Almost Quantum Correlations are Inconsistent
with Specker’s Principle
Tom
´
a
ˇ
s Gonda
1,2
, Ravi Kunjwal
1
, David Schmid
1,2
, Elie Wolfe
1
, and Ana Bel
´
en Sainz
1
1
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5
2
Dept. of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
August 21, 2018
Ernst Specker considered a particular feature of quantum theory to be especially
fundamental, namely that pairwise joint measurability of sharp measurements
implies their global joint measurability [vimeo.com/52923835 (2009)]. To date,
Specker’s principle seemed incapable of singling out quantum theory from the space
of all general probabilistic theories. In particular, its well-known consequence for
experimental statistics, the principle of consistent exclusivity, does not rule out
the set of correlations known as almost quantum, which is strictly larger than the
set of quantum correlations. Here we show that, contrary to the popular belief,
Specker’s principle cannot be satisfied in any theory that yields almost quantum
correlations.
1 Introduction
The advent of quantum theory was accompanied by many conceptual controversies over the
failure of intuitions from classical physics, e.g. the existence of wave-particle duality, the
fundamental indeterminism apparent from the Born rule, and the nonseparability epitomized
by entanglement, as pointed out by Einstein, Podolsky and Rosen [1, 2]. More recently,
we have witnessed the emergence of quantum information theory and a surge of interest
in quantum foundations. As a consequence, there has been remarkable progress in proving
theorems concerning ways in which Nature fails to be classical, given a well-defined notion of
classicality that mathematically formalizes some intuition from classical physics.
This attitude can also be taken towards quantum theory. Are there ways in which Nature
may fail to be quantum? That is, are there deviations from quantum theory in Nature, and
if so, where should one look for them [3]? Recently, there has been much effort to identify
the physical properties that single out the quantum world from the space of hypothetical
Tom
´
a
ˇ
s Gonda: tgonda@perimeterinstitute.ca
Accepted in Quantum 2018-08-17, click title to verify 1
arXiv:1712.01225v2 [quant-ph] 17 Aug 2018
alternatives. By doing this, not only can we learn more about quantum theory itself, but we
also gain insight about where one might or might not hope to find failures of quantum theory.
In the past two decades, there have been two major lines of research tackling this problem.
On the one hand, there is the program of characterizing only the statistical aspect of quantum
theory, i.e. recovering quantum correlations. On the other hand, there is that of deriving the
structure of quantum theory from a set of simple axioms.
Statistical aspects of quantum theory come into play when exploring phenomena such as Bell
nonlocality [4] and Kochen-Specker (KS) contextuality [5]. These cannot be explained by a
classical model of the world, although they arise naturally within quantum theory. Despite
being physically distinct [6], the notions of classicality challenged by these phenomena are
mathematically similar, which makes it possible to study them in a unified manner [7, 8].
Tackling the statistical aspects of quantum theory has provided us with considerable progress
in articulating principles that are satisfied by quantum correlations. However, a key question
remains. Are there any principles that uniquely identify the set of quantum correlations in
the space of all conceivable correlations? The study of this question led to the conception
of a set known as almost quantum correlations [9]. Initially defined only within Bell
scenarios [9], almost quantum correlations have also been subsequently defined within general
(KS-)contextuality scenarios [8]. This set of correlations strictly contains the quantum set,
yet has the following remarkable property: Almost quantum correlations are consistent with
all the principles that to our knowledge have so far been proposed to characterize quantum
correlations
1
[9]. In particular, almost quantum correlations satisfy the principle of consistent
exclusivity (CE) [8], which we define at the end of section 2.4. Therefore, it is desirable to
identify a principle capable of discriminating quantum from almost quantum correlations.
The other approach deals with structural aspects of quantum theory. This has proven
more successful than the previous approach since, to date, there are already derivations
of quantum theory from a set of reasonably simple axioms [10–20]. Hence, lately, there
has been increased interest in how this successful approach could be used to address the
aforementioned issues with almost quantum correlations. The hope is to clarify the relevant
structural differences between quantum theory and physical theories capable of generating
almost quantum correlations. For example, recent results show that any general probabilistic
theory [21] giving rise to almost quantum correlations would have to violate the no-restriction
hypothesis [22, 23].
In this paper, we explore the structural aspects of a hypothetical almost quantum theory
by investigating its statistical aspects—almost quantum correlations. To this end, we use
the connections between two frameworks for describing contextuality scenarios: in terms of
compatible measurements [7], and in terms of operational equivalences among measurement
events [8]. First, in section 2, we review these two formalisms and their relation. Section 3 then
presents Specker’s principle and some of its consequences both for the structure of a theory
and the statistics arising from it. In particular, these include a family of sufficiency properties
which we define therein. In section 4, we show that the ramifications of one of these sufficiency
1
Strictly speaking, almost quantum correlations have not been proven to satisfy the principle of Information
Causality yet; only numerical evidence exists for that claim.
Accepted in Quantum 2018-08-17, click title to verify 2
properties for measurement statistics are violated by almost quantum correlations. Thus,
Specker’s principle cannot be satisfied in any theory yielding almost quantum correlations.
Finally, section 5 provides evidence that not all of the sufficiency properties are in conflict with
almost quantum correlations, demonstrating the subtle nature of the contradiction between
these and Specker’s principle.
2 Marginal Scenarios and the Event-Based Hypergraph Approach to
Contextuality
Contextuality has been studied in a number of different formalisms [7, 8, 24]. The two that
we are interested in here are traditional contextuality scenarios defined from compatibility
structures of measurements, as formalised by Abramsky and Brandenburger in [7], and the
more general approach
2
by Ac´ın, Fritz, Leverrier and Sainz [8]. In this section, we lay out
the key concepts of both formalisms, their connections, and important sets of correlations
(a.k.a. probabilistic models). We begin by making the notion of compatible measurements
explicit.
2.1 Compatibility of Measurements
Compatible measurements: [25, 26]
Let {A
1
, . . . , A
n
} be a set of n measurements with the corresponding outcome sets being
{O
1
, . . . , O
n
}. We say that the elements of the set {A
1
, . . . , A
n
} are compatible (or jointly
measurable) if there exists a measurement A
0
with outcome set O
1
× . . . × O
n
, such that all
the measurements in the set {A
1
, . . . , A
n
} can be simulated by post-processing the outcomes
of A
0
. That is, the statistics of {A
1
, . . . , A
n
} for every preparation
3
ρ can be reconstructed
perfectly by measuring the given preparation ρ with A
0
:
Pr (a
i
|A
i
, ρ) =
X
α\a
i
Pr (α|A
0
, ρ) ∀ρ, ∀i ∈ {1, . . . , n}, (1)
where α ∈ O
1
× . . . × O
n
, and a
i
∈ O
i
is fixed in the sum on the right hand side.
As an example, consider two compatible measurements A
1
and A
2
in quantum theory.
