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 [
37], thermodynamics
naturally adheres to entropic and thus informa-
tional perspectives [812], 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 [
1425], 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, 3041], 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 [
4351] 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 [1425] 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, 3041]. 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