), d(∆t) are nonnegative

(real) D

N

× D

N

matrices. Using the normal-

ization (

4) (with p = 1), one ﬁnds that both

~y

O→S

, ~n

O→S

evolve aﬃnely,

~y

O→S

(∆t) = [a(∆t) − b(∆t)] ~y

O→S

(0) + b(∆t)

~

1,

~n

O→S

(∆t) = [d(∆t) − c(∆t)] ~n

O→S

(0) + c(∆t)

~

1. (49)

We shall now determine relations among the four

matrices a, b, c, d. Firstly, the normalization (

4)

must hold for all initial states and all times.

Hence, for all ∆t ∈ R,

~

1 = ~y

O→S

(∆t) + ~n

O→S

(∆t)

= [a(∆t) − b(∆t)] ~y

O→S

(0) + [d(∆t) − c(∆t)] ~n

O→S

(0) + [b(∆t) + c(∆t)]

~

1

=

(

4)

[a(∆t) − b(∆t) + c(∆t) − d(∆t)] ~y

O→S

(0) + [b(∆t) + d(∆t)]

~

1.

Since the set of all possible initial states ~y

O→S

(0)

contains a basis of R

D

N

, we must conclude

a(∆t) − b(∆t) = d(∆t) − c(∆t),

[b(∆t) + d(∆t)]

~

1 =

~

1. (50)

Next, we note that, by assumption

3, the state of

no information is unique and, by rule

3, clearly

preserved under time evolution. Inserting the

state of no inf ormation, e.g., as ~n

O→S

(∆t) =

1

2

~

1

Accepted in Quantum 2017-11-27, click title to verify 62

into (49), this entails

[c(∆t) + d(∆t)]

~

1 =

~

1. (51)

In conjunction, (50, 51) therefore imply

43

~y

O→S

(∆t) = [a(∆t) − b(∆t)] ~y

O→S

(0) + b(∆t)

~

1,

~n

O→S

(∆t) = [a(∆t) − b(∆t)] ~n

O→S

(0) + b(∆t)

~

1.

Consequently, deﬁning the D

N

× D

N

time evolu-

tion matrix

T (∆t) := a(∆t) − b(∆t), (52)

yields a linear time evolution of what we shall

henceforth call the generalized Bloch v ector

2 ~y

O→S

−

~

1:

2 ~y

O→S

(∆t) −

~

1 = ~y

O→S

(∆t) − ~n

O→S

(∆t)

= T (∆t)



2 ~y

O→S

(0) −

~

1



. (53)

Notice that T (∆t) need neither be nonnegative

nor stochastic in any pair of its components (in

contrast to A(∆t)). Henceforth, we shall only

consider the T (∆t) governing the time evolution

of the Bloch vector

~r

O→S

:= 2 ~y

O→S

−

~

1 (54)

and often employ the latter to parametrize states.

6.5 Time evolution is injective

We continue by demonstrating that time evolu-

tion of S’s states must be injective on Σ

N

. In

subsection

6.6, we will use this result to prove

that time evolution is also surjective on Σ

N

and

thus, in f act, reversible.

Lemma 6.3. Let T (∆t) : Σ

N

→ Σ

N

be a time

evolution as given in (

52) for any ∆t ∈ R. If

I

O→S

is strictly convex, rule

3 entails that T (∆t)

is injective.

Proof. Assume T (∆t) was not injective. T hen

there would exist states ~y

′

O→S

(t

1

) 6= ~y

′′

O→S

(t

1

)

such that

~y

O→S

(t

2

) := T (∆t)~y

′

O→S

(t

1

)

= T (∆t)~y

′′

O→S

(t

1

). (55)

Now consider the coin ﬂip scenario of section

3.2.8. O can prep are S

1

in th e state ~y

O→S

1

(t

1

) :=

~y

′

O→S

(t

1

) an d S

2

in the state ~y

O→S

2

(t

1

) :=

~y

′′

O→S

(t

1

) at time t

1

before tossing the coin. By

(

8),

~y

O→S

12

(t

1

) = λ ~y

O→S

1

(t

1

) + (1 − λ) ~y

O→S

2

(t

1

),

and, on account of (55) and rule 3,

I

O→S

1

(~y

O→S

1

(t

1

)) = I

O→S

2

(~y

O→S

2

(t

1

))

= I

O→S

1

(~y

O→S

1

(t

2

))

= I

O→S

2

(~y

O→S

2

(t

2

)). (56)

Thus, using (56) and s trict convexity of I

O→S

,

this yields for a coin ﬂip with 0 < λ < 1

I

O→S

12

(~y

O→S

12

(t

1

)) < I

O→S

1

(~y

O→S

1

(t

2

)) = I

O→S

2

(~y

O→S

2

(t

2

)). (57)

On the other hand, (55) implies that, at time t

2

,

O would ﬁnd

~y

O→S

12

(t

2

) = ~y

O→S

1

(t

2

) = ~y

O→S

2

(t

2

)

43

As an aside, note that A(∆t) can thus b e replaced by

A(∆t) =



a(∆t) b(∆t)

b(∆t) a(∆t)



,

so th at time evolution is symmetric under a swap of the

‘yes/no’-labeling for all Q ∈ Q

M

N

at once, i.e. under

~y

O→S

↔ ~n

O→S

.

such that

I

O→S

12

(~y

O→S

12

(t

2

)) = I

O→S

1

(~y

O→S

1

(t

2

))

= I

O→S

2

(~y

O→S

2

(t

2

)). (58)

Hence, (

57, 58) entail that O’s information about

S

12

has increased between t

1

and t

2

, despite not

having tossed the coin and asked any qu estions.

This is in contradiction with rule

3. We conclude

that T (∆t) must be injective.