A
1
=
n
A
0
1
, A
1
1
o
:=
n
1
2
+
1
√
2
σ
x
,
1
2
−
1
√
2
σ
x
o
, (2)
A
2
=
n
A
0
2
, A
1
2
o
:=
n
1
2
+
1
√
2
σ
z
,
1
2
−
1
√
2
σ
z
o
, (3)
where σ
x
and σ
z
are two of the Pauli matrices. The outcome set for each measurement is
O = {0, 1} in this case. If by ρ we denote the (arbitrary) state of a quantum system, then for
2
For the particular contextuality scenarios considered in this work, there is no extra generality in the
hypergraph formalism of [8] compared with the sheaf-theoretic formalism of [7].
3
Although we denote a preparation with the symbol ρ, we are not assuming here that the preparation is a
quantum state.
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any a
1
, a
2
∈ O,
Pr (a
1
|A
1
, ρ) = tr
h
1
2
+
(−1)
a
1
√
2
σ
x
ρ
i
, (4)
Pr (a
2
|A
2
, ρ) = tr
h
1
2
+
(−1)
a
2
√
2
σ
z
ρ
i
. (5)
In order to show that A
1
and A
2
are compatible, we have to construct a measurement that
can simulate them both. One such measurement is A
0
= {N
a
1
,a
2
}, with N
a
1
,a
2
defined for any
a
1
, a
2
∈ O as follows:
N
a
1
,a
2
:=
1
4
+
(−1)
a
1
σ
x
+ (−1)
a
2
σ
z
√
2
(6)
This measurement has the following properties:
A
a
1
1
=
X
a
2
∈O
N
a
1
,a
2
A
a
2
2
=
X
a
1
∈O
N
a
1
,a
2
(7)
It follows that, for all ρ, the probabilities of outcomes of A
1
and of A
2
can be simulated by
coarse-graining the probabilities of outcomes of A
0
:
Pr (a
1
|A
1
, ρ) =
X
a
2
tr [N
a
1
,a
2
ρ] =
X
a
2
Pr
a
1
, a
2
|A
0
, ρ
, (8)
Pr (a
2
|A
2
, ρ) =
X
a
1
tr [N
a
1
,a
2
ρ] =
X
a
1
Pr
a
1
, a
2
|A
0
, ρ
, (9)
and thus A
1
and A
2
are compatible.
As a consequence of the definition of compatibility presented above, the marginal statistics
of the set of compatible measurements has to be consistent with the existence of a joint
probability distribution over O
1
× . . . × O
n
. Equivalently, whenever the statistics of a set
of measurements fails to admit a joint probability distribution, there must necessarily be
incompatibility within that set of measurements.
However, it is important to point out that the converse need not hold. That is, if for
each preparation there exists a joint probability distribution over O
1
× . . . × O
n
that gives
the statistics of each of the measurements {A
1
, . . . , A
n
}, it does not follow that these
are compatible measurements. The simplest counter-example is that of two projective
measurements in quantum theory. Obviously, one can construct a joint distribution by taking
the product of the respective distributions for each of the two measurements, regardless of
whether they are compatible (i.e. commuting) or not. These considerations will be important
for understanding the relations that hold among the sufficiency properties in section 3.
2.2 Compatibility Approach to Contextuality
The traditional compatibility approach [7], also known as sheaf-theoretic approach, is
concerned with a set of measurements and the compatibility relations among them. These
relations are described by contexts, each of which is a set of compatible measurements. A
Accepted in Quantum 2018-08-17, click title to verify 4
contextuality scenario is defined by a triplet (X, O, M) and is called a marginal scenario
4
in
this approach. X denotes the set of measurements, each of whose set of outcome labels is
without loss of generality assumed to be identical and equal to O. On the other hand, M is
a subset of the power set of X that corresponds to the maximal contexts, i.e. those contexts
which are not contained within any other one. Hence, all the contexts in a given scenario are
precisely the subsets of elements of M. An example of the marginal scenario usually referred
to as Specker’s triangle is presented in figure 1a.
Given a maximal context x ∈ M, let O
x
denote the set of all families of outcomes of the
measurements in x. That is, O
x
naturally corresponds to the set of functions x → O. An
assignment of probabilities to measurement outcomes, {P
x
(a) : x ∈ M, a ∈ O
x
}, is called an
empirical model. For the purposes of studying the phenomenon of contextuality, we focus
on empirical models that satisfy the no-disturbance principle, which states that for any two
contexts x, x
0
∈ M, P
x|x∩x
0
= P
x
0
|x∩x
0
holds. Here, P
x|x∩x
0
denotes the marginal distribution
of P
x
associated to the measurements in x ∩ x
0
. In words, no-disturbance imposes that the
marginal statistics of a subset of measurements does not depend on the context in which they
could be measured. No-disturbance also provides justification for considering only maximal
contexts: If we are interested in the statistics of a set of compatible measurements that do
not form a maximal context, we can use any maximal context that contains all of them and
get the same result.
An empirical model P is said to be quantum whenever its statistics may be recovered
by performing, on some quantum state, projective measurements which satisfy the given
compatibility relations.
There is a particular family of marginal scenarios, mentioned also in [27], that is of interest to
us with regards to the discussion in this paper. We call them symmetric marginal scenarios
and they are characterized by two parameters (n, k).
Symmetric marginal scenarios:
An (n, k) symmetric marginal scenario is a triple (X, O, M) with |X| = n and
M = {Ω ⊆ X : |Ω| = k} (10)
2.3 Events-Based Hypergraph Approach to Contextuality
The second formalism we discuss is the hypergraph-based formalism from reference [8]. Here, a
contextuality scenario is defined as an events-based hypergraph H = (V, E) with vertices V and
hyperedges E. The vertices correspond to the events in the scenario, each of which represents
an outcome obtained from a device after it receives an input—the measurement choice.
The hyperedges are sets of events representing all the possible outcomes given a particular
measurement choice. The hypergraph approach assumes that every such measurement set is
complete. That is, if the measurement corresponding to hyperedge e is performed, exactly
one of the outcomes in e is obtained. Note that measurement sets may have non-trivial
4
Some works in the literature also refer to these scenarios as measurement scenarios. Here we will stick to
the notation used in [8].
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A
2
A
1
A
3
(a) Specker’s triangle as a marginal sce-
nario.
(00|13)(00|12)
(00|23)
(01|13)(01|12)
(01|23)
(10|13)(10|12)
(10|23)
(11|13)(11|12)
(11|23)
(b) Specker’s triangle as a contextuality
scenario.
Figure 1: Specker’s triangle scenario. (a) Hypergraph representation of its formulation
as a marginal scenario. Each vertex denotes a measurement, and each hyperedge a
maximal context. There are three dichotomic measurements X = {A
1
, A
2
, A
3
} which
are pairwise compatible, hence M = {{A
1
, A
2
}, {A
1
, A
3
}, {A
2
, A
3
}}. The outcome set
is given by O.
