Fermat's Library | q-2020-11-16-363 annotated/explained version.

Information and disturbance in operational probabilistic

theories

Giacomo Mauro D’Ariano, Paolo Perinotti, and Alessandro Tosini

QUIT group, Physics Dept., Pavia University, and INFN Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

Any measurement is intended to provide

information on a system, namely knowl-

edge about its state. However, we learn

from quantum theory that it is generally

impossible to extract information without

disturbing the state of the system or its

correlations with other systems. In this

paper we address the issue of the interplay

between information and disturbance for

a general operational probabilistic theory.

The traditional notion of disturbance con-

siders the fate of the system state after the

measurement. However, the fact that the

system state is left untouched ensures that

also correlations are preserved only in the

presence of local discriminability. Here we

provide the deﬁnition of disturbance that

is appropriate for a general theory. More-

over, since in a theory without causal-

ity information can be gathered also on

the eﬀect, we generalise the notion of no-

information test. We then prove an equiv-

alent condition for no-information without

disturbance—atomicity of the identity—

namely the impossibility of achieving

the trivial evolution—the identity—as the

coarse-graining of a set of non trivial ones.

We prove a general theorem showing that

information that can be retrieved without

disturbance corresponds to perfectly re-

peatable and discriminating tests. Based

on this, we prove a structure theorem for

operational probabilistic theories, showing

that the set of states of any system de-

composes as a direct sum of perfectly dis-

criminable sets, and such decomposition is

preserved under system composition. As

a consequence, a theory is such that any

information can be extracted without dis-

Giacomo Mauro D’Ariano: dariano@unipv.it

Paolo Perinotti: paolo.perinotti@unipv.it

Alessandro Tosini: alessandro.tosini@unipv.it

turbance only if all its systems are classi-

cal. Finally, we show via concrete exam-

ples that no-information without distur-

bance is independent of both local discrim-

inability and puriﬁcation.

1 introduction

The possibility that gathering information on a

physical system may aﬀect the state of the sys-

tem itself was introduced by Heisenberg in his

famous gedanken experiment [1], which became

the ﬁrst paradigm of quantum mechanics. The

issue raised by Heisenberg spawned a vaste lit-

erature up to present days (see [2, 3] as recent

reviews), with a variety of quantiﬁcations of “in-

formation” and “disturbance” and corresponding

tradeoﬀ relations [4, 5, 6, 7]. All these results

are quantitative accounts of a core issue in quan-

tum theory, the no-information without distur-

bance theorem [8, 9]. The proofs of the theorem

rely on the mathematical structure of quantum

theory, and thus do not emphasise the logical

relation between no-information without distur-

bance and other quantum features, such as local

discriminability (the possibility of discriminating

multipartite states via only local measurements)

or puriﬁcation (every mixed state can be obtained

as the marginal state of a pure state).

The framework here used for exploring the rela-

tion between information and disturbance is that

of operational probabilistic theories (OPTs) [10,

11, 9]. In this setting a rigorous formulation of

the notions of system, process, and their compo-

sitions is given, which constitutes the grammar

for the probabilistic description of an experiment.

Quantum theory and classical theory are two in-

stances of OPTs.

For some probabilistic theories which can be

reframed as OPTs, the deﬁnitions of informa-

tion and disturbance have been investigated in

the presence of local discriminability, puriﬁca-

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arXiv:1907.07043v3 [quant-ph] 12 Nov 2020

tion, and causality [12, 13, 14, 15]. For OPTs

satisfying those three axioms the no-information

without disturbance theorem has been proved in

Refs. [10, 9]. In the present paper we point out

a weakness in the existing notion of disturbance,

which is ubiquitous in all past approaches. In-

deed, the conventional deﬁnition of disturbance

asserts that an experiment does not disturb the

system if and only if its overall eﬀect is to leave

unchanged the states of the system, disregard-

ing the eﬀects of the experiment on the environ-

ment. Whilst this captures the meaning of dis-

turbance within quantum theory, we cannot con-

sistently apply the same notion in theories that

violate local discriminability. A signiﬁcative case

is that of the Fermionic theory [16, 17, 18] where,

due to the parity superselection rule, an opera-

tion that does not disturb a bunch of Fermionic

systems still could aﬀect their correlations with

other systems. This issue can be cured asking a

non-disturbing experiment to preserve not only

the system state, but also its puriﬁcations [10, 9].

This extension of the notion of disturbance is gen-

eral enough to capture the operational meaning

of disturbance for Fermionic systems, however, it

is still unsatisfactory, since it cannot be used to

describe disturbance in models that do not enjoy

puriﬁcation, e. g. classical information theory.

Here we will deﬁne non-disturbing operations

only by referring to the OPT framework, thus

providing a notion that holds also for theories

that do not satisfy local discriminability, puriﬁ-

cation, or causality, and even for theories whose

sets of states are not convex. Given a system,

and an operation on it, the fate of any possible

dilation of the states of the system after the oper-

ation is taken into account, where by dilation we

mean any state of a larger system whose marginal

is the dilated state

. Moreover, due to the lack

of causality, eﬀects and states must be treated

on the same footing, and we extend the notion

of information also encompassing the information

about the output. We prove then a necessary

and suﬃcient condition for a theory to satisfy

no-information without disturbance. The condi-

tion is the impossibility of realizing the identity

transformation as a nontrivial coarse-graining of

a set of operations. Technically speaking the

We remind that for non-causal theories the marginal

is not unique, hence more generally, we require that one

of the marginals is the given state.

above condition amounts to atomicity of the iden-

tity. Finally, since a theory might satisfy no-

information without disturbance only when re-

stricted to some collections of preparations and

measurements, we will provide a weaker neces-

sary and suﬃcient condition for this case.

Similarly to the Heisenberg uncertainty rela-

tions, no-information without disturbance has

been considered as a characteristic quantum trait.

Instead, as we will see here, this feature can be

exhibited in the absence of most of the princi-

ples of quantum theory [9], and it is ubiquitous

among OPTs. Moreover, the most general case

is that of an OPT where some information can

be extracted without disturbance, in which case

this information has all the features of a classical

one. On the other hand, the only kind of systems

that allow for extracting any information without

disturbance are classical systems. This observa-

tion provides an alternative way of characterising

classical systems with respect to Ref. [19].

In Section 2 we review the framework of op-

erational probabilistic theories and some relevant

features that characterize quantum theory within

this scenario. In Section 3, after introducing

the deﬁnition of information and disturbance, we

present the main results of this paper: i) the

atomicity of the identity evolution as a necessary

and suﬃcient condition for no-information with-

out disturbance; ii) other equivalent necessary

and suﬃcient conditions in terms of properties of

reversible evolutions of the theory; iii) we prove

a structure theorem for theories where some in-

formation can be extracted without disturbance;

iv) we prove that the information that can be ex-

tracted without disturbance is “classical”, in the

sense that its measurement is a repeatable read-

ing of shareable information; v) ﬁnally we prove

that a theory in which any information can be

extracted without disturbance is a theory where

all systems are classical. In Section 4 we gener-

alize the notion of equality upon input to general

OPTs, including the cases in which local discrim-

inability does not hold. Moreover, dealing also

with non-causal theories, where states and eﬀects

must be considered on the same footing, we intro-

duce the notion of equality upon input and upon

output. This notion can be used when only a sub-

set of the preparations and of the measurements

are accessible, e.g. in resource theories [20, 21].

As a ﬁrst application we generalize the notion of

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information and disturbance to the upon input

and upon output scenario, providing a character-

ization of the no-information without disturbance

also in this case. In Section 5 we deepen the

relation between no-information without distur-

bance and other characteristic properties of quan-

tum theory. We show that no-information with-

out disturbance can be satisﬁed independently of

puriﬁcation and local discriminability, providing

counterexamples based on some of the conditions

mentioned above and other conditions proved in

this section. We end with the conclusions in Sec-

tion 6.

2 The Framework

In this section we review the framework of op-

erational probabilistic theories (OPT) (we refer

to [10, 9, 11] for further details).

The primitives of an operational theory are the

notions of test, event, and system. A test {A

}

i∈X

is the collection of events A

, where i labels the

element of the outcome space X. In the quan-

tum case A

is the ith quantum operation of

the quantum instrument {A

}

i∈X

. The notion of

test bridges the experiment with the theory, with

i ∈ X denoting the objective outcome, and A

the mathematical description of the correspond-

ing event. The notion of system, here denoted by

capital Roman letters A, B, . . ., rules connections

of tests. An input and an output label are asso-

ciated to any test (event). We represent a test

= {A

}

i∈X

and its building events A

by the

diagrams

respectively, with the rule that an output wire

can be connected only to an input wire with the

same label. Thus, given two tests A

and B

can deﬁne their sequential composition (BA)

X×Y

as the collection of events

for i ∈ X and j ∈ Y. A singleton test is a

test containing a single event. We call such an

event deterministic. For every system A there

exists a unique singleton test {I

} such that

A = AI

= A for every event A with input

A and output B, and we call I

identity of sys-

tem A. Besides sequential compositions of tests

and events, a theory is speciﬁed by the rule for

composing them in parallel. For every couple of

systems (A, B) we can form the composite sys-

tem C

= AB, on which we can perform tests

(C ⊗ D)

X×Y

with events C

⊗ D

in parallel com-

position represented as follows

⊗ D

C D

and satisfying the condition (E

⊗ F

)(C

⊗ D

) =

) ⊗ (F

). Notice that we use the tensor

product symbol ⊗ for the parallel composition

rule. Actually, for the quantum and the classical

OPT the parallel composition is the usual tensor

product of linear maps. However, for a general

OPT, the parallel composition may not coincide

with a tensor product.

There exists a special system type I, the trivial

system, such that AI = IA = A for every system

A. The tests with input system I and output A

are called preparation tests of A, while the tests

with input system A and output I are called ob-

servation tests of A. Preparation events of A are

graphically denoted as boxes without the input

wire

(or in formula as round kets |ρ)

), and

the observation events by boxes with no output

wire

(in formula round bras (c|

). For ex-

ample, one can have events of the following kind

⊗ D

We will always use the Greek letters to denote

preparation tests {ρ

}

i∈X

and Latin letters to de-

note observation tests {c

}

j∈X

(we will not spec-

ify the system when it is clear from the context).

