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
1
. 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
1
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
}
iX
is the collection of events A
i
, where i labels the
element of the outcome space X. In the quan-
tum case A
i
is the ith quantum operation of
the quantum instrument {A
i
}
iX
. The notion of
test bridges the experiment with the theory, with
i X denoting the objective outcome, and A
i
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
X
:
= {A
i
}
iX
and its building events A
i
by the
diagrams
A
A
X
B
,
A
A
i
B
,
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
X
and B
Y
we
can define their sequential composition (BA)
X×Y
as the collection of events
A
B
j
A
i
C
=
A
A
i
B
B
j
C
,
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
A
} such that
I
B
A = AI
A
= A for every event A with input
A and output B, and we call I
A
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
i
D
j
in parallel com-
position represented as follows
A
C
i
D
j
B
C D
=
A
C
i
B
C
D
j
D
,
and satisfying the condition (E
h
F
k
)(C
i
D
j
) =
(E
h
C
i
) (F
k
D
j
). 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
ρ
A
(or in formula as round kets |ρ)
A
), and
the observation events by boxes with no output
wire
A
c
(in formula round bras (c|
A
). For ex-
ample, one can have events of the following kind
ρ
i
B
C
D
j
D
=
C
ρ
i
D
j
BD
.
We will always use the Greek letters to denote
preparation tests {ρ
i
}
iX
and Latin letters to de-
note observation tests {c
j
}
jX
(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
i
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after the preparation event ρ
j
corresponds to the
closed circuit (c
i
|ρ
j
)
A
and denotes the probabil-
ity of the outcome (i, j) of the observation test
c
X
after the preparation test ρ
Y
of system A, i.e.
ρ
j
A
c
i
:
= Pr
i, j
ρ
Y
A
c
X
.
For a more complex example, consider the test
T
U
:
=
Ψ
V
A
A
W
B
B
X
C
E
Y
D E
C
Z
F
G
,
with U = V × W × X × Y × Z. Then we define
Ψ
i
A
A
j
B
B
k
C
E
m
D E
C
l
F
G
:
= Pr[i, j, k, l, m|T
U
].
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)
A
|σ)
A
(c|ρ)
A
=
(c|σ)
A
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)
A
and (c|
A
, 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
R
(A) and
Eff
R
(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
R
(AC) to St
R
(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
0
Transf(A, B), one has A = A
0
, if and only
if
Ψ
A
A
B
a
C
=
Ψ
A
A
0
B
a
C
,
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
0
if and only if
Ψ
A
A
B
C
=
Ψ
A
A
0
B
C
, (1)
for every C, and every Ψ St(AC), or
A
A
B
a
C
=
A
A
0
B
a
C
(2)
for every C, and every a Eff(BC).
In the following, the symbols A and
A
A
B
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
R
(A, B). It is now obvious
that a linear map A Transf
R
(A, B) is admissi-
ble if it locally preserves the set of states St(AC),
namely A I
C
(St(AC)) St(BC), for every sys-
tem C. In the following we will write A |Ψ)
AC
instead of A I
C
|Ψ)
AC
, 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
2
.
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
}
iX
from A to B, such
that there exists a test {D
i
}
iY
with X Y and
C =
P
iX
D
i
. We say that a refinement {D
i
}
iX
of C is trivial if D
i
= λ
i
C, λ
i
[0, 1], for every
2
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
}
iX
, which we will also denote
as C = D
X
.
In the following we will often refer to a refine-
ment of C simply as C =
P
iX
D
i
, without speci-
fying the test including the events D
i
.
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
}
iX
of C such that D {D
i
}
iX
.