(b) The events-based representation of the same scenario. The vertices are given
by the events (a
i
a
j
|ij) where a
i
, a
j
∈ O denote the outcomes of the measurements
A
i
, A
j
∈ X from a context {A
i
, A
j
} ∈ M. Blue hyperedges correspond to measuring
maximal contexts, while the remaining hyperedges arise via other, less trivial, measurement
protocols. The Specker’s triangle corresponds to a (3, 2) symmetric marginal scenario,
using the terminology introduced in section 2.2.
intersection. If an event appears in more than one hyperedge, it corresponds to outcomes of
implementing different measurement choices that are operationally equivalent
5
.
There is a close connection between the two approaches. In particular, given a marginal
scenario (X, O, M), we can define an events-based hypergraph H[X, O, M] that corresponds
to the same situation, in accordance with the Appendix D of [8]. In a nutshell, the construction
works as follows.
The vertices of H[X, O, M] are defined as V (H) = {(a|x) : x ∈ M, a ∈ O
x
}, where x ∈ M
is a maximal context and a ∈ O
x
is a family of outcomes of the measurements in x as before.
The vertices of H[X, O, M] thus inherit the underlying structure of the initial scenario, since
their labels contain the information about the maximal contexts and the possible outcomes.
On the other hand, the hyperedges arise via measurements protocols [8, Def. D.1.4] (see next
paragraph for an example), which correspond to adaptive choices of measurements from X.
They are adaptive, because a choice of an individual element of X can depend on the outcomes
of the previously chosen ones.
5
Two measurement events are operationally equivalent if their probabilities coincide for every preparation
of the system being probed [6].
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As an example, consider the case of Specker’s triangle scenario depicted in figure 1.
This scenario consists of three dichotomic measurements that are pairwise compatible but
not triplewise compatible
6
. Hence, as a marginal scenario (X, O, M) Specker’s triangle
features measurements X = {A
1
, A
2
, A
3
} with outcomes O = {0, 1} and maximal contexts
M = {{A
1
, A
2
}, {A
1
, A
3
}, {A
2
, A
3
}}, as shown in figure 1a.
As an events-based hypergraph, Specker’s triangle is shown in figure 1b. The vertices of this
hypergraph are given by the events V = {(a
i
a
j
|ij)}
i<j
, where the indices i and j run from 1
to 3 and refer to the three measurements in X. One can notice that there are two types of
hyperedges. First of all, there are the blue hyperedges of the form {(a
i
a
j
|ij)}
a
i
,a
j
∈O
. These
contain all the possible outcomes for the two measurements in a chosen context. Secondly,
there are hyperedges corresponding to adaptive measurement protocols. As an example,
consider the hyperedge depicted in dark green at the bottom left side of the hypergraph,
containing vertices (00|12), (10|12), (10|23) and (11|23). This hyperedge may be understood
as the following protocol. Initially, one performs measurement of A
2
. If the outcome is a
2
= 0,
then A
1
is measured next. However, if the outcome is a
2
= 1, then the second measurement
one performs is A
3
. The former possibility yields the events (00|12) and (10|12), while the
latter gives (10|23) and (11|23).
2.4 Sets of Probabilistic Models for Events-Based Hypergraphs
The objects of study in the events-based approach are the so-called probabilistic models, which
is a notion analogous to empirical models in the compatibility approach. A probabilistic
model on a contextuality scenario H is a functional p : V → [0, 1] denoting the conditional
probability that an event v occurs when any measurement e with v as one of its outcomes is
performed. The identification of outcomes by operational equivalences in the definition of the
contextuality scenario implies that the probability p(v) is independent of the measurement e.
Moreover, since the measurements are complete, the probabilities are normalized within each
hyperedge. That is,
P
v∈e
p(v) = 1 for all e ∈ E(H). Since the specification of a probabilistic
model is a tuple of real numbers, we often treat a probabilistic model as a vector ~p embedded
in a vector space with coordinates corresponding to the probabilities of events in V .
We will denote the set of all valid probabilistic models on H by G(H). There are various
subsets of G(H) which are of particular interest. Here, we define the subsets corresponding
to classical, quantum, and almost quantum models.
Classical models: [8, Def. 4.1.1]
A probabilistic model p ∈ G(H) is classical if it can be written as a convex combination of the
deterministic ones. Each deterministic probabilistic model p ∈ G(H) satisfies p(v) ∈ {0, 1}
for all v ∈ V . The set of all classical models on H is denoted by C(H) and forms a polytope,
whose vertices are the deterministic probabilistic models.
6
For a discussion on how to realise such a scenario within quantum theory, we refer the reader to [26].
Accepted in Quantum 2018-08-17, click title to verify 7
Quantum models: [8, Def. 5.1.1]
A probabilistic model p ∈ G(H) is quantum if there exists a Hilbert space H, a single quantum
state ρ (density operator on H) and a projection operator P
v
on H associated to every v ∈ V ,
such that
1.
P
v∈e
P
v
=
H
∀e ∈ E,
2. p(v) = tr [P
v
ρ] ∀v ∈ V .
The set of all quantum models on H is denoted by Q(H).
In the specific case of Bell scenarios the set Q(H) coincides with the traditional set of quantum
correlations [8, Prop. 5.2.1].
Almost quantum models: [8, Def. 6.1.2]
A probabilistic model p ∈ G(H) is almost quantum if there exists a Hilbert space H, a single
quantum state ρ (density operator on H) and a projection operator P
v
on H associated to
every v ∈ V , such that
1.
P
v∈e
P
v
≤
H
∀e ∈ E,
2. p(v) = tr [P
v
ρ] ∀v ∈ V .
The set of all almost quantum models on H is denoted by Q
1
(H).
The terminology “almost quantum” for Q
1
models comes from their close connection to the
set of correlations in Bell scenarios known as “almost quantum correlations” [9]. Indeed, it
has been proven that the set Q
1
(H) is equivalent to the set of almost quantum correlations
whenever H is the hypergraph of a Bell scenario
7
.
A common feature shared by quantum and almost quantum models is that they both satisfy
the consistent exclusivity principle [8, 9]. Explicitly, this means that all the CE inequalities
defined below hold for every almost quantum (and quantum) probabilistic model.
CE inequality:
Given a contextuality scenario H = (V, E), an inequality of the form
P
v∈S
p(v) ≤ 1 is a CE
inequality if and only if S ⊆ V is a set of pairwise exclusive events. A set of events S is
pairwise exclusive if, for each pair u, v ∈ S , there is a hyperedge e ∈ E such that u, v ∈ e.
It is worth mentioning that for scenarios of the type H[X, O, M] which arise from marginal
scenarios, the notion of exclusivity introduced in the definition of CE inequalities can be
equivalently stated as follows [8]: two events, (a|x) and (a
0
|x
0
), are exclusive if there is a
measurement in x ∩ x
0
that assigns different outcomes in a and a
0
. Moreover, for these
scenarios the set of empirical models on (X, O, M) is equivalent to the set of probabilistic
models on H[X, O, M].