An arbitrary test obtained by parallel and se-

quential composition of box diagrams is called

circuit. A circuit is closed if its overall input and

output systems are trivial: it starts with a prepa-

ration test and ends with an observation test. An

operational probabilistic theory (OPT) is an op-

erational theory where any closed circuit of tests

corresponds to a probability distribution for the

joint test. Compound tests from the trivial sys-

tem to itself are independent, both for sequential

and parallel composition, namely their joint prob-

ability distribution is given by the product of the

respective joint probability distributions. For ex-

ample the application of an observation event c

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after the preparation event ρ

corresponds to the

closed circuit (c

|ρ

)

and denotes the probabil-

ity of the outcome (i, j) of the observation test

after the preparation test ρ

of system A, i.e.

= Pr



i, j





For a more complex example, consider the test

D E

with U = V × W × X × Y × Z. Then we deﬁne

D E

= Pr[i, j, k, l, m|T

In the following, we will omit the parametric

dependence on the circuit if the latter is clear

from the context.

Summarising: by a closed circuit made of

events we denote their joint probability upon the

connection speciﬁed by the circuit graph, with

nodes being the test boxes, and links being the sys-

tem wires.

Given a system A of a probabilistic theory we

can quotient the set of preparation events of A by

the equivalence relation |ρ)

∼ |σ)

⇔ (c|ρ)

(c|σ)

for every observation event c. Similarly

we can quotient observation events. The equiv-

alence classes of preparation events and observa-

tion events of A will be denoted by the same sym-

bols as their elements |ρ)

and (c|

, respectively,

and will be called state and eﬀect for system A.

For every system A, we will denote by St(A),

Eﬀ(A) the sets of states and eﬀects, respectively.

States and eﬀects are real-valued functionals on

each other, and can be naturally embedded in

reciprocally dual real vector spaces, St

(A) and

Eﬀ

(A), whose dimension dim(A) is assumed to

be ﬁnite.

In Appendix A it is proved that an event A

with input system A and output system B in-

duces a linear map from St

(AC) to St

(BC)

for each ancillary system C. The collection of

all these maps is called transformation from A

to B. More explicitly, given two transformations

A, A

∈ Transf(A, B), one has A = A

, if and only

for every C, every Ψ ∈ St(AC), and every a ∈

Eﬀ(BC), namely they give the same probabilities

within every possible closed circuit. Notice that,

using the fact that two states (eﬀects) are equal

if and only if they give the same probability when

paired to every eﬀect (state), the above condition

amounts to state that A = A

if and only if

, (1)

for every C, and every Ψ ∈ St(AC), or

(2)

for every C, and every a ∈ Eﬀ(BC).

In the following, the symbols A and

will be used to represent the transformation cor-

responding to the event A. The set of transforma-

tions from A to B will be denoted by Transf(A, B),

with linear span Transf

(A, B). It is now obvious

that a linear map A ∈ Transf

(A, B) is admissi-

ble if it locally preserves the set of states St(AC),

namely A ⊗ I

(St(AC)) ⊆ St(BC), for every sys-

tem C. In the following we will write A |Ψ)

instead of A ⊗ I

|Ψ)

, with Ψ ∈ St(AC) and

A ∈ Transf(A, B) when the domains are clear

from the context.

An operational probabilistic theory is now de-

ﬁned as a collection of systems and transforma-

tions with the above rules for parallel and sequen-

tial composition and with a probability associ-

ated to any closed circuit

We introduce now the notions of reﬁnement of

an event and atomic event.

Deﬁnition 1 (Reﬁnement of an event). A re-

ﬁnement of an event C ∈ Transf(A, B) is given by

a collection of events {D

}

i∈X

from A to B, such

that there exists a test {D

}

i∈Y

with X ⊆ Y and

C =

i∈X

. We say that a reﬁnement {D

}

i∈X

of C is trivial if D

= λ

C, λ

∈ [0, 1], for every

Notice that a more detailed account needs a category-

theoretical deﬁnition of parallel and sequential composi-

tion of systems (see Ref. [11]).

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i ∈ X. Conversely, C is called the coarse-graining

of the events {D

}

i∈X

, which we will also denote

as C = D

In the following we will often refer to a reﬁne-

ment of C simply as C =

i∈X

, without speci-

fying the test including the events D

Deﬁnition 2 (Reﬁning event). Given two events

C, D ∈ Transf(A, B) we say that D reﬁnes C, and

write D ≺ C, if there exist a reﬁnement {D

}

i∈X

of C such that D ∈ {D

}

i∈X

Deﬁnition 3 (Non redundant test). We call a

test {A

}

i∈X

non redundant when for every pair

i, j ∈ X one has A

6= λA

for λ > 0.

Notice that a test that is redundant can be in-

terpreted as a non redundant test followed by a

conditional coin tossing. As a consequence a re-

dundant test always gives some spurious infor-

mation, unrelated to the input state. From a

redundant test one can achieve a maximal non

redundant one by taking the test made of coarse

grainings of all the sets of proportional elements.

Deﬁnition 4 (Reﬁnement set). Given an event

C ∈ Transf(A, B) we deﬁne its reﬁnement set

Ref

the set of all events that reﬁne C.

Deﬁnition 5 (Atomic and reﬁnable events). An

event C is atomic if it admits only of trivial

reﬁnements, namely D ≺ C implies D = λC,

λ ∈ [0, 1]. An event is reﬁnable if it is not atomic.

In the special case of states, the word pure is

used as synonym of atomic, with a pure state de-

scribing an event that provides maximal knowl-

edge about the system’s preparation. This means

that the knowledge provided by a pure state can-

not be further reﬁned. As usual a state that is

not pure will be called mixed.

Another important relation between events is

that of coexistence and the consequent notion of

coexistent completion for a set o events.

Deﬁnition 6 (Coexistent events and coexistent

completion). Two events A, B ∈ Transf(A, B) are

coexistent, and we write A ∧ B, if there exists a

test {C

}

i∈X

⊆ Transf(A, B) such that A = C

and B = C

, where Y, Z ⊆ X. Given an event

A we denote by

A the set of all events coexistent

with A, and more generally, given a set of events

X its coexistent completion is deﬁned as

X = {B; B ∧ A, for some A ∈ X}. (3)

We observe that in the present general OPT

framework features that seem intuitive are not

assumed, such as the convex completion of trans-

formations. A remarkable example is that of “no-

restriction of preparation tests” hypothesis, con-

sisting in the requirement that every collection

of states that sum to a deterministic state is a

preparation test. Similarly, we do not assume

the no-restriction hypothesis for transformations,

namely the requirement that every transforma-

tion that preserves the state set belongs to a test.

A fundamental deﬁnition for this manuscript is

that that of dilation.

Deﬁnition 7 (Dilation). We say that Ψ ∈

St(AB) is a dilation of ρ ∈ St(A) if

for some deterministic eﬀect e ∈ Eﬀ(B). Anal-

ogously, c ∈ Eﬀ(AB) is a dilation of a ∈ Eﬀ(A)

for some deterministic state ω ∈ St(B). We de-

note by D

the set of all dilations of the state

ρ. More generally, given a collection of states

S ⊆ St(A) we deﬁne D

ρ∈S

, with D

St(A)

corresponding to the set of all states ψ ∈ St(AB)

for every system B. The same notation is used

for the set of dilations of eﬀects.

We remark that, given σ ∈ S, every state of

the form σ ⊗ ρ belongs to D

Notice that there are generally more than one

deterministic eﬀect for the same system, diﬀer-

ently from quantum theory, where the partial

trace over the Hilbert space of the system is the

only way to discard it. Instead, given a state

Ψ ∈ St(AB), in a theory with more determin-

istic eﬀects for the same system B the marginal

state of system A generally depends on the eﬀect

used to discard the system B. In the following we

will call marginal of a state with deterministic ef-

fect e the speciﬁc marginal obtained by applying

the eﬀect e ∈ Eﬀ(B). Similarly, given an eﬀect

c ∈ Eﬀ(AB) its marginal of system A depends

on the choice of deterministic state on system B

and we will call marginal of an eﬀect with deter-

ministic sate ω the speciﬁc marginal obtained by

applying the deterministic state ω ∈ St(B).

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Given a system A, in the dilations sets D

St(A)

and D

Eﬀ(A)

, there could be states and eﬀects with

the following property.

Deﬁnition 8 (Faithful state and faithful eﬀect).

A state Ψ ∈ St(AC) is faithful for system A if

given two transformations A, A

∈ Transf(A, B),

the condition A |Ψ)

= A

|Ψ)

implies A =

. Analogously, an eﬀect d ∈ Eﬀ(BC) is faith-

ful for B if given two transformations A, A

∈

Transf(A, B), the condition (d|

A = (d|

implies A = A

Remark 1. We observe that in the general

framework, without further assumptions, states

(preparations) and eﬀects (measurements) are on

equal footing, and any proposition proved for

states can be proved in the same way for eﬀects.

Accordingly, since this paper relies only on the

general framework of OPTs, all the results given

in terms of states, dilations of states, and sets of

dilations of states can be mirrored to results on

eﬀects, dilations of eﬀects, and sets of dilations

of eﬀects, respectively. In the next Section 2.1 we

present some signiﬁcant classes of OPTs that are

obtained enriching the present framework with

one or more properties, such as the possibility of

performing the tomography of states using only

local operations, or the possibility of obtaining an

arbitrary mixed state as the marginal of a pure

one. Among the properties discussed in the fol-

lowing there is also causality, which induces an

asymmetry in the structure of states and eﬀects

of the theory. Indeed, as it happens in both clas-

sical and quantum theory, causality forces the ex-

istence of a unique deterministic eﬀect, while the

set of states typically presents several determin-

istic elements also in the presence of causality.

2.1 Relevant classes of OPTs

A frequently highlighted property within the

wider scenario of OPTs is that of multipartite

states discrimination via local measurements:

Deﬁnition 9 (Local discriminability). It is pos-

sible to discriminate between any pair of states

of composite systems using only local measure-

ments. Mathematically, given two joint states

Ψ, Ψ

∈ St(AB) with Ψ 6= Ψ

, there exist two

eﬀects a ∈ Eﬀ(A) and b ∈ Eﬀ(B), such that

Notice that the names local discriminability

and local tomography are used interchangeably

in the literature. Also in this manuscript we will

consider the two names as synonymous.

Two relevant consequences of local discrim-

inability are: i) the local characterization of

transformations, stating that the local behaviour

of a transformation is suﬃcient to fully charac-

terize the transformation itself; ii) the atomicity

of parallel composition. Here we report those two

features for the convenience of the reader.