Since Σ

N

contains a basis of R

D

N

(see subsec-

tion

6.3) it is clear that T (∆t) is also injective

on R

D

N

. For ﬁnite dimensional square matrices,

T (∆t) being injective is equivalent to it being

Accepted in Quantum 2017-11-27, click title to verify 63

bijective on R

D

N

. H ence, to every T (∆t) there

must exist an inverse matrix T

−1

(∆t) such that

T (∆t)T

−1

(∆t) = T

−1

(∆t)T (∆t) = 1.

However, is this T

−1

(∆t) also a legal time evo-

lution?

44

This would be the case if T (∆t) was

also surjective and thereby reversible on Σ

N

⊂

R

D

N

. Indeed, we shall demonstrate this property

next.

6.6 Time evolution is also reversible

To establish reversibility of time evolution we

shall resort to tools from topology and convex

sets.

Lemma 6.4. Restricting any time evolution

T (∆t) : Σ

N

→ Σ

N

to an interior level set L

ε

(

48) deﬁnes a continuous, open and closed map

from L

ε

to itself.

Proof. As an injective square matrix T (∆t) de-

ﬁnes a continuous invertible map from R

D

N

to it-

self and so the pre-image of any open s et in R

D

N

is another open set. Rule

3 implies [T (∆t)](L

ε

) ⊆

L

ε

and that no state from outside L

ε

can evolve

into L

ε

. The open sets of L

ε

in the induced topol-

ogy are the intersections of L

ε

with open sets of

R

D

N

. Consider any such open set V ⊂ L

ε

and

any open set U ⊂ R

D

N

such that V = U ∩ L

ε

.

The pre-image [T

−1

(∆t)](U) is again open as is

therefore [T

−1

(∆t)](U)∩L

ε

which, thanks to rule

3, must be non-empty if V is non-empty. Hence,

T (∆t) deﬁ nes a continuous map from L

ε

to it-

self. For similar reasons, this map is also open.

Finally, the closed map lemma states th at a con-

tinuous map from a compact to a Hausdorﬀ space

is also closed. Thanks to lemma

6.2, we have

L

ε

≃ S

D

N

−1

which is both compact and Haus-

dorﬀ and so the map from L

ε

to itself deﬁned by

T (∆t) is also closed.

Next, we employ this result to show that time

evolution is surjective on interior level sets.

Lemma 6.5. Any allowed time evolution

is surjective on the interior level sets, i.e.

[T (∆t)](L

ε

) = L

ε

and thus also [T (∆t)](L

I≤ε

) =

L

I≤ε

.

Proof. Lemma

6.4 entails that T (∆t) deﬁnes a

continuous, open and closed map from L

ε

to it-

self. It therefore maps all open sets to open sets

and all closed sets to closed sets in L

ε

. In par-

ticular, it must map all sets that are both open

and closed in L

ε

to other sets which are both

open and closed. Since L

ε

≃ S

D

N

−1

is con-

nected, the empty set and the full L

ε

are the

only sets in L

ε

which are both closed and open.

Now, by lemma

6.3, T (∆t) is inj ective so that

[T (∆t)](L

ε

) = L

ε

.

The last result is suﬃcient to ﬁnally establish

that time evolution is surjective also on the state

space.

Theorem 6.6. Any allowed time evolution is

surjective on Σ

N

, i.e. [T (∆t)](Σ

N

) = Σ

N

, and

has | det T (∆t)| = 1.

Proof. Given that an interior fat level set L

I≤ε

is, by lemma

6.2, homeomorphic to the closed

unit ball in R

D

N

it is a convex body and has

volum e [

89]. Lemma 6.5 yields [T (∆t)](L

I≤ε

) =

L

I≤ε

and so its volume is left invariant by

T (∆t) which, according to (53), acts linearly on

states ~r

O→S

in the Bloch vector parametriza-

tion. Theorem 6.2.14 in [89] then implies that

| det T (∆t)| = 1 and therefore also that th e vol-

ume of [T (∆t)](Σ

N

) is equal to the volume of Σ

N

.

Consider now the volume diﬀerence

δ



[T (∆t)](Σ

N

), Σ

N



:= vol



[T(∆t)](Σ

N

) ∪ Σ

N



− vol



[T(∆t)](Σ

N

) ∩ Σ

N



.

Clearly, [T (∆t)](Σ

N

) ⊆ Σ

N

, and

so δ



[T (∆t)](Σ

N

), Σ

N



= vol (Σ

N

) −

vol



[T(∆t)](Σ

N

)



= 0. Exercise 6.2.2 in [89]

then sh ows that δ



[T (∆t)](Σ

N

), Σ

N



= 0 is only

possible if [T (∆t)](Σ

N

) = Σ

N

.

Hence, T

−1

(∆t) maps all legal states to legal

states and must, by rule

4, be legal also.

Corollary 6.7. Any time evolution permitted by

the rules is reversible.

44

I thank a referee for pointing out that addressing this

question was overlooked in a previous draft version.

Accepted in Quantum 2017-11-27, click title to verify 64

6.7 Time evolution deﬁnes a group

Given that any time interval can be decomposed into two time intervals, ∆t = ∆t

1

+ ∆t

2

, and the

evolution of ~r

O→S

= 2 ~y

O→S

−

~

1 is continuous by rule

4, O must ﬁnd

T (∆t

2

) T (∆t

1

) ~r

O→S

(0) = T (∆t

2

) ~r

O→S

(∆t

1

) = ~r

O→S

(∆t) = T (∆t) ~r

O→S

(0),

and thus that multiplication is abelian,

T (∆t

1

+ ∆ t

2

) = T (∆t

2

) T (∆t

1

) = T (∆t

1

) T (∆t

2

).