Consistent Exclusivity principle:
A theory satisfies the principle of consistent exclusivity (CE) if every probabilistic model p
admissible by the theory satisfies all the relevant CE inequalities
8
.
7
For a discussion on how to define such Bell hypergraphs we refer the reader to [8, 28].
8
A stronger version of the Consistent Exclusivity principle imposes constraints on the probabilistic models
of a contextuality scenario by considering its application in larger scenarios that contain the original one (see
[8]).
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CE models:
A probabilistic model p ∈ G(H) is consistently exclusive if it satisfies the CE principle. The set
of all consistently exclusive models on H is denoted by CE(H)
9
and forms a convex polytope.
3 Specker’s Principle and its Consequences
One can derive several constraints on the sets of symmetric marginal scenarios and the
statistics they generate by considering consequences of Specker’s principle, which we now
introduce.
Specker’s principle as originally stated: “If you have several questions and you can
answer any two of them, then you can also answer all of them.” [vimeo.com/52923835 (2009),
also attributed in reference 29.]
As stated, the principle refers to “questions”, i.e. measurements, and imposes that a set of
pairwise compatible measurements is itself compatible. We can formalize this interpretation
of Specker’s principle as follows.
Pairwise sufficiency for measurements: If in a set of measurements every pair is
compatible, then all the measurements are compatible.
Under the principle of pairwise sufficiency for measurements, there would be no distinction
between an (n, 2) symmetric marginal scenario and an (n, n) symmetric marginal scenario, as
those would merely be different representations of the same compatibility relations.
This formalization of Specker’s principle holds true in quantum theory for sharp (i.e. projec-
tive) measurements. Note that Specker’s principle has elsewhere been invoked to motivate
statistical constraints at the level of events, such as the principle of consistent exclusivity
[8, 24]. However, it is important to recognize that these statistical constraints are not equiv-
alent to Specker’s principle. Rather, they are implications thereof. In other words, Specker’s
principle pertains foremost to “questions” (i.e. measurements), and only secondarily to “an-
swers” (i.e. events).
While consistent exclusivity is a statistical constraint implied by Specker’s principle, it is
not the only one. We distinguish consistent exclusivity (which applies to any contextuality
scenario) from the following statistical constraint (which refers to (n, 2) symmetric marginal
scenarios only).
Pairwise sufficiency for probabilistic models: If in a set of measurements every pair is
compatible, then – for every preparation – the statistics generated by these measurements are
marginals of some joint probability distribution.
Equivalently, we can say that in a theory satisfying pairwise sufficiency for probabilistic
models, the set of probabilistic models that arises from an events-based hypergraph that
can be obtained from an (n, 2) symmetric marginal scenario for some integer n is always
classical.
9
The set CE(H) defined here corresponds to the set CE
1
(H) defined in [8].
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Note that every compatible set of measurements admits a joint probability distribution over
the outcomes of the measurements in the set, as implied by the definition of compatibility.
Therefore, if a theory satisfies pairwise sufficiency for measurements, then it also satisfies
pairwise sufficiency for probabilistic models. However, the converse need not hold, as has
been pointed out at the end of section 2.1. This one-way relationship between the principle
for measurements and the principle for probabilistic models is illustrated in figure 2.
Besides the concept of pairwise sufficiency, one can define another property called all-but-one
sufficiency, which might at first appear to be less constraining. All-but-one sufficiency captures
the same idea as pairwise sufficiency, but applies only to a subset of marginal scenarios, namely
(n, n−1) symmetric marginal scenarios.
All-but-one sufficiency for measurements: If in a set of at least three measurements,
the elements of every proper subset are compatible, then all the measurements are compatible.
Under the principle of all-but-one sufficiency for measurements, there would be no distinction
between an (n, n−1) symmetric marginal scenario and an (n, n) symmetric marginal scenario
for n ≥ 3, as those would merely be different representations of the same compatibility
relations. The qualifier n ≥ 3 is needed, because otherwise all pairs of measurements would
be compatible, and hence all contextuality scenarios would be deemed classical.
All-but-one sufficiency for probabilistic models: If in a set of at least three
measurements, the elements of every proper subset are compatible, then – for every preparation
– the statistics generated by these measurements are marginals of some joint probability
distribution.
In other words, we can say that a theory satisfies the principle of all-but-one sufficiency for
probabilistic models if every probabilistic model p is classical whenever the compatibility
scenario is an (n, n−1) symmetric marginal scenario for some integer n ≥ 3.
Notice that if a theory satisfies pairwise sufficiency for measurements, then it also satisfies
all-but-one sufficiency for measurements. Similarly, if a theory satisfies pairwise sufficiency
for probabilistic models, then it satisfies all-but-one sufficiency for probabilistic models as
well. Moreover, we now show that pairwise sufficiency and all-but-one sufficiency are actually
equivalent at the level of measurements, even though they seem to be distinct at the level
of probabilistic models, as the conjecture in section 5 would imply, if true. The implication
relations among the various consequences of Specker’s principle are summarized in figure 2.
Theorem 1. A theory satisfies the all-but-one sufficiency principle for measurements if and
only if it satisfies the pairwise sufficiency principle for measurements.
Proof. The “if” statement holds trivially, so let us focus on the “only if” implication. Assume
that a theory satisfies the all-but-one sufficiency principle for measurements. Let X be a set
of n measurements of which every pair is compatible. We can choose n ≥ 4, because for
n = 3, the argument is trivial. By applying the assumption, we establish that every subset
of X of size 3 is compatible. This follows because every such subset is a (3, 2) symmetric
marginal scenario. With this established, we again apply the assumption in order to establish
that every subset of X of size 4 is compatible. If n > 4, we can further iterate applications
of the assumption until we establish that all subsets of size n, namely the entire set X, is
compatible.
Accepted in Quantum 2018-08-17, click title to verify 10
Specker’s principle
⇐⇒
Pairwise sufficiency
for measurements
⇐⇒
All-but-one sufficiency
for measurements
=⇒
=⇒
=⇒
Consistent exclusivity
Pairwise sufficiency
for probabilistic models
=⇒
All-but-one sufficiency
for probabilistic models
Figure 2: Some of the consequences of Specker’s principle and the implications known
to hold among them. The top row contains principles pertaining to the structure of
measurements, while the bottom row contains principles pertaining to sets of probabilistic
models. Text colour depicts whether a given statement holds in an almost quantum theory:
the green statement is satisfied, the red statements are violated, and the black statement
is the subject of conjecture in section 5.
One can naturally consider two hierarchies of principles akin to pairwise sufficiency and all-
but-one sufficiency. For example, the principle of triplewise sufficiency for measurements
states that triplewise compatibility implies joint compatibility. As another example, the all-
but-two sufficiency principle for probabilistic models states that no nonclassical correlations
can exist in a compatibility scenario wherein every collection of all-but-two measurements is
compatible, i.e. an (n, n−2) symmetric marginal scenario.