Proposition 1 (Local characterization of

transformations). If local discriminability holds,

then for any two transformations A, A

∈

Transf(A, B), the condition A |ρ)

= A

|ρ)

for

every ρ ∈ St(A) implies A = A

See Ref. [10] for the proof.

Proposition 2 (Atomicity of parallel composi-

tion). If an OPT satisﬁes local discriminability

then the parallel composition of atomic transfor-

mations is atomic.

For the proof of the above proposition see

Ref. [22]. We observe that an OPT with local dis-

criminability allows for tomography of multipar-

tite states using only local measurements. In an

OPT with local discriminability, the linear space

of eﬀects of a composite system is the tensor prod-

uct of the linear spaces of eﬀects of the component

systems, namely Eﬀ(AB)

≡ Eﬀ(A)

⊗ Eﬀ(B)

Thus, any bipartite eﬀect c ∈ Eﬀ(AB) can be

written as a linear combination of product ef-

fects, and every probability (c|ρ)

, for ρ ∈

St(AB), can be computed as a linear combina-

tion of the probabilities ((a|

⊗ (b|

) |ρ)

aris-

ing from a ﬁnite set of product eﬀects. The

same holds for the linear space of states and in

an OPT with local discriminability the parallel

composition of two states (eﬀects) can be un-

derstood as a tensor product. Finally, the re-

lation dim (AB) = dim(A) dim(B) between the

linear dimension of the set of states/eﬀects holds,

whereas for theories without local discriminabil-

ity it holds dim (AB) > dim(A) dim(B).

Recently it has been shown that relevant phys-

ical theories, such as the Fermionic theory [16],

can be described in the OPT framework relax-

ing the property of local discriminability [17, 18].

The most general scenario for OPTs that exhibit

a ﬁnite degree of holism is that of OPTs with n-

local discriminability for some n ∈ N [23]:

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Deﬁnition 10 (n-local discriminability). A the-

ory satisﬁes n-local discriminability if whenever

two states ρ and ρ

are diﬀerent, there exist a n-

local eﬀect b such that (b|ρ) 6= (b|ρ

). We say that

an eﬀect is n-local if it can be written as a conic

combination of tensor products of eﬀects that are

at most n-partite.

Two notable examples are indeed Fermionic

quantum theory and real quantum computa-

tion [23, 17, 18] that are both 2-local tomo-

graphic.

Another relevant class of OPTs is that of theo-

ries with puriﬁcation [10, 24]. As a result of this

paper we will show (Proposition 9) that the set of

convex OPTs with puriﬁcation is strictly smaller

than the set of OPTs that satisfy no-information

without disturbance. Moreover, we will see that

a weak version of puriﬁcation, which does not

require the uniqueness (as in quantum theory)

but just the existence of a puriﬁcation for each

state, is enough to imply no-information without

disturbance together with the convexity assump-

tion. Accordingly, we deﬁne the following class of

OPTs.

Deﬁnition 11 (States puriﬁcation). We say that

an OPT satisﬁes states puriﬁcation if for every

system A and for every state ρ ∈ St(A), there

exists a system B and a pure state Ψ ∈ St(AB)

which is a dilation of ρ.

We will prove that also the analogous notion of

puriﬁcation for eﬀects, provided in the following,

is suﬃcient to guarantee no-information without

disturbance.

Deﬁnition 12 (Eﬀects puriﬁcation). We say

that an OPT satisﬁes eﬀects puriﬁcation if for

every system A and for every eﬀect a ∈ Eﬀ(A),

there exists a system B and an atomic eﬀect

c ∈ Eﬀ(AB) that is a dilation of a.

As already noticed, the above deﬁnitions do not

require the puriﬁcation to be unique up to re-

versible transformations on the purifying system.

The last relevant class of OPTs that we point

out is that of causal theories:

Deﬁnition 13 (Causal OPTs). The probability

of preparation events in a closed circuit is inde-

pendent of the choice of observations.

Mathematically, if {ρ

}

i∈X

⊂ St(A) is a prepa-

ration test, then the conditional probability of the

preparation ρ

given the choice of the observation

test {a

}

j∈Y

is the marginal



i|{a

}



j∈Y

|ρ

)

In a causal theory the marginal probability



i|{a

}



is independent of the choice of the

observation test {a

}: if {a

}

j∈Y

and {b

}

k∈Z

are two diﬀerent observation tests, then one has



i|{a

}



= Pr



i|{b

}



The present notion of causality is simply the

Einstein causality expressed in the language of

OPTs. As proved in Ref. [10] causality is equiva-

lent to the existence a unique deterministic eﬀect

. We call the eﬀect e

the deterministic eﬀect

for system A. By deﬁniton in non-causal theories

the deterministic eﬀect cannot be unique.

3 Information and disturbance

Within the general scenario of operational proba-

bilistic theories, and without further assumptions

on the structure of the theory, we aim at deﬁning

the notions of non-disturbing and no-information

test. These notions have already been investi-

gated for causal theories (Deﬁnition 13) that sat-

isfy local discriminability (Deﬁnition 9) or states

puriﬁcation (Deﬁnition 11). We start highlight-

ing the weakness of previous approaches in cases

where the above hypotheses do not hold. The dis-

turbance and the information produced by a test

on a physical system A are commonly deﬁned in

relation to measurements and states of the sys-

tem A only, disregarding the action of the same

test on an enlarged systems AB.

A test {A

}

i∈X

on system A is usually said to

be non-disturbing if for every ρ ∈ St(A) one has

that

|ρ)

= |ρ)

. However, this deﬁnition

is not operationally consistent if applied to the-

ories without local discriminability. A physically

relevant example is that of Fermionic theory [16]

that, due to the parity superselection rule, is non-

local tomographic [17, 18] (it is 2-local tomo-

graphic according to Deﬁnition 10). We can see

via a simple example that, for a Fermionic system

A, a test {A

}

i∈X

such that

|ρ)

= |ρ)

for

every ρ ∈ St(A) still can disturb the states of a

composite system AB.

The parity superselection rule on a system N

of N Fermions forbids any state corresponding

to a superposition of vectors belonging to F

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and F

, representing Fock vector spaces with

total even and odd occupation number, respec-

tively. As a consequence, the linearized set of

states St

) splits in the direct sum of two

spaces, containing the states with even and odd

parity, respectively. It is now convenient to make

use of the projectors onto the well-deﬁned parity

subspaces P

, for the even space, and P

, for the

odd one. Notice that, since P

= P

= 0

any Fermionic state ρ will be of the form ρ =

ρP

+ P

ρP

. Consequently the parity test

·P

, P

·P

} leaves every state ρ ∈ St(N

) un-

changed. Intuitively, this seems to suggest that

parity can be measured without disturbing. In-

deed, this view is in agreement with the notion

of disturbance that has been considered in the

literature so far.

Consider now a mixed state ρ ∈ St(N

), with

ρ = p

+ p

, ρ

and ρ

an even and an odd

pure state respectively, and p

+ p

= 1. For

example, consider the states

= |00ih00| , ρ

= |01ih01| ,

and p

= p

= 1/2, so that

ρ =

(|00ih00| + |01ih01|).

Since Fermionic theory allows for states puriﬁ-

cation [17] (see Deﬁnition 11), we can always ﬁnd

a state Ψ ∈ St(M

), with M > N that puriﬁes ρ.

Since Ψ is pure, it has a deﬁnite parity, say even.

In our example one can choose

Ψ =

(|000i + |011i)(h000| + h011|). (4)

Therefore, the local test on the system N

that

measures the parity of the system will not disturb

the states of N

but will decohere the state Ψ to a

mixed state, then introducing a disturbance. For

example, in our case

⊗ I)Ψ(P

⊗ I) + (P

⊗ I)Ψ(P

⊗ I)

(|000ih000| + |011ih011|).

In order to avoid the above issue, and to in-

troduce a deﬁnition of non-disturbing test that

works also for theories without local discrim-

inability, one could say that a test {A

}

i∈X

on sys-

tem A is non-disturbing upon input of ρ ∈ St(A),

if for every σ in the reﬁnement set of ρ and ev-

ery puriﬁcation Ψ

∈ St(AB) of σ one has that

|Ψ)

= |Ψ)

. This route, which has

been proposed in Refs. [10, 9], captures the oper-

ational meaning of disturbance also for Fermionic

systems. However, the deﬁnition of Refs. [10, 9]

requires puriﬁcation, and thus cannot be used in

theories without puriﬁcation, e. g. the cases of PR

boxes, or the classical theory of information.

Based on the above motivations our proposal

is to deﬁne the disturbance (and the information)

produced by a test in terms of its action on di-

lations, both of states and eﬀects. This leads to

notions of information and disturbance that are

completely general and thus do not depend on

causality, local discriminability, or puriﬁcation.

This will allow us to prove the no-information

without disturbance theorem for a very large class

of OPTs. In this Section we ﬁrst consider the dis-

turbance and the information provided by a test

when no restrictions are posed on the states and

eﬀects of the theory. The generalization to a sce-

nario where both preparations and measurements

are limited to given subsets is presented in Sec-

tion 4.2.

Deﬁnition 14 (Non-disturbing test). Consider

a test {A

}

i∈X

on system A. We say that the test

is non-disturbing if

= I

. (5)

Notice that, following the above deﬁnition, the

test {A

}

i∈X

is disturbing if there exist Ψ ∈

St(A)

, and c ∈ D

Eﬀ(A)

, such that

i∈X

. (6)

This deﬁnition of disturbance thus stresses the

eﬀect of a transformation on correlations with

remote systems, indeed a test {A

}

i∈X

is non-

disturbing if it is operationally equal to the iden-

tity transformation of system A, namely it acts

as the identity on any possible state and eﬀect of

any composite system.

Remark 2. We could have deﬁned a non-

disturbing test from A to C as follows

= R, (7)

where R ∈ Transf(A, C) is a reversible transfor-

mation, namely there exists another transforma-

tion W ∈ Transf(C, A) such that WR = I

, and

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RW = I

. Indeed if the test provides a system-

atic reversible transformation on the inputs, then

its eﬀect can be trivially corrected by inverting it.

The classiﬁcation of non-disturbing test accord-

ing to this deﬁnition is trivially provided by the

classiﬁcation according to Deﬁnition 14. Indeed,

the most general non-disturbing test from A → C

is the sequence of tests of the form {A

i∈X

with {A

}

i∈X

non-disturbing according to Deﬁ-

nition 14, and R ∈ Transf(A, C) reversible.