(The last equality follows from time translation

invariance.) From the last equation and T (0) =

1, using ∆t − ∆t = 0, we can also infer that

T

−1

(∆t) = T (−∆t). If we permit O to consider

the time evolution of S for any duration ∆t ∈ R,

it f ollows that the product of any two time evolu-

tion matrices is again a time evolution matrix. In

summary, we therefore gather that a given time

evolution, as perceived by O and under the cir-

cumstances to which he has subjected S, is such

that:

(i) T (0) = 1,

(ii) to every T (∆t) there exists an inverse

T

−1

(∆t) = T (−∆t),

(iii) the multiplication of any two time evolution

matrices is again a time evolution matrix,

and

(iv) matrix multiplication is obviously associa-

tive.

In conclusion, under the assumptions of section

3.2 and rules 3 and 4, a given time evolution de-

ﬁnes therefore an abelian, one-parameter matrix

group. Hence, a given time evolution is described

by a single evolution generator and single param-

eter ∆t parametrizing the duration. We note,

however, that a multiplicity of time evolutions of

S is possible, depending on the physical circum-

stances (interactions) to which O may subject S.

Diﬀerent time evolutions will be generated by dif-

ferent generators, but each time evolution will

form a one-parameter group as discussed above.

In fact, the full set of time evolutions T

N

which

O is able to implement must likewise be a group.

Namely, if T (∆t), T

′

(∆t

′

) correspond to two dis-

tinct interactions, then rule

4 implies that also

T (∆t) · T

′

(∆t

′

) is a legal time evolution for any

state and since both T, T

′

are invertible their full

set must be a group. This implies, in particular,

that T

N

is a group. We shall return to this further

below and in [

1].

6.8 The squared length of the Bloch vector as

information measure

We now have suﬃcient structure in our hand

to determine the functional relation between α

i

and y

i

. Given that the generalized Bloch vector

2 ~y

O→S

−

~

1 transforms nicely under time evolution

(

53), it is useful to parametrize α

i

by 2y

i

− 1, i.e.

α

i

= α(2y

i

− 1). Rule

3 entails that O’s total in-

formation about (an otherwise non-interacting) S

is a ‘conserved charge’ of time evolution

I

O→S

(~y

O→S

(∆t)) = I

O→S

(~y

O→S

(0))

which translates into the condition

I

O→S



T (∆t)



2 ~y

O→S

(0) −

~

1



=

D

N

X

i=1

α





D

N

X

j= 1

T

ij

(∆t) (2y

j

(0) − 1)





=

D

N

X

i=1

α (2y

i

(0) − 1) = I

O→S

(2 ~y

O→S

(0) −

~

1). (59)

If T (∆t) was a permutation matrix, (59) would

hold for any function α(2y

i

− 1). For exam-

ple, N class ical bits are governed by the evolu-

tion group Z

2

× · · · × Z

2

. However, permutations

Accepted in Quantum 2017-11-27, click title to verify 65

form a discrete group, while in our present case

{T (∆t), ∆t ∈ R} constitutes a continuous one-

parameter group. This is where continuity of time

evolution, as asserted by rule

4, becomes crucial.

Under a reasonable assumption on the informa-

tion measure, we shall now show that continuity

of time evolution, together with conditions (i)–

(vi) of subsection

6.2, enforces the quadratic re-

lation α

i

= (2y

i

− 1)

2

. To this end, we once more

invoke the coin ﬂip scenario.

45

Given our parametrization in terms of the

Bloch vector 2 ~y

O→S

−

~

1, O’s information about

the outcomes of his questions in the coin ﬂip sce-

nario can be written as follows

I

O→S

12



2 (λ ~y

O→S

1

+ (1 − λ) ~y

O→S

2

) −

~

1



= I

O→S

12



λ (2 ~y

O→S

1

−

~

1) + (1 − λ)(2 ~y

O→S

2

−

~

1)



=

D

N

X

i=1

α



λ (2y

1

i

− 1) + (1 − λ)(2y

2

i

− 1)



.

It is instructive to consider the case in which O

is entirely oblivious about S

2

such that the lat-

ter is in the state of no information ~y

O→S

2

=

1

2

~

1

relative to him, but that O has some informa-

tion about S

1

. In this case, 2 ~y

O→S

12

−

~

1 =

λ(2 ~y

O→S

1

−

~

1) and (assuming the outcome of the

coin ﬂip is not certain) strict convexity of I

O→S

(see subsection

6.2) implies

I

O→S

12

(λ (2 ~y

O→S

1

−

~

1)) < λ I

O→S

1

(2 ~y

O→S

1

−

~

1)

or, equivalently,

I

O→S

12

(λ (2 ~y

O→S

1

−

~

1)) (60)

= f · I

O→S

1

(2 ~y

O→S

1

−

~

1),

where f < 1 is a factor parametrizing O’s in-

formation loss relative to the case in which he

does not toss a coin and, instead, directly asks

S

1

.

46

The reason O experiences such a relative

information loss about the outcome of his inter-

rogation is, of course, entirely due to the random-

ness of the coin ﬂip. But the coin ﬂip is indepen-

dent of the s ystems S

1,2

and, in particular, of the

states in which these are relative to O; the fac-

tor λ by which the probabilities ~y

O→S

1

become

45

We suspect that this result may be derivable from

purely group theoretic arguments without an operational

setup by employing the mathematical fact that to every

continuous matrix group acting linearly on some space

there corresponds a conserved inner produ ct which is

quadratic in the components of the vectors.

46

In fact, one can equivalently interpret the situation as

follows: O only considers a system S

1

which would be in

the state ~y

O→S

1

if it was present with certainty. However,

λ in the state λ(2 ~y

O→S

1

−

~

1) represents the probability

that S

1

‘is there’ at all. Indeed, in this case, (

4) can be

written as λ(~y

O→S

1

+~n

O→S

1

) = λ·

~

1 such that λ(~y

O→S

1

−

~n

O→S

1

) = λ(2 ~y

O→S

1

−

~

1).

rescaled is state independent. For that reason,

the relative information loss should likewise de-

pend only on the coin ﬂip, quantiﬁed by λ, and

not on the state ~y

O→S

1

. For instance, if we also

considered the case that the coin ﬂip was certain,

i.e. λ = 0, 1, then clearly for λ = 1 we must have

f = 1 ∀ ~y

O→S

1

∈ Σ

N

and for λ = 0 it must hold

f = 0 ∀ ~y

O→S

1

∈ Σ

N

. We shall make this into

a requirement on the information measure f or all

values of λ:

Requirement 1. The relative information loss

factor f in (

61) is a state independent (contin-

uous) function of the coin ﬂip probability λ with

f(λ) < 1 for λ ∈ (0, 1).