4 Almost Quantum Models Violate Specker’s Principle
Since projective measurements in quantum theory satisfy the principle of pairwise sufficiency,
the set Q(H) for an events-based hypergraph H obtained from an (n, 2) symmetric marginal
scenario coincides with C(H). Here, we demonstrate that there are probabilistic models
in Q
1
(H) for H obtained from an (4, 2) symmetric marginal scenario (see figure 3), which
are outside of C(H). This fact shows that almost quantum correlations violate “pairwise
sufficiency for probabilistic models”, and therefore any physical theory consistent with them
violates “pairwise sufficiency for measurements” and, by doing so, also Specker’s principle.
In a (4, 2) symmetric marginal scenario, each facet of the classical polytope is either a CE
constraint or a pentagonal inequality [30]. One instance of the pentagonal inequalities reads
I
pent
= − hX
1
X
2
i − hX
1
X
3
i + hX
1
X
4
i + hX
1
i − hX
2
X
3
i (11)
+ hX
2
X
4
i + hX
2
i + hX
3
X
4
i + hX
3
i − hX
4
i ≤ 2 ,
where hX
i
X
j
i =
P
a
i
,a
j
∈O
(−1)
a
i
+a
j
p(a
i
a
j
|ij) and hX
j
i =
P
a
j
∈O
(−1)
a
j
p(a
j
|j). Notice that
the upper bound of 2 is satisfied for both classical and quantum probabilistic models [30].
This is expected, because it is well known that both classical and quantum theories respect
Specker’s principle.
A linear program optimisation over the maximum values of equation (11) for general
probabilistic models yields a value of I
∗
pent
= 6. Similarly, optimising equation (11) via
Accepted in Quantum 2018-08-17, click title to verify 11
A
1
A
2
A
3
A
4
Figure 3: Marginal scenario consisting of four pairwise compatible measurements. The
four measurements are X = {A
1
, A
2
, A
3
, A
4
}, and the compatible subsets (contexts) are
M = {{A
1
, A
2
}, {A
1
, A
3
}, {A
1
, A
4
}, {A
2
, A
3
}, {A
2
, A
4
}, {A
3
, A
4
}}.
a semidefinite program [8] yields a maximum value of I
AQ
pent
= 2.5 for almost quantum
probabilistic models, which violates the classical and quantum bound of the inequality. This
is despite the fact that almost quantum models satisfy all the CE inequalities.
This result reveals a counterintuitive aspect of almost quantum models. Namely, there exist
scenarios in which all the measurements are pairwise compatible yet almost quantum models
are strictly more general than classical models. Therefore, almost quantum models are
inconsistent with Specker’s principle, in that any set of measurements that gives rise to them
cannot satisfy it.
Theorem 2. There is no set of measurements, in any generalized probabilistic theory, which
both gives rise to almost quantum models and also satisfies Specker’s principle.
By theorems 1 and 2, we learn that there is no set of measurements, in any theory, which both
gives rise to almost quantum models and also satisfies the principle of all-but-one sufficiency
for measurements.
As a final comment, the research scope that tackles the formulations of general physical
theories beyond the quantum one has particularly focused on the definition of a subset of
measurements, known as sharp measurements, which correspond to projective measurements
in the case of quantum theory. There is still no consensus on how sharp measurements should
be defined in a general theory [31, 32]. In the following we argue that (i) any definition of sharp
measurements in a theory that yields almost quantum correlations must violate Specker’s
principle, and (ii) any notion of sharpness in an almost quantum theory must deviate from
the candidates proposed so far [31, 32].
Corollary 2.1. If in any theory there is a notion of sharpness, relative to which the sharp
measurements yield almost quantum correlations, then there are some sharp measurements in
the theory which violate Specker’s principle.
This corollary is an instance of theorem 2, whereby the set of measurements giving rise to
almost quantum correlations is exactly the one that satisfies some (unspecified) notion of
‘sharpness’.
This specific case can be motivated by analogy with quantum theory. As in quantum theory, in
a hypothetical almost quantum theory one must also restrict the set of allowed measurements
Accepted in Quantum 2018-08-17, click title to verify 12
to a strict subset of the set of all measurements in order to pick out exactly the almost
quantum correlations in Kochen-Specker contextuality scenarios. If all measurements were
allowed – in either quantum or almost quantum theory – then one could realize any logically
possible probabilistic model, using the completely noisy effects
10
[33]. Furthermore, the set of
measurements allowed in quantum theory is exactly the set of sharp quantum measurements.
In an almost quantum theory, it is unclear what this allowed set of measurements would be,
but one might expect such measurements to correspond to some notion of sharpness as well.
For example, Corollary 2.1 sheds light on the notion of sharpness for general probabilistic
theories that was proposed by Chiribella and Yuan [31, 32]. The notion of sharpness there
does imply Specker’s principle, and as such no theory can give rise to the almost quantum
models using only the sharp measurements in the sense of references [31, 32]. In particular,
such a theory cannot violate the pentagonal inequality presented in equation (11).
5 Almost Quantum Models and the All-But-One Sufficiency Principle
The example presented in the previous section teaches us that any almost quantum theory
must violate the all-but-one sufficiency principle for measurements. Incidentally, it also
demonstrates that almost quantum models fail to satisfy the all-but-two sufficiency principle
for probabilistic models, since C(H) 6= Q
1
(H) for an event-based hypergraph H obtained from
a (4, 2) symmetric marginal scenario.
Now, is it possible that almost quantum models nevertheless satisfy the principle of all-but-one
sufficiency, despite its violation at the level of measurements by any almost quantum theory?
It is conceivable that a physical theory might satisfy the principle of all-but-one sufficiency
at the level of probabilistic models while violating the principle at the level of measurements.
It might feel unnatural to divorce a statistical constraint from any restriction on the structure
of measurements. Nevertheless, this unexpected satisfaction of the all-but-one sufficiency
principle for probabilistic models without motivation from the corresponding measurement-
based principle appears to be a feature of any almost quantum theory.
We state this as a conjecture, and provide suggestive evidence in terms of a theorem that
confirms the conjecture for the special case of dichotomic measurements.
Conjecture. Almost quantum correlations satisfy the all-but-one sufficiency principle for
probabilistic models. In other words, the sets Q
1
(H) and C(H) coincide, whenever H
corresponds to an (n, n−1) symmetric marginal scenario.
Theorem 3. If H is an events-based hypergraph constructed from an (n, n−1) symmetric
marginal scenario with n ≥ 2 and dichotomic measurements, then the sets C(H), Q(H),
Q
1
(H) and CE(H) all coincide.
The proof of theorem 3 is presented in appendix A. This theorem tells us that the sets Q
1
(H)
and C(H) coincide whenever H corresponds to an (n, n−1) symmetric marginal scenario with
binary outcome measurements, in agreement with the conjecture above.
10
In either case, one can always construct a measurement generating the desired set of probabilities by
multiplying the unit effect of the theory with the probabilities one aims to generate.