In the same spirit we can establish if a test

provides information. Again a test could provide

information both on the input (preparation) and

on the output (observation).

Let us consider the task in which Bob wants

to extract information on states (eﬀects) of Alice

via the test A

= {A

}

i∈X

⊆ Transf(AC). Op-

erationally, in the most general case the test A

can be used in composite tests made of a prepa-

ration test Ψ

= {Ψ

}

j∈Y

⊆ St(AB), the test

= {A

}

i∈X

⊆ Transf(AC) and an observa-

tion test C

= {c

}

k∈Z

⊆ Eﬀ(CB), leading to

the joint probabilities

= p(j, i, k|Ψ

, A

, C

)

(8)

associated with possible outcomes j, i, k, and

where we explicitly show the dependence of the

joint probability distribution p(j, i, k) on the tests

composing the circuit.

Within this scenario, Bob can use both the test

and any observation test C

in order to ex-

tract the information on the inputs, while he can

use both the test A

and any preparation test

in order to extract the information on the

outputs. This leaves room for two inequivalent

conditions for a no-information test A

1. Strong condition for no-information test.

(a) No-information on inputs: the test A

is no-information on inputs if for every

preparation test Ψ

and for every obser-

vation tests C

, the joint probability in

Eq. (8) factorizes as

p(j, i, k|Ψ

, A

, C

)

= r(i|k; A

, C

)s(j, k|Ψ

, A

, C

namely we impose that

r(i|j, k; Ψ

, A

, C

) = r(i|k; A

, C

where the probability distribution r does

not depend on the preparation test Ψ

(we remind that it may happen that a

probability distribution depends on a

given test but not on its outcomes). The

interpretation of this condition is that the

outcomes of A

and their correlations

with the outcomes of any observation

test do not provide information on the

preparation.

(b) No-information on outputs: the test A

is no-information on outputs if for every

and for every Ψ

the joint probability

in Eq. (8) factorizes as

p(j, i, k|Ψ

, A

, C

)

= r(i|j; Ψ

, A

)s(j, k|Ψ

, A

, C

where the probability distribution r does

not depend on the observation test C

This condition ensures that the outcomes

of A

and their correlations with the out-

comes of any preparation test do not pro-

vide information on the observation.

2. Weak condition for no-information test.

(a) No-information on inputs: the test A

is no-information on inputs if for every

preparation test Ψ

and for every obser-

vation test C

, the joint probability in

Eq. (8) is such that

p(j, i, k|Ψ

, A

, C

)

= r(i|A

, C

)s(j|Ψ

, A

, C

(9)

where the probability distribution r does

not depend on the preparation test Ψ

The interpretation of this condition is

that the outcomes of A

do not provide

information on the preparation, whenever

we ignore the outcome of the observation

test.

(b) No-information on outputs: the test A

is no-information on outputs if for every

observation test C

and for every prepa-

ration test Ψ

, the joint probability in

Eq. (8) is such that

p(j, i, k|Ψ

, A

, C

) =

r(i|Ψ

, A

)s(k|Ψ

, A

, C

(10)

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where the probability distribution r does

not depend on the observation test C

This means that the outcomes of A

not provide information on the observa-

tion, whenever we ignore the outcome of

the preparation test.

It is elementary to see that 1a⇒2a

(and 1b⇒2b), namely the strong no-information

condition implies the weak one, both on inputs

and outputs. In the literature on no-information

without disturbance in quantum theory the au-

thors take the weak notion 2a of no-information

test (only on inputs since the quantum theory is

causal). Here, we also choose the conditions 2 as

expressed in the next deﬁnition (the equivalence

between the weak conditions 2 and Deﬁnition 15

is proved in Appendix B). The motivation for this

choice is that if an OPT satisﬁes no-information

without disturbance according to conditions 2,

then it also satisﬁes no-information without

disturbance in the strongest sense of conditions 1

(see Remark 4 in the following).

Deﬁnition 15 (No-information test). A test

}

i∈X

with events A

∈ Transf(A, C) is a no-

information test if, for every choice of determin-

istic eﬀect e

and deterministic state ω

, there

exists a deterministic eﬀect f

and a determin-

istic state ν

such that for every i ∈ X one has

(e|

= p

(e) (f|

, (11)

|ω)

= q

(ω) |ν)

. (12)

According to Eq. (11) (that coincides with

Eq. (9) in the weak condition of item 2a), the

test {A

}

i∈X

does not provide information upon

any possible input state. However, the proba-

bility distribution p

(e) might in principle pro-

vide information about the eﬀect e. On the other

hand according to Eq. (12) (that coincides with

Eq. (10) in the weak condition of item 2b), the

test {A

}

i∈X

does not provide information upon

output of any possible eﬀect, while the probabil-

ity distribution q

(ω) might in principle provide

information about the state ω. The conjunction

of the two conditions implies that no-information

is provided by the test about D

St(A)

and D

Eﬀ(A)

namely about any possible input state and output

eﬀect of any dilated system. The last statement

is proved in the following lemma.

Lemma 1. Let the test {A

}

i∈X

with events A

∈

Transf(A, C) be a no-information test. Then one

has

(e|

= r

(f|

, (13)

|ω)

= r

|ν)

. (14)

Proof. By Eqs. (11) and (12) one has

(e|

|ω)

= p

(e) = q

(ω) = r

where we used the fact that e, f and ω, ν are re-

spectively deterministic eﬀects and deterministic

states. 

Remark 3. Notice that in Eq. (11) the probabil-

ity of the transformation A

∀i ∈ X generally

depends on the deterministic eﬀect e

, this ac-

counting for non-causal theories. In the more

general case in which also the deterministic ef-

fect f

on the right hand side of Eq. (11) de-

pends on i ∈ X, the test {A

}

i∈X

would provide

information on the system state (this would hap-

pen, however, only for probabilistic states). An

analogous argument holds for ν in Eq. (12).

3.1 No-information without disturbance

In this section we state the condition of no-

information without disturbance and introduce

criteria for it to be satisﬁed by an OPT.

Deﬁnition 16 (OPT with no-information with-

out disturbance). We say that an OPT satisﬁes

no-information without disturbance if, for every

system A, and every test {A

}

i∈X

⊆ Transf(A),

if the test is non-disturbing then it is a no-

information test.

Theorem 1. An OPT satisﬁes no-information

without disturbance if and only if the identity

transformation is atomic for every system of the

theory.

Proof. We start proving that if an OPT satis-

ﬁes no-information without disturbance then the

identity transformation is atomic. Consider a

system A of the theory, and a reﬁnement {A

}

i∈X

∈ Transf(A) for every i ∈ X) of the identity

map I

for system A. The test {A

}

i∈X

is clearly non-disturbing, therefore by hypothesis

it is a no-information test. By deﬁnition of no-

information test, and using Lemma 1, we know

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that for every deterministic eﬀect e

, and de-

terministic state ω

, there exists a determinis-

tic eﬀect f

and a deterministic state ν

such

that for every i ∈ X one has (e|

= r

(f|

and A

|ω)

= r

|ν)

. Summing both sides of

the last equation over the index i ∈ X, and re-

membering that

i∈X

= 1, we ﬁnd that e = f

and ω = ν. Therefore, the no-information condi-

tion is

(e|

= r

(e|

, (15)

|ω)

= r

|ω)

, (16)

for every deterministic eﬀect e

and for every

deterministic state ω

. Consider now an arbi-

trary pure state Ψ ∈ St(AB) (the same proof

can be done choosing an arbitrary atomic eﬀect

c ∈ Eﬀ(AB)). Since

|Ψ)

= |Ψ)

, it

follows that

|Ψ)

= λ

(Ψ) |Ψ)

(Ψ) = 1, (17)

where the coeﬃcients λ

(Ψ) generally depend on

the state Ψ. However, for each pure state Ψ there

exists a deterministic eﬀect e

∈ Eﬀ(AB) such

that (e

|Ψ) 6= 0. Upon applying the determinis-

tic eﬀect e

on both sides of Eq. (17), we get

|Ψ)

= λ

(Ψ)(e

|Ψ)

. (18)

Now, applying both sides of Eq. (15) to Ψ, we

get

|Ψ)

= r

|Ψ)

, (19)

and comparing the last two identities, consider-

ing that (e

|Ψ)

6= 0, we obtain

(Ψ) = r

, ∀i ∈ X. (20)

Since this holds true for every pure state Ψ, we

conclude that λ

(Ψ) is independent of Ψ. Then

|ρ)

= r

|ρ)

, ∀ρ ∈ St(AB), proving that

= r

. Notice that we implicitly assumed

that the probabilities r

do not depend on the

choice of the system B. Actually this can be

proven as shown in Appendix C.

The converse implication, namely that if in an

OPT the identity transformation is atomic then

a non-disturbing test is no-information, is trivial.



Remark 4. Eq. (8) shows the most general sce-

nario in which a test {A

}

i∈X

can be used to

extract information on its inputs or on its out-

puts. We noticed that two inequivalent deﬁ-

nitions of no-information tests are possible, a

strong condition 1, and a weak condition 2, de-

pending on the features of the joint probabil-

ity distribution p(j, i, k) of Eq. (8). However,

due to the above theorem, if a theory satis-

ﬁes no-information without disturbance in the

weak sense, then a non-disturbing test {A

}

i∈X

∈

Transf(A) is such that A

= q

, with

= 1.

It follows that in Eq. (8) the joint probability

distribution p(j, i, k) is of the form p(j, i, k) =

p(j, k), and the test is also no-information in

the strong sense.

Besides the atomicity of the identity, we can

provide other two equivalent necessary and suf-

ﬁcient conditions for no-information without dis-

turbance.

Proposition 3. An OPT satisﬁes no-

information without disturbance if and only

if for every system there exists an atomic trans-

formation which is either left- or right-reversible.

Proof. We start proving that a theory with an

atomic reversible transformation for each system

satisﬁes no-information without disturbance. Let

R ∈ Transf(A, C) be atomic and left-reversible

(the right-reversible case is analogous). Then

consider a reﬁnement I

, with A

∈

Transf(A) for i ∈ X, of the identity transforma-

tion. By deﬁnition of identity map we have that

= R, and due to the atomic-

ity of R it must be RA

∝ R for every i ∈ X.