The binary Shannon entropy H(y) =

−y log y − (1 − y) log(1 − y) in the role of α would

fail this requirement. Namely, the measure α thus

factorizes, α(λ (2y

i

− 1)) = f (λ) α(2y

i

− 1), while

H(y) would not. Setting λ = λ

1

· λ

2

yields

47

f(λ

1

· λ

2

) α(2 y

i

− 1) = α(λ

1

· λ

2

(2 y

i

− 1))

= f (λ

1

) α(λ

2

(2 y

i

− 1))

= f (λ

1

)f(λ

2

) α(2 y

i

− 1)

and therefore f(λ

1

) · f(λ

2

) = f (λ

1

· λ

2

), which

implies f(λ) = λ

p

, for some power p ∈ R. But

then, α must be a homogeneous function α(2 y

i

−

47

Such a factorization of coin ﬂip probabilities could be

achieved, e.g., if O decided to use one coin, with ‘heads’

probability λ

1

, to ﬁrstly decide which of two possible con-

vex mixtures to prepare where both possible mix tures are

generated with a second coin with ‘heads’ probability λ

2

.

If t hree of the four states within the two mixtures are

chosen as th e state of no information, one would obtain

precisely such an equation.

Accepted in Quantum 2017-11-27, click title to verify 66

1) = k (2 y

i

− 1)

p

with some constant k ∈ R. In

consequence, the information I

O→S

(2 ~y

O→S

−

~

1) is

(up to k) the p-norm of the Blo ch vector 2 ~y

O→S

−

~

1.

We can rule out that p ∈ (−∞, 0] because in

this case, as one can easily check, it is impossi-

ble to s atisfy all the consistency conditions (i)–

(vi) of subsection

6.2. Hence, p > 0. At this

stage we can make us e of (

59) and a result by

Aaronson [

90] which implies that the only vector

p-norm with p > 0 which is preserved by a contin-

uous matrix group is the 2-norm. Since any given

time evolution of the Bloch vector 2 ~y

O→S

−

~

1 is

governed by a continuous, one-parameter matrix

group, we conclude that α(2 y

i

−1) = k (2 y

i

−1)

2

.

Imposing condition (iv) yields k = 1 and there-

fore ultimately

I

O→S

(~y

O→S

) =

D

N

X

i=1

(2 y

i

− 1)

2

. (61)

It is straightforward to convince oneself that all

of (i)–(vi) of subsection

6.2 are satisﬁed by this

quadratic information measure. O’s total amount

of information about S is thus the sq uared length

of the generalized Bloch vector, thereby assuming

a geometric ﬂavour. It is important to emphasize

that, had we not imposed continuity of time evo-

lution in rule

4, we would not have been able to

arrive at (

61); if time evolution was not continu-

ous, many solutions to α in terms of y

i

would be

possible.

The quadratic information measure (

61) has

been proposed earlier by Brukner and Zeilinger

in [32,34,91,92] from a diﬀerent perspective, em-

phasizing that this is the most natural measure

taking into account an observer’s uncertainty –

due to statistical ﬂuctuations – about the out-

come of the next trial of measurements on a

system in a multiple shot experiment. Further-

more, taking the formalism of quantum theory

as given, Brukner and Zeilinger [

73] later singled

out the quadratic measure from the set of Tsal-

lis entropies by imp osing an ‘information invari-

ance principle’, according to which a continuous

transformation among any two complete sets of

mutually complementary measurements in quan-

tum theory should leave an observer’s informa-

tion about the system invariant. While [

73] is

certainly compatible with the present framework,

here we come from farther away to the same re-

sult: we do not pre-suppose quantum theory and

derive the quadratic measure more generally by

starting from the landscape of inf ormation infer-

ence theories and imposing rule

3 of information

preservation and rule

4 of maximality of time evo-

lution thereon. In this regard, the present deriva-

tion may similarly be taken as a strong justiﬁca-

tion for the original Brukner-Zeilinger proposal.

6.9 The set of all time evolutions is a subgroup

of SO(D

N

)

Thus far, we only gathered that a given time evo-

lution is described by a one-parameter group and,

by theorem

6.6, must have | det T (∆t)| = 1. Now,

given (

61), we are in the position to say q uite a bit

more. The full matrix group leaving (

61) invari-

ant is O(D

N

). However, a given time evolution

is a continuous one-parameter group and must

therefore be connected to the identity. Hence,

T (∆t) ∈ SO(D

N

), ∀ T (∆t). Consequently, the

group corresponding to any ﬁxed time evolution

is a one-parameter subgroup of SO(D

N

). We

also noted before that the set of all possible time

evolutions T

N

is a group thanks to rule

4. It

must therefore likewise satisfy T

N

⊂ SO(D

N

).

This topic is thoroughly discussed in the com-

panion articles, where it is shown that the rules

imply T

N

= PSU(2

N

) for the D

1

= 3 [

1] and

T

N

= PSO(2

N

) for the D

1

= 2 case [

2]. Note

that PS U(2

N

) is a prop er subgroup of SO(D

N

=

4

N

− 1) for N > 1. The generators of T

N

are the

set of possible time evolution generators of S’s

states.

6.10 Pure and mixed states

The explicit quantiﬁcation of O’s information

now permits us to render the distinction between

three informational classes of S’s states – which

we already loosely referred to as ‘states of max-

imal knowledge’ or ‘states of non-maximal infor-

mation’ in previous sections – precise.