Accepted in Quantum 2018-08-17, click title to verify 13
Thus, we conclude that almost quantum models satisfy some statistical consequences of
Specker’s principle, but not others. For example, they satisfy the principle of consistent
exclusivity, as well as the principle of all-but-one sufficiency for probabilistic models (at least
for binary outcome measurements). On the other hand, almost quantum models satisfy neither
the principle of pairwise sufficiency nor that of all-but-two sufficiency, be it for measurements
or probabilistic models. This dichotomy challenges our ability to understand the statistical
predictions of any almost quantum theory in terms of Specker-like restrictions at the level of
measurements.
6 Conclusions
In this work, we have explored some consequences of Specker’s principle and their consistency
with almost quantum correlations. By studying contextuality scenarios on a single system,
we have found a fundamental difference between quantum theory and any potential almost
quantum theory, complementing the results by Sainz et al. [23]. There, a different no-
go theorem (pertaining to almost quantum theories and the no-restriction hypothesis) was
inferred based on considerations of Bell scenarios involving multiple subsystems.
Our results imply that in any general probabilistic theory, the structure of measurements
reproducing almost quantum models is in contradiction with Specker’s principle. Accordingly,
the notion of sharpness proposed in references [31, 32] cannot be used to recover the almost
quantum correlations. This result runs counter to sentiments implicit in earlier literature.
Previously, almost quantum models appeared to be the purest embodiment of Specker’s
principle [9, 24, 29, 34–36], in the sense that they are uniquely identified
11
by consistent
exclusivity and closure under wirings [8, Thrm. 7.6.2].
However, consistent exclusivity is not the only constraint on probabilistic models implied by
Specker’s principle. Another such principle is pairwise sufficiency for probabilistic models,
as defined in section 3. Despite the “success” of the almost quantum correlations relative to
consistent exclusivity, we nevertheless witness their failure to satisfy Specker’s principle when
analyzed through the lens of pairwise sufficiency. As such, the above preconceptions must
be reversed. Almost quantum models do not exemplify Specker’s principle. Rather, they are
antithetical to it.
For advocates of Specker’s principle, our results challenge the possible physical significance
of the almost quantum correlations [37]. More importantly, however, our findings restore the
prominence of Specker’s principle as a potential means for identifying the essence of quantum
theory. The violation of Specker’s principle by almost quantum models demonstrates that
holistic considerations of Specker’s principle enable greater insight than consistent exclusivity
alone.
All the results in this manuscript were found solely by studying a rather limited set of
contextuality scenarios, namely symmetric scenarios with binary outcome measurements.
Consideration of asymmetric compatibility structures or scenarios beyond binary-outcome
11
Almost quantum correlations are uniquely identified under the assumptions that the set of correlations
allowed in Nature contains the quantum one, and by considering the ramifications of consistent exclusivity
even in the limit of many independent copies of the scenario.
Accepted in Quantum 2018-08-17, click title to verify 14
measurements might illuminate stronger constraints and richer intuitions about almost
quantum theory than the results presented here.
7 Acknowledgements
We thank Emily Kendall for fruitful discussions. This research was supported by Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is supported by the
Government of Canada through the Department of Innovation, Science and Economic
Development Canada and by the Province of Ontario through the Ministry of Research,
Innovation and Science.
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A Proof of Theorem 3
For the proof of theorem 3, it is useful to introduce the concept of a generalized event in relation
to the events as vertices of some hypergraph H. This notion refers to a coarse-graining of
events in H. For instance, in the case of figure 1b, a generalized event g = (a
1
|1) can be defined
as either the coarse graining of {(a
1
, a
2
|1, 2), (a
1
, ¯a
2
|1, 2)} or that of {(a
1
, a
3
|1, 3), (a
1
, ¯a
3
|1, 3)},
since the measurement A
1
is compatible with both A
2
and A
3
and the no-disturbance condition
holds. Each of these sets {(a
1
, a
j
|ij), (a
1
, ¯a
j
|ij)} is then a refinement of g. The probability
assigned to a generalized event is then given by summing the probabilities of all the events in
any of its refinements.
The notion of exclusivity of events can be also extended to generalized events. Two generalized
events g, g
0
are said to be exclusive if events in a refinement of g are pairwise exclusive with
events in a refinement of g
0
. Equivalently, one can say that the union of any refinements of g
and g
0
is a set of pairwise exclusive events in V .
In this appendix we study almost quantum probabilistic models in a particular family of
contextuality scenarios, which we call binary n-Specker scenarios. These are the events-
based hypergraphs corresponding to (n, n−1) symmetric marginal scenarios introduced in
definition 2.2. Moreover, we restrict our attention to cases with the measurements in X each
having two outcomes, so that we can choose O = {0, 1}. The set of contexts M consists of
every subset of X with n−1 elements. That is, each proper subset of X is jointly measurable,
but the whole set X of n measurements is not necessarily jointly measurable. Figure 1a
depicts a 3-Specker scenario and figure 4 a 4-Specker scenario.
From the very definition of the scenario it immediately follows that every quantum empirical
model in (X, O, M) has a realisation in terms of deterministic non-contextual hidden variable
models, since in quantum theory a set of pairwise compatible measurements is jointly
measurable. Here we show that the set of almost quantum models has a similar behaviour.
The proof relies on showing that every facet of the classical polytope corresponds to a CE
inequality, each of which is satisfied by both quantum and almost quantum models.
Let us introduce some extra notation. The measurements are labelled by X = {1, 2, . . . , n}.
The events in H
:
= H[X, O, M] are denoted by (a
i
1
, . . . , a
i
n−1
), where {i
1
, . . . , i
n−1
} ∈ M
Accepted in Quantum 2018-08-17, click title to verify 17
A
1
A
2
A
3
A
4
Figure 4: The marginal scenario representation of the 4-Specker scenario, i.e. (4, 3)
symmetric marginal scenario. There are four dichotomic measurements X =
{A
1
, A
2
, A
3
, A
4
} which are triple-wise compatible, so that O = {0, 1}, and M =
{{A
1
, A
2
, A
3
}, {A
1
, A
2
, A
4
}, {A
1
, A
3
, A
4
}, {A
2
, A
3
, A
4
}}.
and a
i
k
∈ O is the outcome of measurement i
k
. By (a
i
1
, . . . , a
i
k
), we denote a generalized
event with the remaining n−k−1 measurements in the context {i
1
, . . . , i
n−1
} disregarded. A
probabilistic model
12
~p is then a family of probabilities indexed by the contexts and outcomes
of the measurements in the respective context
~p =
p(a
i
1
, . . . , a
i
n−1
) : Ω
:
= {i
1
, . . . , i
n−1
} ∈ M, (a
i
1
, . . . , a
i
n−1
) ∈ O
Ω
. (12)
In this notation, the no-disturbance condition guarantees that the marginal distribution
p(a
i
1
, . . . , a
i
k
) is independent of the choice of context from which it is computed.
The rest of this appendix is dedicated to the proof of the theorem, which can written in the
language of n-Specker scenarios as follows.