Definition 3 (Non redundant test). We call a
test {A
i
}
iX
non redundant when for every pair
i, j X one has A
i
6= λA
j
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
C
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
}
iX
Transf(A, B) such that A = C
Y
and B = C
Z
, where Y, Z X. Given an event
A we denote by
b
A the set of all events coexistent
with A, and more generally, given a set of events
X its coexistent completion is defined as
b
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
ρ
A
=
Ψ
A
B
e
for some deterministic effect e Eff(B). Anal-
ogously, c Eff(AB) is a dilation of a Eff(A)
if
A
a
=
A
c
ω
B
for some deterministic state ω St(B). We de-
note by D
ρ
the set of all dilations of the state
ρ. More generally, given a collection of states
S St(A) we define D
S
:
=
S
ρS
D
ρ
, 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
S
.
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
0
Transf(A, B),
the condition A |Ψ)
AC
= A
0
|Ψ)
AC
implies A =
A
0
. Analogously, an effect d Eff(BC) is faith-
ful for B if given two transformations A, A
0
Transf(A, B), the condition (d|
BC
A = (d|
BC
A
0
implies A = A
0
.
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
0
St(AB) with Ψ 6= Ψ
0
, there exist two
effects a Eff(A) and b Eff(B), such that
Ψ
A
a
B
b
6=
Ψ
0
A
a
B
b
.
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
0
Transf(A, B), the condition A |ρ)
A
= A
0
|ρ)
A
for
every ρ St(A) implies A = A
0
.
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)
R
Eff(A)
R
Eff(B)
R
.
Thus, any bipartite effect c Eff(AB) can be
written as a linear combination of product ef-
fects, and every probability (c|ρ)
AB
, for ρ
St(AB), can be computed as a linear combina-
tion of the probabilities ((a|
A
(b|
B
) |ρ)
AB
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
0
are different, there exist a n-
local effect b such that (b|ρ) 6= (b|ρ
0
). 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
}
iX
St(A) is a prepa-
ration test, then the conditional probability of the
preparation ρ
i
given the choice of the observation
test {a
j
}
jY
is the marginal
Pr
i|{a
j
}
:
=
X
jY
(a
j
|ρ
i
)
A
.
In a causal theory the marginal probability
Pr
i|{a
j
}
is independent of the choice of the
observation test {a
j
}: if {a
j
}
jY
and {b
k
}
kZ
are two different observation tests, then one has
Pr
i|{a
j
}
= Pr
i|{b
k
}
.
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
e
A
. We call the effect e
A
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
}
iX
on system A is usually said to
be non-disturbing if for every ρ St(A) one has
that
P
i
A
i
|ρ)
A
= |ρ)
A
. 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
}
iX
such that
P
i
A
i
|ρ)
A
= |ρ)
A
for
every ρ St(A) still can disturb the states of a
composite system AB.
The parity superselection rule on a system N
F
of N Fermions forbids any state corresponding
to a superposition of vectors belonging to F
e
N
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and F
o
N
, representing Fock vector spaces with
total even and odd occupation number, respec-
tively. As a consequence, the linearized set of
states St
R
(N
F
) 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
e
, for the even space, and P
o
, for the
odd one. Notice that, since P
e
P
o
= P
o
P
e
= 0
any Fermionic state ρ will be of the form ρ =
P
e
ρP
e
+ P
o
ρP
o
. Consequently the parity test
{P
e
·P
e
, P
o
·P
o
} leaves every state ρ St(N
F
) 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
F
), with
ρ = p
e
ρ
e
+ p
o
ρ
o
, ρ
e
and ρ
o
an even and an odd
pure state respectively, and p
e
+ p
o
= 1. For
example, consider the states
ρ
e
= |00ih00| , ρ
o
= |01ih01| ,
and p
e
= p
o
= 1/2, so that
ρ =
1
2
(|00ih00| + |01ih01|).
Since Fermionic theory allows for states purifi-
cation [17] (see Definition 11), we can always find
a state Ψ St(M
F
), with M > N that purifies ρ.
Since Ψ is pure, it has a definite parity, say even.