Since R is left-reversible (namely there exists

W ∈ Transf(C, A) such that WR = I

) it fol-

lows that A

∝ I

for every i ∈ X, which proves

the atomicity of I

The other implication, that in a theory that

satisﬁes no-information without disturbance for

every system there exists an atomic transfor-

mation which is either left- or right-reversible,

is trivial. Indeed, in a theory that satisﬁes

no-information without disturbance the identity,

which is both right- and left-reversible, is atomic

as proved in Theorem 1. 

Proposition 4. An OPT satisﬁes no-

information without disturbance if and only

if for every system every reversible transforma-

tion is atomic.

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Proof. We prove that if the theory satisﬁes no-

information without disturbance, then every re-

versible transformation is atomic. Indeed, let

R ∈ Transf(A) be reversible, and suppose that

R =

i∈X

for test {R

}

i∈X

. Then, one has

i∈X

−1

= I

, (21)

and by Theorem 1 one has that R

−1

= p

Finally, multiplying by R to the right, we con-

clude that R

= p

R, namely the reﬁnements of

R must be trivial. For the converse, it is suf-

ﬁcient to observe that the identity is reversible.



3.2 Information without disturbance

In this section we provide the general structure of

the state spaces and eﬀect spaces of any theory

where some information can be extracted from

a system without introducing disturbance. Such

information is “classical” in the sense that the

measurement is the reading of information that

is repeatable and shareable. In particular, for the

classical OPT the whole information encoded on

a system can be read in this way. The proof of

the above statements are based on the following

theorem.

Theorem 2. The non redundant atomic reﬁne-

ment of the identity is unique for every system.

Moreover, given the non redundant atomic re-

ﬁnement {A

}

i∈X

⊆ Transf(A) of the identity

, one has A

= A

Proof. Suppose that the identity transformation

of system A allows for two atomic reﬁnements

i∈X

, and I

j∈Y

. Since

= B

, from the atomicity of the transfor-

mations B

we get A

= c

, for some c

≥ 0

such that

i∈X

= 1 ∀j ∈ Y. Similarly we

get A

= d

for some d

≥ 0 such that

j∈Y

= 1 ∀i ∈ X. Then c

= d

. By

non redundancy one has that for ﬁxed j there is

only one value of i = i(j) such that c

> 0, and

normalisation gives c

i(j)j

= 1. By a similar argu-

ment for a ﬁxed i there is j(i) such that d

ij(i)

= 1.

Then one has B

= A

i(j)

. This proves the unique-

ness of the non redundant atomic reﬁnement of

the identity.

By the same argument as before, for the non

redundant atomic reﬁnement of the identity one

has A

= c

= d

, for some c

, d

≥ 0

such that

i∈X

j∈X

= 1 ∀i, j ∈ X.

By atomicity and non redundancy one must have

= d

= δ

i,j

. 

The above theorem has as a consequence the

following structure theorem for OPTs.

Corollary 1. For any pair of systems A, B of

an OPT one has the following decomposition of

the set of states and of the set of eﬀects of AB

St(AB) =

(i,j)∈X×Y

(AB),

Eﬀ(AB) =

(i,j)∈X×Y

Eﬀ

(AB),

(22)

where for non redundant atomic decompositions

}

i∈X

, {B

}

j∈Y

of the identities I

and I

one has

⊗ B

)





= δ









⊗ B

) = δ





(23)

for all Ψ

∈ St

(AB) and c

∈ Eﬀ

(AB).

Remark 5. Notice that from Eq. (22) it trivially

follows that for any system A the block decom-

position holds

St(A) =

i∈X

(A), Eﬀ(A) =

i∈X

Eﬀ

(A).

(24)

However, Eq. (22) contains the additional infor-

mation that the decomposition holds in that spe-

ciﬁc form also for composite systems. This is

not a straightforward consequence of the decom-

position of local states and local eﬀects, as wit-

nessed by the Fermionic case. Indeed, the state

in Eq. (4) does not have deﬁnite parity for the

two subsystems corresponding to two Fermions

on the left and one on the right, hence the state

space cannot be of the form in Eq. (22).

Remark 6. For a theory without atomicity of par-

allel composition it is possibile that the reﬁne-

ment A

⊗ B

in Eq. (23) of I

is not atomic.

In such a case one has St(AB) =

k∈Z

(AB),

and St

(AB) =

k∈Z

(AB), for some parti-

tion Z

of Z.

3.2.1 Full information without disturbance

In the following we formalise the fact that a the-

ory where any information can be extracted via

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a non-disturbing test must have only classical

systems. Let us ﬁrst deﬁne the notion of full-

information without disturbance.

Deﬁnition 17 (Full-information without distur-

bance). An OPT satisﬁes full-information with-

out disturbance if for every system A and ev-

ery test {B

}

j∈Y

⊆ Transf(A) there exists a non

disturbing test {A

}

i∈X

⊆ Transf(A) (namely

= I

) such that

p(j|i) V

, (25)

for some probability distribution p and reversible

transformations V

, R

∈ Transf(A).

As a consequence of the above deﬁnition we

have the following lemma on the structure of

atomic maps in a theory with full-information

without disturbance.

Lemma 2. Consider an OPT with full-

information without disturbance. Any atomic

transformation B ∈ Transf(A) is of the form

B = λ UA

= λ A

where A

, A

are atomic transformations in the

unique non-redundant reﬁnement of the identity

of Theorem 2, U ∈ Transf(A) is a

reversible transformation and λ ≥ 0.

Proof. Consider a test {B

}

j∈Y

⊆ Transf(A) such

that B = B

for some j ∈ Y. Due to full-

information without disturbance (see Eq. (25)

in Deﬁnition 17 where we now take as non dis-

turbing test {A

}

i∈X

the unique non-redundant

atomic reﬁnement of the identity I

)

and to the atomicity of B one has B = λ VA

for some i ∈ X, λ ≥ 0 and V, R ∈ Transf(A) re-

versible transformations. Consider now the three

tests {VA

i∈X

, {A

VR}

i∈X

and {VRA

}

i∈X

and observe that

R =

VRA

VR = VR = U, with U ∈ Transf(A) re-

versible. We then conclude the proof noticing

that by Theorem 2 the non redundant atomic re-

ﬁnement of a reversible transformation is unique.



We can now state the main Theorem of this

section.

Theorem 3. If an OPT is full-information with-

out disturbance then every system of the theory is

classical.

Proof. Consider an arbitrary system A of the

theory. Since by hypothesis the identity is not

atomic, let {A

}

i∈X

be the unique non-redundant

atomic reﬁnement of the identity I

Theorem 2. Due to Corollary 1 (and the imme-

diately following remark) the sets of states and

eﬀects decompose as in Eq. (24). We now prove

that all the blocks in such decompositions must

be one-dimensional. To this end we show that

any pair of states ρ, ρ

∈ St

(A) is such that

∝ ρ.

First we show that if an OPT satisﬁes full-

information without disturbance then for every

non null atomic ρ ∈ St

(A) and atomic a ∈

Eﬀ

(A), one has (a|ρ) 6= 0. Given such ρ ∈ St

(A)

and a ∈ Eﬀ

(A) consider the transformation

|ρ) (a| ∈ Transf(A), that is generally not atomic.

Due to Lemma 2 all the atomic reﬁnements of

the above transformation are of the form

|ρ) (a| =

, (26)

where each A

is the element of the non re-

dundant reﬁnement of the identity I

such that

|ρ) = |ρ), and (a| A

= (a|, λ

> 0, and the

are reversible trasformations. Applying both

sides of Eq. (26) to the state ρ ∈ St

(A) one has

|ρ) (a|ρ) =

|ρ). Reminding that the U

are all reversible, λ

> 0 and ρ is non-null, one

concludes that the right hand side cannot be null,

and this proves that also the pairing (a|ρ) is non-

null.

Let us apply the transformation in Eq. (26) to

another arbitrary atomic state ρ

∈ St

(A). Since









one ﬁnds |ρ) (a|ρ

) =





for some λ

> 0 and U

reversible transforma-

tions. As shown above it is (a|ρ

) 6= 0, and using

the atomicity of ρ one has





∝ U

−1

|ρ) for ev-

ery j. Since this holds true for every atomic state

∈ St

(A), one has proved that all atomic states

(and then all states) in St

(A) are proportional to

the same atomic state, let’s say U

−1

|ρ) for some

. Via an analogous argument one can see that

all eﬀects in Eﬀ

(A) are proportional to the same

atomic eﬀect. 

Remark 7. We remind that a system is classi-

cal when all its pure states are jointly perfectly

discriminable. In this case the base of the conic

hull of the pure states of each system is a sim-

plex, which corresponds to a subset of the set of

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states for a convex theory. A special case of the-

ory whose systems are all classical is the usual

classical information theory, where indeed one

has full-information without disturbance. On the

other hand, even when all systems are classical,

the theory can diﬀer from classical information

theory e. g. in the rule for systems composition.

For example there exist OPTs whose systems are

all clssical but that do not satisfy local discrim-

inability (see Ref. [25]).

4 Information and disturbance with re-

stricted input and output

In this section we extend our previous results to

study the relation between disturbance and in-

formation when both input states and output ef-

fects are limited to some given subsets. To this

end we ﬁrst introduce the basics notion of iden-

tical transformations upon restricted input and

output resources.

4.1 Operational identities between transforma-

tions

As expressed in Eq. (1), two transformations

A, A

∈ Transf(A, B) of an OPT are said to be

operationally equal if for every system C and

for every state Ψ ∈ St(AC) one has A |Ψ)

|Ψ)

. However, two non-identical maps

A, A

∈ Transf(A, B) could behave in the same

way when their action is restricted to a relevant

subclass of states.

The notion of identical transformation upon in-

put of a state ρ ∈ St(A) has been already in-

troduced in the literature (see Refs. [26, 9] and

references therein):

Deﬁnition 18 (Equal transformations upon in-

put of ρ). We say that two transformations

A, A

∈ Transf(A, B) are equal upon input of

ρ ∈ St(A), and write A =

, if for every

σ ∈ Ref

we have that Aσ = A

σ.