Firstly, we determine the maximally attain-

able (independent and dependent) information

content within a state ~y

O→S

of a sy stem of N

gbits. This can be easily counted: once O knows

the answers to N mutually independent questions

(these do not need to be individuals), he will also

know the answers to all their bipartite, tripar-

tite,... and N-partite correlation questions – all

of which are contained in Q

M

N

too by theorems

Accepted in Quantum 2017-11-27, click title to verify 67

5.27 and 5.31. But these are then

N

1

!

+

N

2

!

+ · · ·

N

!

=

N

X

i=1

N

i

!

= 2

N

− 1

answered questions from Q

M

N

, while all remain-

ing q uestions in Q

M

N

will be maximally comple-

mentary to at least one of the known ones. O’s to-

tal information, as quantiﬁed by I

O→S

(

47), thus

contains plenty of dependent bits of information

– a result of the fact that the questions in Q

M

N

are pairwise but not necessarily mutually inde-

pendent.

Using this observation, we shall characterize

S’s states according to their information content,

i.e. squared length of the Bloch vector. By rules

3

and 4, this distinction applies to all states which

are connected via some time evolution to the

states above, including those for which the infor-

mation may be distributed partially over many

elements of a ﬁxed Q

M

N

. Speciﬁcally, we shall

refer to a state ~y

O→S

as a

pure sta te: if it is a state of maximal information content, i.e. maximal length

I

O→S

(~y

O→S

) =

D

N

X

i=1

(2 y

i

− 1)

2

= (2

N

− 1) bits,

mixed state: if it is a state of non-extremal information content, i.e. non-extremal length

0 bit < I

O→S

(~y

O→S

) =

D

N

X

i=1

(2 y

i

− 1)

2

< (2

N

− 1) bits,

totally mixed state: if it is the state of no information ~y

O→S

=

1

2

~

1 with zero length

I

O→S



~y

O→S

=

1

2

~

1



= 0 bit.

We note that these characterizations of states in

terms of their length are indeed true in quantum

theory. In particular, N qubit pure states actu-

ally have a Bloch vector sq uared length equal to

2

N

− 1. Our reconstruction gives this peculiar

fact a clear informational interpretation.

One may wonder whether the above deﬁnition

of a pure state is directly equivalent to being ex-

tremal in Σ

N

and thereby to the usual deﬁni-

tion of pure states in GPTs or quantum theory.

This is not obvious from what we have estab-

lished thus far. However, since clearly we as-

sume I

O→S

= (2

N

− 1) bits to be the maxi-

mally attainable information, we can already con-

clude that pure states so deﬁned must lie on the

boundary of Σ

N

because a non-constant convex

function cannot have its maximum in the inte-

rior of its convex domain [

89]. Showing that the

pure states, as deﬁned above, of the theory sur-

viving the impos ition of all rules – quantum the-

ory – are, indeed, the extremal states, requires

more work. For N = 1 we shall demonstrate this

shortly, while the discussion for N > 1 is deferred

to [

1].

7 The N = 1 case and the Bloch ball

Before closing this toolkit for now, we quickly give

a ﬂavor of the capabilities of the newly developed

concepts and tools by applying them to show that

in the simplest case of a single gbit (N = 1) rules

1–4 indeed only have two solutions within L

gbit

,

namely the qubit and the rebit state space includ-

ing their respective time evolution groups. This

proves the claim of section

4.1 for N = 1.

To this end, recall theorem

5.6 which asserts

that the dimension of the N = 1 state space Σ

1

is either D

1

= 2 or D

1

= 3 which thus far we

suggestively referred to as the ‘rebit’ and ‘qubit

case’, respectively.

Accepted in Quantum 2017-11-27, click title to verify 68

7.1 A single qubit and the Bloch ball

We begin with the D

1

= 3 case. Σ

1

will be parametrized by a three-dimensional vector

~y

O→S

=







y

1

y

2

y

2







,

where y

1

, y

2

, y

3

are the ‘yes’-probabilities of a mutually maximally complementary question set

Q

1

, Q

2

, Q

3

constituting an informationally complete Q

1

(

11).

The informational distinction of states introduced in section

6.10 reads in this case

pure sta tes: ~y

O→S

such that

I

O→S

= (2 y

1

− 1)

2

+ (2 y

2

− 1)

2

+ (2 y

3

− 1)

2

= 1 bit,

mixed states: ~y

O→S

such that

0 bit < I

O→S

= (2 y

1

− 1)

2

+ (2 y

2

− 1)

2

+ (2 y

3

− 1)

2

< 1 bit,

totally mixed state: ~y

O→S

=

1

2

~

1 such that

I

O→S

= (2 y

1

− 1)

2

+ (2 y

2

− 1)

2

+ (2 y

3

− 1)

2

= 0 bit.

Recall from section 6.9 that the set of possible

time evolutions T

1

must be contained in SO(3)

and that the time evolution rule 4 requires the

set of states into which any state ~y

O→S

can evolve

to be maximal, while being compatible with the

rules. Thus, in particular, the image of the cer-

tainly legal pure state (1, 0, 0) (rules

1 and 2 im-

ply its existence in Σ

1

) under T

1

must be maxi-

mal. Applying all of SO(3) to this state generates

all states with |2~y −

~

1|

2

= 1 bit and these cer-

tainly abide by the rules. Consequently, the set

of all possible time evolutions, compatible with

rules

1–4, is the rotation group

T

1

≃ SO(3) ≃ PSU(2).

This is the component of the isometry group of

the Bloch ball which is connected to the identity

and it is required in full in order to maximize the

number of states into which (1, 0, 0) can evolve.

However, since time evolution is state indep en-

dent (c.f. assumption

8), this is the time evolu-

tion group f or all states. PSU(2) is precisely the

adjoint action of SU(2) on density matrices ρ

2×2

over C

2

, ρ

2×2

7→ U ρ

2×2

U

†

, U ∈ SU(2) and thus

coincides with the set of all possible unitary time

evolutions of a single qubit in standard quantum

theory.