In our example one can choose
Ψ =
1
2
(|000i + |011i)(h000| + h011|). (4)
Therefore, the local test on the system N
F
that
measures the parity of the system will not disturb
the states of N
F
but will decohere the state Ψ to a
mixed state, then introducing a disturbance. For
example, in our case
(P
e
I)Ψ(P
e
I) + (P
o
I)Ψ(P
o
I)
=
1
2
(|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
}
iX
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
AB
St(AB) of σ one has that
P
i
A
i
|Ψ)
AB
= |Ψ)
AB
. 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
}
iX
on system A. We say that the test
is non-disturbing if
X
i
A
i
= I
A
. (5)
Notice that, following the above definition, the
test {A
i
}
iX
is disturbing if there exist Ψ
D
St(A)
, and c D
Eff(A)
, such that
X
iX
Ψ
A
A
i
A
c
B
6=
Ψ
A
c
B
. (6)
This definition of disturbance thus stresses the
effect of a transformation on correlations with
remote systems, indeed a test {A
i
}
iX
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
X
i
A
i
= 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
A
, and
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RW = I
C
. 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
R}
iX
,
with {A
i
}
iX
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
X
:
= {A
i
}
iX
Transf(AC). Op-
erationally, in the most general case the test A
X
can be used in composite tests made of a prepa-
ration test Ψ
Y
:
= {Ψ
j
}
jY
St(AB), the test
A
X
:
= {A
i
}
iX
Transf(AC) and an observa-
tion test C
Z
:
= {c
k
}
kZ
Eff(CB), leading to
the joint probabilities
Ψ
j
A
A
i
C
c
k
B
= p(j, i, k|Ψ
Y
, A
X
, C
Z
)
(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
A
X
and any observation test C
Z
in order to ex-
tract the information on the inputs, while he can
use both the test A
X
and any preparation test
Ψ
Y
in order to extract the information on the
outputs. This leaves room for two inequivalent
conditions for a no-information test A
X
.
1. Strong condition for no-information test.
(a) No-information on inputs: the test A
X
is no-information on inputs if for every
preparation test Ψ
Y
and for every obser-
vation tests C
Z
, the joint probability in
Eq. (8) factorizes as
p(j, i, k|Ψ
Y
, A
X
, C
Z
)
= r(i|k; A
X
, C
Z
)s(j, k|Ψ
Y
, A
X
, C
Z
),
namely we impose that
r(i|j, k; Ψ
Y
, A
X
, C
Z
) = r(i|k; A
X
, C
Z
),
where the probability distribution r does
not depend on the preparation test Ψ
Y
(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
X
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
X
is no-information on outputs if for every
C
Z
and for every Ψ
Y
the joint probability
in Eq. (8) factorizes as
p(j, i, k|Ψ
Y
, A
X
, C
Z
)
= r(i|j; Ψ
Y
, A
X
)s(j, k|Ψ
Y
, A
X
, C
Z
),
where the probability distribution r does
not depend on the observation test C
Z
.
This condition ensures that the outcomes
of A
X
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
X
is no-information on inputs if for every
preparation test Ψ
Y
and for every obser-
vation test C
Z
, the joint probability in
Eq. (8) is such that
X
k
p(j, i, k|Ψ
Y
, A
X
, C
Z
)
= r(i|A
X
, C
Z
)s(j|Ψ
Y
, A
X
, C
Z
),
(9)
where the probability distribution r does
not depend on the preparation test Ψ
Y
.
The interpretation of this condition is
that the outcomes of A
X
do not provide
information on the preparation, whenever
we ignore the outcome of the observation
test.
(b) No-information on outputs: the test A
X
is no-information on outputs if for every
observation test C
Z
and for every prepa-
ration test Ψ
Y
, the joint probability in
Eq. (8) is such that
X
j
p(j, i, k|Ψ
Y
, A
X
, C
Z
) =
r(i|Ψ
Y
, A
X
)s(k|Ψ
Y
, A
X
, C
Z
),
(10)
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where the probability distribution r does
not depend on the observation test C
Z
.
This means that the outcomes of A
X
do
not provide information on the observa-
tion, whenever we ignore the outcome of
the preparation test.