Remark 8 (Operational interpretation of equality

upon input). The equality upon input of a state ρ

was originally introduced for quantum theory in

Ref. [26], where the authors extended the equal-

ity to the whole support of the chosen density

matrix ρ. Within the OPT framework the equal-

ity upon input of ρ is instead extended to the

reﬁnement set Ref

[9]. This choice can be eas-

ily motivated in operational terms: the equality

A =

means that the two maps A and A

are

indistinguishable on the state ρ, independently of

how it has been prepared. Suppose that the state

ρ is prepared by Alice as ρ =

i∈X

, for some

reﬁnement of ρ. Even Alice, using her knowledge

of the preparation cannot distinguish between A

and A

From Proposition 1 we know that the local ac-

tion of a map is suﬃcient to determine the map

itself if the OPT satisﬁes local discriminability

(see Deﬁnition 9). However, for theories without

local discriminability the local action of a trans-

formation might not be suﬃcient to character-

ize it. According to Deﬁnition 18, then, even if

A =

, still the maps A and A

could act diﬀer-

ently upon input of dilations of ρ, namely it could

be A |Ψ)

6= A

|Ψ)

, for some Ψ ∈ Ref

. In

this case the diﬀerence between A and A

would

go undetected if their action on system A only

is considered. For this reason we introduce the

notion of equal transformations upon input of di-

lations of a state ρ.

Deﬁnition 19 (Equal transformations upon in-

put of D

). Given a state ρ ∈ St(A), we say

that two transformations A, A

∈ Transf(A, B)

are equal upon input of D

, and write A =

if A |Ψ)

= A

|Ψ)

for every Ψ ∈ Ref

Notice that the above deﬁnition requires that

two transformations act in the same way on the

set Ref

. Due to the absence of no-restriction

of preparation tests, it is not true in general that

Ref

= Ref

, as one might expect. The only

inclusion that can be proved without further as-

sumptions is Ref

⊆ D

Ref

, (see Lemma 5 in

Appendix D).

Here we show that the two Deﬁnitions 18 and

19 coincide for causal OPTs with local discrim-

inability. For this purpose we ﬁrst need the fol-

lowing lemma.

Lemma 3. In a causal OPT, if Ψ ∈ St(AB),

with Ψ ∈ Ref

for some ρ ∈ St(A), then

{(b|

|Ψ)

|b ∈ Eﬀ(B)} ⊆ Ref

. (27)

Proof. Since Ψ ∈ Ref

there exists a prepara-

tion test {Ψ,

Ψ, Λ} ⊆ St(AB) such that Ψ +

Ψ ∈

. For an arbitrary b ∈ Eﬀ(B), thanks to

causality that ensures the existence of a unique

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deterministic eﬀect (see Deﬁnition 13), one can

construct the test

{(b|

|Ψ)

, (e − b|

|Ψ)

(b|





, (e − b|





(b|

|Λ)

, (e − b|

|Λ)

where e

is the unique deterministic eﬀect. Since

the coarse-graining of the ﬁrst four elements is ρ,

we conclude that (b|

|Ψ)

∈ Ref

. 

Proposition 5. In a causal OPT with local dis-

criminability, given two transformations A, A

∈

Transf(A, B), the two conditions A =

and

A =

are equivalent.

Proof. We ﬁrst prove that A =

⇒ A =

. Consider an arbitrary Ψ ∈ Ref

, with

ρ ∈ St(A). Let for example be Ψ ∈ St(AC).

By hypothesis we have that A =

, namely

A |σ)

= A

|σ)

for every σ ∈ Ref

. Then, due

to Lemma 3, ∀b ∈ Eﬀ(B), ∀c ∈ Eﬀ(C), we have

that (b|

(c|

(A ⊗ I

) |Ψ)

= (b|

(c|

⊗

) |Ψ)

, and by local discriminability (see Def-

inition 9) we conclude that (A ⊗ I

) |Ψ)

⊗ I

) |Ψ)

. Since this holds true for every

Ψ ∈ Ref

, we conclude that A =

. The

converse implication A =

⇒ A =

trivial. 

Dealing also with non-causal theories, where

the role of states and eﬀects is interchangeable

(see Remark 1) it is in order to introduce the

counterpart of the above deﬁnition with eﬀects re-

placing states. Accordingly we deﬁne equal trans-

formations upon output of dilations of an eﬀect

Deﬁnition 20 (Equal transformations upon out-

put of D

). Given an eﬀect b ∈ Eﬀ(B), we say

that two transformations A, A

∈ Transf(A, B)

are equal upon output of D

, and write A =

if (c|

A = (c|

for every c ∈ Ref

In the most general case one can deﬁne equal-

ity of two transformations when both states and

eﬀects are limited to two given subsets.

Deﬁnition 21 (Equal transformations upon

X, Y ). We say that two transformations A, A

∈

Transf(A, B) are equal upon input of X ⊆ D

St(A)

and upon output of Y ⊆ D

Eﬀ(B)

—or simply upon

X, Y —and write A =

Y X

, if

(28)

for every Ψ ∈ Ref

and for every c ∈ Ref

As a special case, given a state ρ ∈ St(A), and

an eﬀect b ∈ Eﬀ(B), two transformations A, A

∈

Transf(A, B) are equal upon D

, D

when in the

last deﬁnition we take X = D

and Y = D

Accordingly, also Deﬁnitions 19 and 20 are spe-

cial cases of Deﬁnition 21, with A =

cor-

responding to the choice X = D

, Y = D

Eﬀ(B)

and A =

corresponding to the choice X =

St(A)

, Y = D

. Naturally, also the notion of

equal transformations A = A

is the one of Def-

inition 21 with no restrictions on the set states

and eﬀects, that is X = D

St(A)

, Y = D

Eﬀ(B)

Based on the above identities of transforma-

tions, any property of an OPT can be generalized

in the upon X, Y scenario. Here we only present

the case of two properties that will be used to

derive the following results.

In Deﬁnition 5 we introduced the notion of

atomic events, and we can provide a weaker ver-

sion of the property of atomicity for transforma-

tions.

Deﬁnition 22 (Atomic and reﬁnable transfor-

mation upon X, Y .). A transformation A ∈

Transf(A, B) is atomic upon input of X ⊆ D

St(A)

and upon output of Y ⊆ D

Eﬀ(B)

—or simply upon

X, Y —if all its reﬁnements are trivial upon X, Y ,

namely B ≺ A implies B =

Y X

λA, λ ∈ [0, 1].

Conversely, we say that an event is reﬁnable upon

X, Y whenever it is not atomic upon X, Y

Again, we mention as a special case the atom-

icity upon X = D

, Y = D

for some state

ρ ∈ St(A) and eﬀect b ∈ Eﬀ(B), which in turn

reduces to atomicity upon input of D

(atomicity

upon output of D

) when X = D

, Y = D

Eﬀ(B)

(X = D

St(A)

, Y = D

). Similarly, the usual

notion of atomicity corresponds to the choice

X = D

St(A)

, Y = D

Eﬀ(B)

Finally, the usual notion of faithful state and

faithful eﬀect (see Deﬁnition 8) can be generalized

to the notion of faithful state and faithful eﬀect

upon input of X and upon output of Y .

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Deﬁnition 23 (Faithful state and faithful ef-

fect upon X, Y ). Consider two arbitrary trans-

formations A, A

∈ Transf(A, B) and two sub-

sets X ⊆ D

St(A)

and Y ⊆ D

Eﬀ(B)

. A state

Ψ ∈ St(AC) is faithful upon input of X and upon

output of Y —or simply upon X, Y —if the con-

dition A |Ψ)

= A

|Ψ)

implies A =

Y X

Analogously, an eﬀect d ∈ Eﬀ(BC) is faithful

upon X, Y if the condition (d|

A = (d|

implies A =

Y X

The case of faithful state and faithful eﬀect

of Deﬁnition 8 corresponds to the choice X =

St(A)

, Y = D

Eﬀ(B)

in the above deﬁnition.

4.2 Information and disturbance upon X,Y

We start generalizing Deﬁnition 14 of non-

disturbing test:

Deﬁnition 24 (Non-disturbing test upon X, Y ).

Consider a test {A

}

i∈X

on system A. We say

that the test is non-disturbing upon input of X ⊆

St(A)

and upon output of Y ⊆ D

Eﬀ(A)

—or sim-

ply upon X, Y —if

Y X

. (29)

According to Deﬁnition 14 a test {A

}

i∈X

non-disturbing if it is operationally equal to the

identity transformation of system A; this is a spe-

cial case of the above deﬁnition when X = D

St(A)

and Y = D

Eﬀ(A)

We can now determine if a test {A

}

i∈X

⊆

Transf(A, C) provides information upon X ⊆

St(A)

, Y ⊆ D

Eﬀ(C)

. We ﬁrst observe that the

prescription upon X, Y establishes that (i) we are

only interested in getting information on states in

X and on eﬀects in Y , and (ii) we can only use

preparations that involve states in X and mea-

surements that involve eﬀects in Y in order to

extract the information. However, a test contain-

ing a state in X (or an eﬀect in Y ) may involve

events in

X (

Y ), where the set

Z is the coexistent

completion of the set Z as in Deﬁnition 6.

Let us focus on the scheme in Eq. (8), with the

state in the set X and eﬀect in the set Y . A test

that directly provides information on X may give

diﬀerent probability distributions for diﬀerent el-

ements of X, given that one measures an eﬀect

c ∈ Y . However, the test could also provide infor-

mation about X indirectly. This is the case, for

example, when for every test {c

}

k∈Z

⊆

Y , one

has p(j, i|k) = p(i)p(j) for Ψ

∈ X, while factori-

sation does not occur for Ψ

∈

X \ X. A similar

situation can occur when the information is about

Y , with the roles of preparation and measurement

exchanged.

We thus generalize Deﬁnition 15 of no-

information test as follows:

Deﬁnition 25 (No-information test upon X, Y ).

A test {A

}

i∈X

with events A

∈ Transf(A, C) is

a no-information test upon input of X ⊆ D

St(A)

and upon output of Y ⊆ D

Eﬀ(C)

—or simply upon

X, Y —if for every choice of deterministic eﬀect

∈

Y and deterministic state ω

∈

X, there

exists a deterministic eﬀect f

and a determin-

istic state ν

such that for every i ∈ X one has

(e|

(e) (f|

, (30)

|ω)

(ω) |ν)

. (31)

As a special case the above deﬁnition coincides

with Deﬁnition 15, corresponding to X = D

St(A)

and Y = D

Eﬀ(A)

. Notice that in this case it is

also

X = X and

Y = Y .

According to Eq. (30), the test {A

}

i∈X

does

not provide information upon input of

X once

the observations are limited to

Y (e

∈

Y ).