The set of allowed states populates the entire

unit ball in the three-dimensional Bloch ball, i.e.

Σ

1

≃ B

3

. This follows from the fact that apply-

ing the full group T

1

= SO(3) to (1, 0, 0) gener-

ates the entire Bloch sphere as a closed set of ex-

tremal states and the fact that Σ

1

must be closed

convex according to assumption

2. Hence, we

recover the well-known three-dimensional Bloch

ball state space of a single qubit of standard quan-

tum theory with the set of all pure states deﬁning

the boundary sphere S

2

, the totally mixed state

(as the state of no information) constituting the

center and the set of mixed states ﬁlling the inte-

rior in between, as illustrated in ﬁgure

6a. This

is precisely the geometry of the set of all normal-

ized density matrices on C

2

. Notice that the pure

state space S

2

≃ CP

1

indeed coincides with the

set of all unit vectors in C

2

(modulo phase).

Given the complete symmetry of the Blo ch ball

as the state space for N = 1, there should not ex-

ist a distinguished informationally complete ques-

tion set Q

M

1

, corresponding to a distinguished or-

thonormal Bloch vector basis, by means of which

O can interrogate S. While it is an additional

assumption, it is thus natural to stipulate that

Accepted in Quantum 2017-11-27, click title to verify 69

PSfrag rep lacements

totally mixed state

mixed states

pure states

~r

~r = 2 ~y

O→S

−

~

1

(a) 3D qubit Bloch ball

PSfrag rep lacements

totally mixed state

mixed states

pure states

~r

~r = 2 ~y

O→S

−

~

1

(b) 2D rebit Bloch disc

Figure 6: The three-dimensional Blo ch ball (a) and the

two- dimensional Bloch disc (b) are the correct state

spaces Σ

1

of a single qubit in standard quantum theory

and a single rebit in real quantum theory, respectively.

The vector ~r parametrizing the state s is the Bloch vector

2 ~y

O→S

−

~

1.

to e very Q ∈ Q

1

there exists a unique pure state

in Σ

1

which represents the truth value Q = ‘yes’

and, conversely, that every pure state of this sys-

tem corresponds to the deﬁnite answer to one

question in Q

1

. But then Q

1

≃ S

2

which also

coincides with the set of all possible projective

measurements onto the +1 (or, equivalently, the

−1) eigenspaces of the Pauli operators ~n · ~σ over

C

2

which is parametrized by ~n ∈ R

3

, |~n|

2

= 1,

and where ~σ = (σ

x

, σ

y

, σ

z

) are the usual Pauli

matrices. This set of permissible questions Q

N

for all N will be discussed more thoroughly in [

1]

together with a derivation of the Born rule for

projective measurements.

The ‘ballness’ and three-dimensionality of the

state space of a single qubit can also b e derived

from various operational axioms within G PTs

[

15, 16, 18, 21, 23, 59, 93] and constitutes a cru-

cial step in most GPT based reconstructions of

quantum theory [14–16, 20, 21]. The principle of

continuous reversibility, according to which every

pure state of the convex set can be mapped into

any other by means of a continuous and reversible

transformation, usually assumes a crucial role in

such derivations. Here we oﬀer a novel perspec-

tive on the origin of the Bloch ball by deriving it

from elementary rules for the informational rela-

tion between O and S; in particular, we recover

continuous reversibility as a by-product.

7.2 A single rebit and the Bloch disc

The analogous result holds for the D

1

= 2 case:

Σ

1

will be parametrized by a two-dimensional

vector

~y

O→S

=

y

1

y

2

!

,

where y

1

, y

2

are the ‘yes’-probabilities of two

maximally complementary questions Q

1

, Q

2

con-

stituting an informationally complete Q

1

(

11).

We then have

pure sta tes: ~y

O→S

such that

I

O→S

= (2 y

1

− 1)

2

+ (2 y

2

− 1)

2

= 1 bit,

mixed states: ~y

O→S

such that

0 bit < I

O→S

= (2y

1

− 1)

2

+ (2y

2

− 1)

2

< 1 bit,

and the

totally mixed state: ~y

O→S

=

1

2

~

1 such that

I

O→S

= (2 y

1

− 1)

2

+ (2 y

2

− 1)

2

= 0 bit.

In analogy to the qubit case, the time evolution

rule

4 implies that

(a) the set of all possible time evolutions is the

(projective) rotation group

T

1

≃ SO(2)

#

≃ PSO(2) ≃ SO(2)/Z

2

because SO(2) is the full (connected com-

ponent of the orientation preserving) isom-

etry group of the Bloch disc and all time

Accepted in Quantum 2017-11-27, click title to verify 70

evolutions are permitted which preserve the

Bloch vector length, representing O’s total

information about S, and the orientation of

the Bloch vector basis. Orientation preser-

vation means that O’s convention about the

‘yes’-‘no’-labelling of the question outcomes

is preserved: The isomorphism denoted with

a # requires some explaining because SO(2)

is actually a double cover of PSO(2) as indi-

cated on the right. Indeed, the real density

matrix of a single rebit, ρ

2×2

=

1

2

(1 +r

x

σ

x

+

r

z

σ

z

) on R

2

where r

i

= (2y

i

− 1), evolves

under the adjoint action of SO(2), ρ

2×2

7→

O ρ

2×2

O

T

for O = e

iσ

y

t

∈ SO(2). This is

equivalent to an action of PS O(2) because

the non-trivial center element −1 ∈ SO(2)

acts trivially on ρ

2×2

and thus factors out.

However, as a subcase of the single qubit

case this adjoint action of SO(2) on ρ

2×2

is

also equivalent to O

′

~r for O

′

∈ SO(2) in the

Bloch vector representation.