It is elementary to see that 1a2a
(and 1b2b), 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
{A
i
}
iX
with events A
i
Transf(A, C) is a no-
information test if, for every choice of determin-
istic effect e
CB
and deterministic state ω
AB
, there
exists a deterministic effect f
AB
and a determin-
istic state ν
CB
such that for every i X one has
(e|
CB
A
i
= p
i
(e) (f|
AB
, (11)
A
i
|ω)
AB
= q
i
(ω) |ν)
CB
. (12)
According to Eq. (11) (that coincides with
Eq. (9) in the weak condition of item 2a), the
test {A
i
}
iX
does not provide information upon
any possible input state. However, the proba-
bility distribution p
i
(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
}
iX
does not provide information upon
output of any possible effect, while the probabil-
ity distribution q
i
(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
}
iX
with events A
i
Transf(A, C) be a no-information test. Then one
has
(e|
CB
A
i
= r
i
(f|
AB
, (13)
A
i
|ω)
AB
= r
i
|ν)
CB
. (14)
Proof. By Eqs. (11) and (12) one has
(e|
CB
A
i
|ω)
AB
= p
i
(e) = q
i
(ω) = r
i
,
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
i X generally
depends on the deterministic effect e
CB
, this ac-
counting for non-causal theories. In the more
general case in which also the deterministic ef-
fect f
AB
on the right hand side of Eq. (11) de-
pends on i X, the test {A
i
}
iX
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
}
iX
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
}
iX
(A
i
Transf(A) for every i X) of the identity
map I
A
=
P
i
A
i
for system A. The test {A
i
}
iX
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
CB
, and de-
terministic state ω
AB
, there exists a determinis-
tic effect f
AB
and a deterministic state ν
AB
such
that for every i X one has (e|
AB
A
i
= r
i
(f|
AB
,
and A
i
|ω)
AB
= r
i
|ν)
AB
. Summing both sides of
the last equation over the index i X, and re-
membering that
P
iX
r
i
= 1, we find that e = f
and ω = ν. Therefore, the no-information condi-
tion is
(e|
AB
A
i
= r
i
(e|
AB
, (15)
A
i
|ω)
AB
= r
i
|ω)
AB
, (16)
for every deterministic effect e
AB
and for every
deterministic state ω
AB
. 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
P
i
A
i
|Ψ)
AB
= |Ψ)
AB
, it
follows that
A
i
|Ψ)
AB
= λ
i
(Ψ) |Ψ)
AB
,
X
i
λ
i
(Ψ) = 1, (17)
where the coefficients λ
i
(Ψ) generally depend on
the state Ψ. However, for each pure state Ψ there
exists a deterministic effect e
Ψ
Eff(AB) such
that (e
Ψ
|Ψ) 6= 0. Upon applying the determinis-
tic effect e
Ψ
on both sides of Eq. (17), we get
(e
Ψ
|A
i
|Ψ)
AB
= λ
i
(Ψ)(e
Ψ
|Ψ)
AB
. (18)
Now, applying both sides of Eq. (15) to Ψ, we
get
(e
Ψ
|A
i
|Ψ)
AB
= r
i
(e
Ψ
|Ψ)
AB
, (19)
and comparing the last two identities, consider-
ing that (e
Ψ
|Ψ)
AB
6= 0, we obtain
λ
i
(Ψ) = r
i
, i X. (20)
Since this holds true for every pure state Ψ, we
conclude that λ
i
(Ψ) is independent of Ψ. Then
A
i
|ρ)
AB
= r
i
|ρ)
AB
, ρ St(AB), proving that
A
i
= r
i
I
A
. Notice that we implicitly assumed
that the probabilities r
i
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
}
iX
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
}
iX
Transf(A) is such that A
i
= q
i
I
A
, with
P
i
q
i
= 1.