Theorem 3. Consider a binary n-Specker scenario H with n ≥ 2. Then the sets C(H),
Q(H), Q
1
(H) and CE(H) all coincide.
First of all notice that there are inclusions C(H) ⊆ Q(H) ⊆ Q
1
(H) ⊆ CE(H), which hold in
general for any contextuality scenario H [8]. Therefore, in order to prove theorem 3, we only
need to show CE(H) ⊆ C(H). This can be broken down into three parts. Firstly, we define
a finite set of linear inequality constraints I
n
on G(H) and the corresponding polytope P,
which satisfies them. Then we show in lemma 3.1 that all these inequalities are actually CE
inequalities, so that CE(H) ⊆ P holds. The proof concludes with lemma 3.2, which establishes
the equality of C(H) and P.
Consider a set of measurements S ⊆ X with k
:
= |S|. Let s = {a
r
: r ∈ S} ∈ O
S
be a
collection of outcomes of the k measurements S. Similarly o = {a
r
: r ∈ X \ S} ∈ O
X\S
is
a collection of outcomes of the n − k measurements X \ S. We define the set I
n
in terms of
linear functionals I
k,S,s,o
n
on the vector space in which the probabilistic models are embedded.
These depend on the choice of S, s and o in the following way.
I
k,S,s,o
n
:
= p(o) +
k−1
X
j=1
(−1)
j
X
{i
1
,...,i
j
}⊆S
p(a
i
1
, . . . , a
i
j
, o), (13)
12
For simplicity we will denote by p(a
i
1
, . . . , a
i
k
) the conditional probability distribution
p(a
i
1
, . . . , a
i
k
| i
1
, . . . , i
k
). Similarly, the events (a
i
1
, . . . , a
i
n−1
| i
1
, . . . , i
n−1
) in H[X, O, M] are simply
denoted by (a
i
1
, . . . , a
i
n−1
).
Accepted in Quantum 2018-08-17, click title to verify 18
with p(o) = 1 whenever k = n. We use the notation I
k
n
for the set of corresponding inequalities
with a given k.
I
k
n
:
=
n
I
k,S,s,o
n
≥ 0 : s ∈ O
S
, o ∈ O
X\S
, S ⊆ X, |S| = k
o
(14)
We define the full set of inequalities, I
n
, as the union of all I
k
n
for odd k between 1 and n.
In fact, the right hand side of equation (13) can be expressed in simpler terms, as we illustrate
next. Given a probabilistic model ~p, which specifies a probability distribution for each proper
subset of X, there is a function f : O
X
→ R that recovers ~p by marginalization
13
[7]. That
is, by marginalizing f over any measurement i, we obtain the original probabilistic model for
the context X \{i} Even though this function is not unique, we can write I
k,S,s,o
n
in terms of
f uniquely. In particular, we have
I
k,S,s,o
n
= f(s, o) + f (
¯
s, o). (15)
Notice that one can choose f to be a probability distribution over O
X
if and only if ~p is a
classical probabilistic model.
Example 1. As an example, consider the 3-Specker scenario, also known as the Specker’
triangle. The two sets of inequalities that give rise to I
3
are
I
1
3
=
n
p(a
i
, a
j
) ≥ 0 : {i, j} ∈ {1, 2, 3}, (a
i
, a
j
) ∈ O
{i,j}
o
, (16)
I
3
3
=
n
1 − p(a
1
) − p(a
2
) − p(a
3
) + p(a
1
, a
2
) + p(a
1
, a
3
) + p(a
2
, a
3
) ≥ 0 : a
i
∈ O
{i}
o
. (17)
The first ones are just the positivity constraints, which are always CE inequalities. Likewise,
inequalities in I
3
3
can be written in a form that is manifestly CE, as
I
3
3
=
n
1 ≥ p(a
1
, ¯a
2
) + p(a
2
, ¯a
3
) + p(¯a
1
, a
3
) : a
i
∈ O
{i}
o
, (18)
where ¯a
i
:
= a
i
+ 1 (mod 2). A generalization of the fact that I
3
consists of CE inequalities
to all I
n
is the content of the next lemma.
Lemma 3.1. Every element of I
n
defined as above is a CE inequality.
Proof. We prove this result by induction on n. However, in order to perform the inductive
step, we need the induction hypothesis to be slightly stronger than just the statement of
lemma 3.1 for a particular value of n.
Induction hypothesis (IH
m
). For all m,
(a) every element of I
k
m
is a CE inequality for any odd k between 1 and m − 1.
Moreover, if m is odd, then given any inequality
14
I
m,X,s,∅
m
≥ 0 in I
m
m
, there exists a set D
s
m
of generalized events (which by a slight abuse of notation we refer to simply as events), such
that the following properties hold:
13
The marginal of f over Ω ⊆ X is computed by summing the values of f over O
Ω
, which yields a function
O
X\Ω
→ R.
14
Note that when k is equal to n in I
k,S,s,o
n
, S has to be X itself and o contains no outcomes.
Accepted in Quantum 2018-08-17, click title to verify 19
(b) One can express I
m,X,s,∅
m
as
I
m,X,s,∅
m
= 1 −
X
α∈D
s
m
p(α). (19)
(c) Each of the events in D
s
m
involves outcomes of at most m − 1 measurements among X.
(d) The set D
s
m
consists of pairwise exclusive events.
(e) Each event g in the set D
s
m
has the following property. There exists a measurement in
X whose outcome for g is different from the one for s, and another measurement in X
whose outcome for g is different from the one for
¯
s.
Notice that the statement that all elements of I
m
are CE inequalities is equivalent to
property (a) whenever m is even and equivalent to the conjunction of properties (a) through
(d) whenever m is odd. Therefore, Lemma 3.1 follows once we show that IH
2
and IH
3
both
hold (base case), and that IH
m-2
with IH
m-1
together imply IH
m
(inductive step).
Base case. The fact that IH
2
holds follows because I
2
consists of positivity constraints only.
Since 3 is odd, we need to check all five properties in order to establish IH
3
. Let’s analyze
them one by one.
BC(a) As we have seen in example 1, I
1
3
consists of positivity constraints, which are CE
inequalities.
BC(b) Following the discussion in example 1, we can define
D
(a
1
,a
2
,a
3
)
3
:
= {(a
1
, ¯a
2
), (a
2
, ¯a
3
), (¯a
1
, a
3
)}, (20)
whence equation (19) holds for m = 3 and s = (a
1
, a
2
, a
3
).
BC(c) This property is immediate from the definition of D
(a
1
,a
2
,a
3
)
3
above.
BC(d) For every pair of events in D
(a
1
,a
2
,a
3
)
3
, there is a measurement with different outcome
for each of the two events in the pair. Therefore, the events in D
(a
1
,a
2
,a
3
)
3
are pairwise
exclusive.