However, the probability distribution p

(e) might

in principle provide information about the eﬀect

e. On the other hand according to Eq. (31), the

test {A

}

i∈X

does not provide information upon

output of

Y , once the preparations are limited

X (ω

∈

X), while the probability distri-

bution q

(ω) might in principle provide informa-

tion about the state ω. The conjunction of the

two conditions implies that no-information is pro-

vided by the test about

X and

Y :

Lemma 4. Let the test {A

}

i∈X

with events A

∈

Transf(A, C) be a no-information test upon X, Y .

Then one has

(e|

(f|

, (32)

|ω)

|ν)

. (33)

Proof. By Eqs. (30) and (31), and remembering

that ω

∈

X, e

∈

Y , one has

(e|

|ω)

= p

(e) = q

(ω) = r

where we used the fact that e, f and ω, ν are re-

spectively deterministic eﬀects and deterministic

states. 

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Here we state the condition of no-information

without disturbance upon X, Y :

Deﬁnition 26 (No-information without distur-

bance upon X, Y ). Consider a system A and two

subsets X ⊆ D

St(A)

, Y ⊆ D

Eﬀ(A)

. Then the OPT

satisﬁes no-information without disturbance upon

input of X and upon output of Y —or simply upon

X, Y —if for every test {A

}

i∈X

⊆ Transf(A) that

is non-disturbing upon X, Y , the test does not

provide information upon X, Y .

Clearly the above deﬁnition generalizes Deﬁ-

nition 16 which corresponds to the choice X =

St(A)

, and Y = D

Eﬀ(A)

, namely the theory sat-

isﬁes no-information without disturbance and any

informative test necessary disturbs.

All the results presented in Section 3.1 on no-

information without disturbance can now be ex-

tended to the “upon X, Y ” scenario. The result

of Theorem 1, proving the atomicity of the iden-

tity as a necessary and suﬃcient condition for no-

information without disturbance, is generalized

by the following theorem:

Theorem 4. An OPT satisﬁes no-information

without disturbance upon

Y , with X ⊆ D

St(A)

Y ⊆ D

Eﬀ(A)

, if and only if the identity I

atomic upon

Y .

Proof. The proof follows the lines of that of The-

orem 1. In the present case Eq. (17) for pure

state in

X holds upon

Y . Now, if we apply on

both sides a deterministic eﬀect e

∈

Y , using

Eq. (32) it may happen that (e|Ψ) = 0. The case

where this happens for every e ∈

Y corresponds

to a state Ψ which is equal to the null state

upon

Y . Considering the remaining cases

one concludes that A

|ρ) =

|ρ), ∀ρ ∈

Similarly, one concludes that (b| A

(b|,

∀b ∈

Y . These two last conditions imply that

, corresponding to atomicity of I

upon

Y . Also in this case the proof follows

straightforwardly from that of Theorem 1. The

opposite implication is trivial. 

Analogously, one can generalize the other nec-

essary and suﬃcient conditions of Section 3.1.

One can also provide only suﬃcient conditions for

no-information without disturbance. An example

is given in the following proposition:

Proposition 6. An OPT satisﬁes no-

information without disturbance upon

Y ,

with X ⊆ D

St(A)

, Y ⊆ D

Eﬀ(A)

, if there exists a

pure state Ψ ∈ Ref

that is faithful upon

Y .

Similarly, an OPT satisﬁes no-information

without disturbance upon

Y if there exists an

atomic eﬀect b ∈ Ref

that is faithful upon

Y .

Proof. We explicitly prove the case of faithful

state, since that of faithful eﬀect follows by anal-

ogy. Given a system A, let Ψ ∈ Ref

be pure and

faithful upon

Y (see Deﬁnition 23). Now let

the test {A

}

i∈X

∈ Transf(A) be non-disturbing

upon

Y , namely

. Then, since

Ψ ∈ Ref

we have

|Ψ) = |Ψ), and since Ψ

is pure, there exists a set of probabilities {p

}

i∈X

such that A

|Ψ) = p

|Ψ). However, due to the

faithfulness of Ψ, the map A 7→ A |Ψ) is injective

upon

Y , and we conclude that A

and, by deﬁnition, the test {A

}

i∈X

∈ Transf(A)

does not extract information upon

Y . 

As a corollary one has a suﬃcient condition

for no-information without disturbance with no

restrictions on inputs and outputs:

Corollary 2. An OPT satisﬁes no-information

without disturbance if for every system A there

exists a pure state faithful for A or an atomic

eﬀect faithful for A.

5 Outlook on no-information without

disturbance

In this last section we analyse the relation be-

tween no-information without disturbance and

other properties of operational probabilistic the-

ories. We focus on local discriminability (see

Deﬁnition 9) and puriﬁcation (see Deﬁnitions 11

and 12) that, being typical quantum features, are

commonly associated with no-information with-

out disturbance. Here instead we show that no-

information without disturbance can actually be

satisﬁed independently of the above two proper-

ties.

No-information without disturbance vs. puriﬁca-

tion

The following proposition proves that the proba-

bilistic theory [27, 28, 29, 30, 31] corresponding

to the PR-boxes model, originally introduced in

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Ref. [32], satisﬁes no-information without distur-

bance.

Proposition 7. The PR-boxes theory satisﬁes

no-information without disturbance.

Proof. This can be proved in several ways. For

example we show that any system of the theory

allows for a reversible atomic transformation and

then use Proposition 3. The fact that any sys-

tem has a reversible atomic transformation fol-

lows from the following three points. I) The re-

versible transformations of the elementary sys-

tem A of the theory (the convex set of normalized

states of A is represented by a square, and the set

of reversible transformations of A coincides with

the set of symmetries of a square, the dihedral

group of order eight D

, containing four rotations

and four reﬂections) are atomic [29]. II) From

Refs. [33, 34] we know that the set of reversible

maps of the N-partite system A

⊗N

is generated

by local reversible operations plus permutations

of the systems. Accordingly, the system A

⊗N

al-

lows for a multipartite reversible transformation

⊗ U

made of local reversible transfor-

mations U

, i = 1, . . . N. III) Since PR-boxes sat-

isfy local discriminability, the chosen multipartite

transformation is atomic due to the atomicity of

parallel composition (see Proposition 2). 

As a corollary one has that also PR-boxes with

minimal tensor product satisfy no-information

without disturbance.

Corollary 3. The PR-boxes theory with mini-

mal tensor product satisﬁes no-information with-

out disturbance.

Proof. Consider PR-boxes theory with minimal

tensor product [14]. We remind that in a prob-

abilistic framework, the minimal tensor product

of state sets (or eﬀect sets) of two systems is the

operation that yields the set of states (or eﬀects)

of the composite system as containing only prod-

uct states (or product eﬀects) and their prob-

abilistic mixtures. The PR-boxes theory with

minimal tensor product has the same elemen-

tary system as the PR-boxes theory (with the

convex set of normalized states represented by a

square), but with composite systems constrained

by the minimal tensor product for both states

and eﬀects. The proof that this probabilistic the-

ory satisﬁes no-information without disturbance

is as in Proposition 7 (one proves that every sys-

tem has a reversible atomic transformation and

no-information without disturbance follows from

Proposition 3). 

We can now establish the independence be-

tween no-information without disturbance and

puriﬁcation (see also Figure 1).

Proposition 8. No-information without distur-

bance and states or eﬀects puriﬁcation are inde-

pendent.

Proof. We prove that:

1. Puriﬁcation ; No-information without dis-

turbance.

2. No-information without disturbance ; Pu-

riﬁcation

1. Consider deterministic classical theory.

This is classical theory where the probabilities

of outcomes in any test are either 0 or 1 (see also

Ref. [10]). One can easily see that in this prob-

abilistic theory all states are pure and all eﬀects

are atomic. As a consequence the theory satisﬁes

both states and eﬀects puriﬁcation according to

Deﬁnitions 11 and 12. On the other hand, in this

theory all information can be extracted without

disturbance.

2. Consider PR-boxes theory with minimal

tensor product. On one hand this probabilis-

tic theory satisﬁes no-information without distur-

bance (see Corollary 3). On the other hand, one

can check that the theory satisﬁes neither states

puriﬁcation, nor eﬀects puriﬁcation. First notice

that, due to the minimal tensor product prescrip-

tion, multipartite pure states and atomic eﬀects

are tensor products of local pure states and lo-

cal atomic eﬀects, respectively. Therefore, given

a multipartite pure state (atomic eﬀect) all its

marginals are also pure states (atomic eﬀects).

It follows that a state (eﬀect) that is not pure

(atomic) cannot admit of any pure (atomic) di-

lation. Since the elementary system of PR-boxes

with minimal tensor product includes both non

pure states and non atomic eﬀects, the theory

does not satisfy states or eﬀects puriﬁcation. 

While puriﬁcation is not enough to imply no-

information without disturbance, in the next

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•

FQT

•

RQT

•

Local discriminability

•

BCT

•

DCT

Puriﬁcation

No-info w. disturbance

Puriﬁcation

& convexity

Figure 1: Comparing OPTs that satisfy no-information

without disturbance (grey set), local discriminability (red

set) and states or eﬀects puriﬁcation (blue set). Quan-

tum theory (QT) lies at the intersection of the three

sets. As proved in Proposition 8 no-information with-

out disturbance and puriﬁcation are independent fea-

tures. An example of OPT that satisﬁes no-information

without disturbance but violates puriﬁcation is the PR-

boxes theory with minimal tensor product (PR). More-

over, PR-boxes satisfy local discriminability, providing

a non-trivial intersection between local discriminability

and no-information without disturbance in the absence

of puriﬁcation. On the other hand determinisctic clas-

sical theory (DCT) is an example of OPT that satisﬁes

puriﬁcation but violates no-information without distur-

bance. As proved in Proposition 9 the set of convex

OPTs with puriﬁcation is a proper subset of OPTs with

no-information without disturbance. We observe that

no-information without disturbance is also independent

of local discriminability, as proved in Proposition 10. In-

deed classical theory (CT) satisﬁes only local discrim-

inability while Fermionic quantum theory (FQT) and real

quantum theory (RQT) satisfy only no-information with-

out disturbance. Finally, it has been shown in Ref. [35]

that there exist OPTs without local discriminability, that

have all systems classical, thus retaining the possibility

of extracting all the information without disturbance.