It follows that in Eq. (8) the joint probability
distribution p(j, i, k) is of the form p(j, i, k) =
q
i
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
A
=
P
i
A
i
, with A
i
Transf(A) for i X, of the identity transforma-
tion. By definition of identity map we have that
RI
A
=
P
i
RA
i
= R, and due to the atomic-
ity of R it must be RA
i
R for every i X.
Since R is left-reversible (namely there exists
W Transf(C, A) such that WR = I
A
) it fol-
lows that A
i
I
A
for every i X, which proves
the atomicity of I
A
.
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 =
P
iX
R
i
for test {R
i
}
iX
. Then, one has
X
iX
R
i
R
1
= I
A
, (21)
and by Theorem 1 one has that R
i
R
1
= p
i
I
A
.
Finally, multiplying by R to the right, we con-
clude that R
i
= p
i
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
}
iX
Transf(A) of the identity
I
A
=
P
i
A
i
, one has A
i
A
j
= A
i
δ
ij
.
Proof. Suppose that the identity transformation
of system A allows for two atomic refinements
I
A
=
P
iX
A
i
, and I
A
=
P
jY
B
j
. Since
P
i
A
i
B
j
= B
j
, from the atomicity of the transfor-
mations B
j
we get A
i
B
j
= c
ij
B
j
, for some c
ij
0
such that
P
iX
c
ij
= 1 j Y. Similarly we
get A
i
B
j
= d
ij
A
i
for some d
ij
0 such that
P
jY
d
ij
= 1 i X. Then c
ij
B
j
= d
ij
A
i
. By
non redundancy one has that for fixed j there is
only one value of i = i(j) such that c
ij
> 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
j
= 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
i
A
j
= c
ij
A
j
= d
ij
A
i
, for some c
ij
, d
ij
0
such that
P
iX
c
ij
=
P
jX
d
ij
= 1 i, j X.
By atomicity and non redundancy one must have
c
ij
= d
ij
= 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) =
M
(i,j)X×Y
St
ij
(AB),
Eff(AB) =
M
(i,j)X×Y
Eff
ij
(AB),
(22)
where for non redundant atomic decompositions
{A
i
}
iX
, {B
j
}
jY
of the identities I
A
and I
B
,
one has
(A
i
B
j
)
Ψ
i
0
j
0
= δ
ii
0
δ
jj
0
Ψ
ij
,
c
i
0
j
0
(A
i
B
j
) = δ
ii
0
δ
jj
0
c
ij
,
(23)
for all Ψ
i
0
j
0
St
i
0
j
0
(AB) and c
i
0
j
0
Eff
i
0
j
0
(AB).
Remark 5. Notice that from Eq. (22) it trivially
follows that for any system A the block decom-
position holds
St(A) =
M
iX
St
i
(A), Eff(A) =
M
iX
Eff
i
(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
i
B
j
in Eq. (23) of I
AB
is not atomic.
In such a case one has St(AB) =
L
kZ
St
k
(AB),
and St
ij
(AB) =
L
kZ
ij
St
k
(AB), for some parti-
tion Z
ij
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
Accepted in Quantum 2020-11-05, click title to verify. Published under CC-BY 4.0. 12
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
}
jY
Transf(A) there exists a non
disturbing test {A
i
}
iX
Transf(A) (namely
P
i
A
i
= I
A
) such that
B
j
=
X
i
p(j|i) V
ij
A
i
R
ij
, (25)
for some probability distribution p and reversible
transformations V
ij
, R
ij
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
k
= λ A
l
U,
where A
k
, A
l
are atomic transformations in the
unique non-redundant refinement of the identity
I
A
=
P
i
A
i
of Theorem 2, U Transf(A) is a
reversible transformation and λ 0.