(b) the s tate space Σ

1

coincides with the two-

dimensional Bloch disc, as depicted in ﬁgure

6b, with the totally mixed state in the center,

the pure states on the boundary circle S

1

and

the mixed states in the interior in between.

This is precisely the geometry of the set of

unit trace, positive-semideﬁnite, symmetric

matrices over R

2

– the space of density ma-

trices of a single rebit. Similarly, the pure

state space S

1

≃ RP

1

coincides with the set

of all unit vectors in the Hilbert space R

2

modulo Z

2

.

Again, given the s ymmetry of the Bloch disc,

there is no reason for a distinguished informa-

tionally complete question basis Q

M

1

on Q

1

, in-

carnated as a distinguished Bloch vector basis,

to exist. Accordingly, we assume that any pure

state on S

1

corresponds to the deﬁnite answer

of some question in Q

1

which O may ask the

rebit. In this case, Q

1

≃ S

1

, which coincides

with the s et of projective measurements onto the

+1 eigenspaces of the Pauli operators ~n · ~σ over

R

2

which are parametrized by ~n ∈ R

2

, |~n|

2

= 1

and where ~σ = (σ

x

, σ

z

) are the two real and sym-

metric Pauli matrices.

8 Discussion and outlook

Based on the premise to only speak about in-

formation which an observer (or more generally

system) has access to by interaction with other

systems, we have developed a novel framework

for characterizing and (re-)constructing the quan-

tum theory of qubit sys tems. We have laid down,

without ontological statements, the mathemati-

cal and conceptual foundations for a landscape

of theories describing how an observer acquires

information about physical systems through in-

terrogation with yes-no-questions.

Within this landscape four elementary rules

have been given which constrain the observer’s

acquisition of information f or qubit (and rebit)

systems. The rule of limited information and the

complementarity rule imply an independence and

compatibility structure of the binary questions

which, in fact, reproduces that of projective mea-

surements onto the +1 eigenspaces of Pauli op-

erators in quantum theory. In particular, these

rules entail in a constructive and simple way

1. a novel argument for the dimensionality of

the Bloch ball,

2. a new method for determining the

correct number of independent ques-

tions/measurements necessary to describe a

system of N qubits (or rebits),

3. a natural explanation for entanglement,

monogamy of entanglement and quantum

non-locality,

4. the explicit correlation s tructure of two

qubits and rebits, and

5. more generally the correlation structure for

arbitrarily many qubits and rebits.

Furthermore, the rules of information preserva-

tion and maximality of time evolution are shown

to result

6. in a reversible time evolution, and

7. under elementary consistency conditions, in

a quadratic information measure, quantify-

ing the observer’s prior information about

the answers to the various q uestions he may

ask the system.

This measure has been earlier proposed by

Brukner and Zeilinger from a diﬀerent perspec-

tive [

32,34,73,91,92], complementing our present

derivation.

Combining these results, we then show, as the

simplest example, how the rules entail that

Accepted in Quantum 2017-11-27, click title to verify 71

8. the Bloch ball and Bloch disc are recovered

as state spaces for a single qubit and rebit,

respectively, together with the correct time

evolution groups and question sets.

The full reconstruction of qubit quantum the-

ory, following from the present four rules (and two

additional ones), is completed in the companion

paper [

1] (the rebit case which violates one of the

additional rules is considered separately [

2]). In

conjunction, this derivation highlights the partial

interpretation of quantum theory as a law book,

describing and governing an observer’s acquisi-

tion of information about physical systems. In

particular, it highlights the quantum state as the

observer’s ‘catalogue of knowledge’.

Certainly, there are some limitations to the

present approach: First of all, we employ a clear

distinction between observer and system which

cannot be considered as fundamental. Secondly,

the construction is speciﬁcally engineered for the

quantum theory of qubit systems. While arbi-

trary ﬁnite dimensional quantum systems could,

in principle, be immediately encompassed by im-

posing the so-called subspace axiom of GPTs

[

14, 16], the latter does not naturally ﬁt into our

set of principles, mostly because rules

1 and 2

quantify the information content of a system in

terms of N ∈ N bits which is only suitable for

comp os ite gbit systems. More generally, the lim-

itation is that the current approach only encom-

passes ﬁnite dimensional quantum theory, but not

quantum mechanics. As it stands, the mechani-

cal phase space language does not naturally ﬁt

into the present framework and more sophisti-

cated tools are required in order to cover mechan-

ical systems, let alone anything beyond that.

While this informational construction recovers

the state spaces, the set of possible time evolu-

tions and projective measurements of qubit quan-

tum theory, in other words, the architecture of

the theory, it clearly does not tell us much about

the concrete physics (other than demanding it to

ﬁt into this general construction). This purely

informational framework is rather universal and

information carrier independent. But qubits as

information carriers can be physically incarnated

in many diﬀerent ways: as electron or muon spins,

photon polarization, quantum dots, etc. The

framework cannot distinguish among the diﬀer-

ent physical incarnations, underlining the obser-

vation that not everything can be reduced to in-

formation and additional inputs are necessary in

order to do proper physics.

Nevertheless, despite its current limitations,

this informational approach teaches us something

non-trivial about the structure and physical con-

tent of q uantum theory.

While this work is more generally motivated by

the eﬀort to understand physics from an infor-

mational and operational perspective and high-

lights a partial interpretation of quantum the-

ory, it clearly does not single out ‘the right one’

(see also the discussion in [

94]). For example, by

speaking exclusively about the information acces-

sible to an observer, we are by default silent on

the f ate of hidden variables and on whether they

could give rise to the as sumptions and rules which

we impose. The status of hidden variables is sim-

ply not relevant here (other than that local hidden

variables are ruled out).