An example is the bilocal classical theory (BCT) of the

same Ref. [35], which satisﬁes 2-local discriminability

(see Deﬁniton 10).

proposition we show that convex OPTs

that

satisfy states or eﬀect puriﬁcation are a subset

of the OPTs with no-information without distur-

bance (actually they are a proper subset, see also

Fig. 1). This provides another useful suﬃcient

condition for no-information without disturbance

Actually one might relax the convexity hypothesis

with the following weaker condition: If two states belong

to a jointly perfectly discriminable set, there exists an el-

ement of the set that can be convexly mixed with both. In

particular, this is the case if some convex combination of

the two states is a state of the theory.

(see also Figure 1).

Proposition 9. A convex OPT with states

puriﬁcation or eﬀects puriﬁcation satisﬁes no-

information without disturbance.

Proof. We consider the case of states puriﬁcation

(see Deﬁnition 11), with the proof for eﬀects pu-

riﬁcation (see Deﬁnition 12) following by analogy

according to Remark 1.

Given a convex OPT with states puriﬁcation

suppose that it violates no-information without

disturbance, namely there exists a system A such

that I

is not atomic. Then let I

for some atomic non redundant test {A

}

i∈X

⊆

Transf(A). Let us consider a mixed state St(A) 3

|ρ) =

i∈X

|σ

) with p

|σ

)

= A

|ρ), and

}

i∈X

a probability distribution with p

> 0

∀i. Then by Theorem 2 we have A





|σ

). Since the theory allows for puriﬁcation,

let Ψ ∈ St(AB) be a puriﬁcation of ρ for deter-

ministic eﬀect e ∈ Eﬀ(B). Now, one one hand

since the test {A

}

i∈X

reﬁnes the identity, it is

|Ψ)

i∈X

|Ψ)

, and being Ψ pure it

must be A

|Ψ)

= q

|Ψ)

, with {q

}

i∈X

probability distribution. On the other hand, for

every i 6= j the marginals with deterministic ef-

fect e ∈ Eﬀ(B) of A

|Ψ)

and A

|Ψ)

are

perfectly discriminable. But this contradicts the

fact that A

|Ψ)

and A

|Ψ)

are both pro-

portional to Ψ. 

No-information without disturbance vs. local dis-

criminability

Turning to the case of local discriminability (see

Deﬁnition 9), we now show that it is independent

of no-information without disturbance (see also

Figure 1).

Proposition 10. No-information without dis-

turbance and local discriminability are indepen-

dent.

Proof. We prove that:

1. Local discriminability ; No-information

without disturbance,

2. No-information without disturbance ; Lo-

cal discriminability.

1. Classical theory satisﬁes local discriminabil-

ity but violates no-information without distur-

bance, since in this theory all information can be

extracted without disturbance.

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2. Fermionic [16, 18, 17] and real quantum the-

ory [23, 17, 18] both violate local discriminabil-

ity as proved in Refs. [23, 17, 18] where it is

also shown that they are both 2-local theories

according to Deﬁnition 10. On the other hand

Fermionic and real quantum theories are con-

vex theories with states puriﬁcation and, due to

Proposition 9, both satisfy no-information with-

out disturbance. 

Finally, we observe that as a consequence of

Corollary 3, and subsequent Remark 7, the clas-

sical theory of information is the only theory with

local discriminability in which all the informa-

tion can be extracted without disturbance. How-

ever, in the absence of local discriminability, it

is still possible to have other theories where all

the information can be extracted without distur-

bance. This has been proved in Ref. [35] where

the authors describe an OPT whose systems of

any dimension are classical (and then violate no-

information without disturbance), but with a par-

allel composition that diﬀers from the usual clas-

sical one, leading to a violation of local discrim-

inability, more precisely to a 2-local theory ac-

cording to Deﬁnition 10. This theory is inter-

esting because it provides an example of OPT

that violates simultaneously no-information with-

out disturbance, local discriminability and puriﬁ-

cation (see bilocal classical theory (BCT) in Fig-

ure 1).

6 Conclusions

We have analysed the interplay between infor-

mation and disturbance for a general operational

probabilistic theory, considering the eﬀect of mea-

surements also on correlations with the environ-

ment, diﬀerently from the traditional approach

focused only on the measured system. Indeed,

the two resulting notions of disturbance coincide

only in special cases, such as quantum theory,

as well as every theory that satisﬁes local dis-

criminability. Our approach is universal for any

OPT, including theories without causality, puriﬁ-

cation or convexity. In this setting we proved that

the atomicity of the identity transformation is an

equivalent condition for no-information without

disturbance.

We have characterized the structure of theo-

ries where the identity is not atomic, showing

that in this case the information that can be ex-

tracted without disturbance is “classical”, in the

sense that it is sharable and repeatable. On the

other hand, we have established that every OPT

entails information whose extraction requires dis-

turbance, with the only exception of theories with

all systems classical.

While no-information without disturbance is a

consequence of convexity along with puriﬁcation

(puriﬁcation of states or of eﬀects), we proved

that puriﬁcation and no-information without dis-

turbance are independent. Similarly, we have

shown that no-information without disturbance

and local discriminability are independent prop-

erties.

Our results are expected to have immediate

applicability to secure key-distribution. Indeed,

a physical theory including a system (or even

just a set of states of a system) that satisﬁes

no-information without disturbance can guaran-

tee a private and reliable channel for distribut-

ing messages. The idea of studying secure key-

distribution in a framework more general than the

classical and the quantum ones has been proposed

in Refs. [12, 14]. In Ref. [12] it has been conjec-

tured that in every theory that is not classical

secure key-distribution is possible. The present

generalisation of no-information without distur-

bance to arbitrary OPTs is a ﬁrst step in proving

such a conjecture.

Acknowledgments

This publication was made possible thanks to the

ﬁnancial support of Elvia and Federico Faggin

Foundation.

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A Transformations induced by events

In the operational framework any event A induces a map between states. Consider for example the

event

For every choice of ancillary system C, and for every state Ψ ∈ St(AC), the event A maps the state Ψ

to the state given by the following circuit

Accordingly, while states and eﬀects are linear functionals over each other, we can always look at an

event as a map between states

|Ψ)

∈ St(AC) 7→ A |Ψ)

∈ St(BC), (34)

(and similarly as a map between eﬀects, from Eﬀ(AC) to Eﬀ(BC)). The map above can be linearly

extended to a map from St

(AC) to St

(BC) (we denote the extended map with the same symbol A)

and this extension is unique. Indeed a linear combination of states of AC is null, say

|Ψ

), if and

only if

(a|Ψ

) = 0 for any a ∈ Eﬀ(BC). Moreover, since (b|

A ∈ Eﬀ(AC) for every b ∈ Eﬀ(B),

then ∀b ∈ Eﬀ(B) we have that 0 = (b|

|Ψ

)

) =

(b|

A |Ψ

)

= (b|

A |Ψ

)

and we ﬁnally get

A |Ψ

)

= 0.

B No-information test

We show that the weak condition 2 for no-information test is equivalent to one in Deﬁnition 15.

We focus now on no-information on the input (the case of no-information on the output following by

analogy) and show that Eqs. (9) and (11) are equivalent. We ﬁrst prove that Eq. (11) implies Eq. (9).

To this end we evaluate the left hand side of Eq. (9) using Eq. (11), namely

p(j, i, k|Ψ

, A

, C

) =

| A





= (e| A





= p

(e)(f|Ψ

Now we notice that p

(e), which is a probability distribution on the outcomes of the test A

, also

depends on the deterministic eﬀect e =

. Therefore p

(e) is a probability distribution that depends

on both the test A

and on the observation test C

while it does not depend on the preparation

test Ψ

, exactly as the probability distribution r on the right hand side of Eq. (9). Finally, we

notice that (f|Ψ

) is a probability distribution on the outcomes of the preparation test Ψ

, and

the deterministic eﬀect f can depend on both tests A

and C

. We then conclude that (f|Ψ

) is

a probability distribution that generally depends on all test Ψ

, A

and C

, as the probability

distribution s on the right hand side of Eq. (9). Now we check the other implication, namely that

Eq. (9) implies Eq. (11). Due to Eq. (9), one has

(e| A





| A





p(j, i, k|Ψ

, A

, C

)

= r(i|A

, C

)s(j|Ψ

, A

, C

(35)

First we notice that summing over the index i in the ﬁrst and last member of Eq. (35) we get

s(j|Ψ

, A

, C

) = (f|Ψ

), where (f| =

(e| A

is a deterministic eﬀect depending on tests A

, C

(indeed

is a deterministic transformation). Therefore one has (e| A

= r(i|A

, C

) (f|. Since

on the left hand side of the last identity the only dependence on the observation test C

is through

the deterministic eﬀect e, one ﬁnally gets that r(i|A

, C

) is of the form p

(e), as in Eq. (11).

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C Techical observation

Given a transformation A ∈ Transf(A) such that for every system B there exists p

such that

A |ρ)

= p

|ρ)

, (36)

then actually p

≡ p cannot depend on the system B. Indeed, choosing ρ = τ ⊗ σ

with arbitrary

τ ∈ St(A) and normalised σ ∈ St(B), and discarding system B on both sides of Eq. (36) ∀ρ ∈ St(AB),

one obtains A |τ ) = p

|τ). This clearly shows that p

≡ p.

D Techical lemma

Lemma 5. Given a state ρ ∈ St(A) one has Ref

⊆ D

Ref

, where Ref

denotes the union of the

reﬁnements of any state in D

. Given an eﬀect a ∈ Eﬀ(A) one has Ref

⊆ D

Ref

, where Ref

denotes the union of the reﬁnements of any eﬀect in D

Proof. Since the proof in the case of states and eﬀects is exactly the same (see Remark 1) we focus

on the former. Consider a system B and a state Ψ ∈ St(AB):

1. Ψ ∈ D

Ref

iﬀ (e|

|Ψ)

= |σ)

, with σ ∈ Ref

, and e ∈ Eﬀ(B) deterministic.

2. Ψ ∈ Ref

iﬀ there exists a state Ω ∈ St(AB), with Ω ∈ D

such that Ψ ∈ Ref

Ω

, i.e. there exists

a reﬁnement {Ψ

}

i∈X

of Ω such that Ψ ∈ {Ψ

}

i∈X

By hypothesis {Ψ

}

i∈X

is a reﬁnement of Ω, namely

|Ω)

i∈X

|Ψ

)

, (37)

and Ω ∈ D

, namely (e|

|Ω)

= |ρ)

. Accordingly, marginalising both sides of Eq. (37) one has

that for every i ∈ X, (e|

|Ψ

)

= |σ

)

∈ Ref

. Since Ψ ∈ {Ψ

}

i∈X

, this concludes the proof. 

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