Proof. Consider a test {B
j
}
jY
Transf(A) such
that B = B
j
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
}
iX
the unique non-redundant
atomic refinement of the identity I
A
=
P
i
A
i
)
and to the atomicity of B one has B = λ VA
i
R,
for some i X, λ 0 and V, R Transf(A) re-
versible transformations. Consider now the three
tests {VA
i
R}
iX
, {A
i
VR}
iX
and {VRA
i
}
iX
,
and observe that
P
i
VA
i
R =
P
i
VRA
i
=
P
i
A
i
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
}
iX
be the unique non-redundant
atomic refinement of the identity I
A
=
P
i
A
i
of
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
0
St
i
(A) is such that
ρ
0
ρ.
First we show that if an OPT satisfies full-
information without disturbance then for every
non null atomic ρ St
i
(A) and atomic a
Eff
i
(A), one has (a|ρ) 6= 0. Given such ρ St
i
(A)
and a Eff
i
(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| =
X
j
λ
j
U
j
A
i
, (26)
where each A
i
is the element of the non re-
dundant refinement of the identity I
A
such that
A
i
|ρ) = |ρ), and (a| A
i
= (a|, λ
j
> 0, and the
U
j
are reversible trasformations. Applying both
sides of Eq. (26) to the state ρ St
i
(A) one has
|ρ) (a|ρ) =
P
j
λ
j
U
j
|ρ). Reminding that the U
j
are all reversible, λ
j
> 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
0
St
i
(A). Since
A
i
ρ
0
=
ρ
0
one finds |ρ) (a|ρ
0
) =
P
j
λ
j
U
j
ρ
0
,
for some λ
j
> 0 and U
j
reversible transforma-
tions. As shown above it is (a|ρ
0
) 6= 0, and using
the atomicity of ρ one has
ρ
0
U
1
j
|ρ) for ev-
ery j. Since this holds true for every atomic state
ρ
0
St
i
(A), one has proved that all atomic states
(and then all states) in St
i
(A) are proportional to
the same atomic state, let’s say U
1
j
0
|ρ) for some
j
0
. Via an analogous argument one can see that
all effects in Eff
i
(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
Accepted in Quantum 2020-11-05, click title to verify. Published under CC-BY 4.0. 13
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
0
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)
AC
=
A
0
|Ψ)
AC
. However, two non-identical maps
A, A
0
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
0
Transf(A, B) are equal upon input of
ρ St(A), and write A =
ρ
A
0
, if for every
σ Ref
ρ
we have that Aσ = A
0
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
ρ
[9]. This choice can be eas-
ily motivated in operational terms: the equality
A =
ρ
A
0
means that the two maps A and A
0
are
indistinguishable on the state ρ, independently of
how it has been prepared. Suppose that the state
ρ is prepared by Alice as ρ =
P
iX
σ
i
, for some
refinement of ρ. Even Alice, using her knowledge
of the preparation cannot distinguish between A
and A
0
.
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 =
ρ
A
0
, still the maps A and A
0
could act differ-
ently upon input of dilations of ρ, namely it could
be A |Ψ)
AC
6= A
0
|Ψ)
AC
, for some Ψ Ref
D
ρ
. In
this case the difference between A and A
0
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
ρ
). Given a state ρ St(A), we say
that two transformations A, A
0
Transf(A, B)
are equal upon input of D
ρ
, and write A =
D
ρ
A
0
,
if A |Ψ)
AC
= A
0
|Ψ)
AC
for every Ψ Ref
D
ρ
.
Notice that the above definition requires that
two transformations act in the same way on the
set Ref
D
ρ
. Due to the absence of no-restriction
of preparation tests, it is not true in general that
D
Ref
ρ
= Ref
D
ρ
, as one might expect. The only
inclusion that can be proved without further as-
sumptions is Ref
D
ρ
D
Ref
ρ
, (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
D
ρ
for some ρ St(A), then
{(b|
B
|Ψ)
AB
|b Eff(B)} Ref
ρ
. (27)
Proof. Since Ψ Ref
D
ρ
there exists a prepara-
tion test {Ψ,
¯
Ψ, Λ} St(AB) such that Ψ +
¯
Ψ
D
ρ
. For an arbitrary b Eff</