Let us, nonetheless, make a few possible

(but not inescapable) interpretational state-

ments. Given the absence of ontological commit-

ments, this informational approach is generally

compatible with (but does not rely on) the re-

lational [

28, 29], Brukner-Zeilinger informational

[

31–35], or QBist [36–39] interpretations of quan-

tum mechanics. They depart f rom the traditional

idea that systems necessarily have, absolute, i.e.

observer independent properties (or, more gen-

erally, properties independent of their relations

with other systems). Instead, many physical

properties are interpreted as relational; the inter-

action between systems establishes a relation be-

tween them, permitting an information ex change

which reveals certain physical properties relative

to one another and in the absence of hidden vari-

ables this would be all there is. Certainly, in or-

der not to render such a view hopelessly solip-

sistic, systems ought to have certain intrinsic at-

tributes, e.g. a corresponding state space or set

of permissible interactions/measurements, such

that diﬀerent observers have a basis f or agree-

ing or disagreeing on the description of physical

objects. However, a state of a system or measure-

ment/question outcome is taken relative to who-

ever performs the measurement or asks the q ues-

tion. Diﬀerent observers may agree on states or

measurement outcomes by communication (i.e.,

physical interaction) but if one rejects the idea of

an absolute and omniscient observer it is natural

to also abandon the idea of an absolute and ex-

Accepted in Quantum 2017-11-27, click title to verify 72

ternal standard by means of which properties of

systems could be deﬁned.

From such a perspective, one could say that

the relational character of the qubits’ proper-

ties is a consequence of a universal limit on the

amount of information accessible to the observer

and the mere existence of complementary infor-

mation such that the observer can not know the

answers to all his questions simultaneously. It is

the observer who determines which questions he

will ask and thereby what k ind of system prop-

erty the interrogation will reveal and thus, ulti-

mately, which kind of information he will acquire

(although clearly he does not determine what the

question outcome is). But relative to the observer

the system of qubits does not have properties

other than those accessible to him.

8.1 An operational alternative to the wave

function of the universe

Pushing an informational interpretation of quan-

tum theory to the extreme, one may speculate

whether the quantum state could also represent

a state of information in a gravitational or cosmo-

logical context. For instance, is such an interpre-

tation adequate f or the ‘wave function of the uni-

verse’ which is ubiquitous in standard approaches

to quantum cosmology (e.g., see [95–97])? Such

an interpretation would require the existence of

an absolute and omniscient observer, an idea

which we just abandoned.

Alternatively, one could adopt one of the cen-

tral ideas of relational quantum mechanics [

28,

29], according to which all physical systems can

assume the role of an ‘observer’, recording infor-

mation about other systems, thereby relieving the

clear distinction used in this manuscript. Ex-

tending this idea to a space-time context, one

could interpret the universe as an abstract net-

work of subsystems/subregions, viewed as infor-

mation registers, which can communicate and ex-

change information through interaction (see ﬁg-

ure

7). In this background independent context,

any information acquisition by any register is in-

ternal, i.e. occurs within the network; a global

observer outside the network becomes meaning-

less. This may appear as a purely philosophical

observation, but it implies concrete consequences

for the description of the network: there should

be no global state (aka ‘wave function of the uni-

verse’) for the entire network at once. Indeed,

PSfrag rep lacements

absolute observer

S

1

S

2

S

3

S

4

S

5

S

6

S

7

communication

Figure 7: T he universe as an information exchange net-

work of subsytems without a b solute observer.

admitting any register in the network to act as

‘observer’, the self-reference problem [

98,99] im-

pedes a given register to infer the global state of

the entire network – including itself – from its in-

teractions with the rest. Accordingly, relative to

any subsystem, one could assign a state to the

rest of the network, but a global state and thus

a global Hilbert space would not arise. The ab-

sence of a global state in quantum gravity has

been proposed before [100–103] – albeit from a

diﬀerent, less informational perspective.

This oﬀers an operational alternative to the

problematic concept of the ‘wave function of the

universe’ which is ubiquitous in quantum cosmol-

ogy. However, while a ‘wave function of the uni-

verse’ might not make sense as a fundamental

concept, it could still be given meaning as an ef-

fective notion, namely as the state of the large

scale structure of our universe on whose descrip-

tion ‘late time’ observers better agree.

Clearly, if this was to oﬀer a coherent picture

of physics, there would need to exist non-trivial

consistency relations among the diﬀerent regis-

ters’ descriptions such as, e.g., the requirement

that ‘late time observers’ agree on the state of

the large scale structure. This is not a practically

unrealistic expectation as a concrete playground

for this idea has recently been constructed from

a diﬀerent motivation [

104]: a scalar ﬁeld on the

background geometry of elliptic de Sitter space

can only be quantized in an observer dependent

manner. A global Hilbert space for the quantum

ﬁeld does not exist, but consistency conditions

between diﬀerent observers’ descriptions can be

derived. (Observer consistency has also been dis-

Accepted in Quantum 2017-11-27, click title to verify 73

cussed in general terms within Relational Quan-

tum Mechanics [

28, 29] and QBism [64,65].) Al-

though not being a quantum gravity model, it

may serve as a platform for concretizing this pro-

posal further.

Acknowledgements

This work has beneﬁted from interactions with

many colleagues. First and foremost, it is a

pleasure to thank Markus Müller for uncount-

ably many insightf ul discussions, patient expla-

nations of his own work and suggestions. I am

grateful to Chris Wever for collaboration on [

1,2]

and important discussions, to Sylvain Carrozza

for drawing my attention to Relational Quantum

Mechanics in the ﬁrst place and many related

conversations, to Tobias Fritz and Lucien Hardy

for asking numerous clarifying questions, and to

Rob Spekkens for many careful comments and

suggestions. I would also like to thank Caslav

Brukner, Borivoje Dakic, Martin Kliesch, Matt

Leifer, Miguel Navascués, Matteo Smerlak and

Rafael Sorkin for useful discussions. Research at

Perimeter Institute is supp orted by the Govern-

ment of Canada through Industry Canada and by

the Province of Ontario through the Ministry of

Research and Innovation.

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