BC(e) Each event in D
(a
1
,a
2
,a
3
)
3
is of the form g = (a
i
, ¯a
j
), for some i and j in {1, 2, 3}. Hence,
measurement i assigns different outcomes to g and
¯
s, whereas measurement j assigns
different outcomes to g and s. Property (e) then follows.
Inductive step. In this part of the proof of lemma 3.1, we aim to show that the conjunction
of IH
m-2
and IH
m-1
implies IH
m
. Let’s check that the properties (a) through (e) are satisfied
if we assume IH
m-2
and IH
m-1
in turn.
IS(a) Consider an arbitrary inequality I
k,S,s,o
m
≥ 0 from I
k
m
, where k is odd and smaller than
m. Since o is non-empty, we can choose a measurement i ∈ X \S and one of its outcomes
a
i
∈ o. If we define o
0
:
= o \ {a
i
}, then the inequality I
k,S,s,o
0
m−1
≥ 0 is an element of
I
k
m−1
. Notice that we have restricted X to X \ {i}, but S can remain unchanged, since
we have chosen i in a way such that S ⊆ X \ {i}.
It follows from IH
m-1
that I
k,S,s,o
0
m−1
≥ 0 is a CE inequality. Moreover, appending an
outcome a
i
of one extra measurement i to a set of pairwise exclusive events doesn’t
Accepted in Quantum 2018-08-17, click title to verify 20
affect their pairwise exclusivity. Therefore, I
k,S,s,o
m
≥ 0 is also a CE inequality, which
completes the proof of the implication IH
m-1
=⇒ IH
m
(a).
In order to address properties (b) through (e), we first present a candidate for D
s
m
m
, given an
arbitrary inequality I
m,X
m
,s
m
,∅
m
≥ 0 in I
m
m
. It is defined recursively, by constructing D
s
m
m
from
D
s
m−2
m−2
, which corresponds to the inequality I
m−2,X
m−2
,s
m−2
,∅
m−2
≥ 0 in I
m−2
m−2
. The labels for the
measurements and events in the two relevant scenarios, (X
m
, O, M
m
) and (X
m−2
, O, M
m−2
),
are
X
m
= {1, . . . , m} s
m
= (a
1
, . . . , a
m
) (21)
X
m−2
= {3, . . . , m} s
m−2
= (a
3
, . . . , a
m
) , (22)
so that s
m−2
is just s
m
with the (outcomes of the) first two measurements omitted. With
these defined, we can proceed to define D
s
m
m
as
D
s
m
m
:
=
(¯a
1
, a
2
), (¯a
2
, s
m−2
), (a
1
,
¯
s
m−2
)
∪
h
(a
1
, a
2
), (a
1
, ¯a
2
), (¯a
1
, ¯a
2
)
× D
s
m−2
m−2
i
. (23)
In the above expression, the Cartesian product has the effect of concatenating the corre-
sponding strings of measurement outcomes. Now we can move on to proving that D
s
m
m
satisfies
properties (b) through (e).
IS(b) Recall the expression (15), which corresponds to
I
m,X
m
,s
m
,∅
m
= f(s
m
) + f (
¯
s
m
) (24)
in our case. If we define B
s
m
m
:
= O
m
\ {s
m
,
¯
s
m
}, then we have
15
I
m,X
m
,s
m
,∅
m
= 1 −
X
α∈B
s
m
m
f(α). (25)
In order to match equation (25) with equation (19), we identify the sum over B
s
m
m
with
a sum over D
s
m
m
via the following:
X
α∈B
s
m
m
f(α) =
X
α∈O
m−2
f(a
1
, ¯a
2
, α) + f (¯a
1
, a
2
, α)
+
X
α∈B
s
m−2
m−2
f(a
1
, a
2
, α) + f (¯a
1
, ¯a
2
, α)
(26)
+ f (¯a
1
, ¯a
2
, a
3
, . . . , a
m
) + f (a
1
, a
2
, ¯a
3
, . . . , ¯a
m
)
=
X
α∈B
s
m−2
m−2
[f(a
1
, a
2
, α) + f (a
1
, ¯a
2
, α) + f (¯a
1
, ¯a
2
, α)] (27)
+ f (¯a
1
, a
2
) + f (¯a
2
, a
3
, . . . , a
m
) + f (a
1
, ¯a
3
, . . . , ¯a
m
)
=
X
β∈D
s
m
m
f(β) (28)
=
X
β∈D
s
m
m
p(β) (29)
Equation (26) is just an expansion of the sum over B
s
m
m
to sums over B
s
m−2
m−2
. In
equation (27) we rearrange the terms and perform marginalization. Equation (28) uses
15
This is a consequence of the function f being normalized, which follows from the normalization of the
probabilistic model p obtained from f by marginalization.
Accepted in Quantum 2018-08-17, click title to verify 21
the fact that the set of events being summed over is precisely D
s
m
m
. Moreover, we use
property (b) for D
s
m−2
m−2
, which is a part of IH
m-2
. Finally, equation (29) replaces f with
p. This is possible, because D
s
m
m
satisfies property (c), which is justified in the following
step.
IS(c) Since D
s
m−2
m−2
involves outcomes of no more than m−3 measurements, it is easy to verify
that D
s
m
m
refers to events with at most m−1 measurement outcomes.
IS(d) The fact that D
s
m−2
m−2
satisfies both properties (d) and (e) implies that D
s
m
m
defined by
equation (23) consists of pairwise exclusive generalized events. One can take a step
further and take D
s
m−2
m−2
to be the union of the refinements of the generalized events that
compose it, without changing its relevant properties. By doing this, it follows that D
s
m
m
is a set of pairwise exclusive events in H[X
m
, O, M
m
].
IS(e) Direct inspection shows that every event in D
s
m
m
comprises a measurement that assigns
a different outcome to s
m
, and one that assigns a different outcome to
¯
s
m
. Hence, D
s
m
m
satisfies property (e) as well.
Thus the proof that IH
m-2
and IH
m-1
imply IH
m
is complete. As a consequence, the induction
hypothesis holds for all integers greater than 1, which proves lemma 3.1.
An immediate corollary of lemma 3.1 is the inclusion CE(H) ⊆ P. The rest of the proof of
theorem 3 consists of demonstrating P = C(H).
Lemma 3.2. The set of probabilistic models satisfying all inequalities in I
n
coincides with
the polytope of classical probabilistic models, i.e. P = C(H).
Proof. The proof is divided into three parts, each corresponding to one of the claims below.
Firstly, we establish the dimension of the polytopes, showing that it is the same for both
polytopes. Next, we show that every facet of P contains a facet of C(H). Finally, we prove
that C(H) does not have any facets other than those contained within facets of P. These
facts together imply P = C(H).
Claim 3.2.1: The dimension of both C(H) and P is 2
n
−2.
Notice that every element of the probabilistic model p ∈ G(H) can be written as a linear
combination of terms that refer to outcome(s) 0 only. There are
P
n−1
j=1
n
j
= 2
n
−2 such
terms, which gives an upper bound on the dimensions of G(H), P and C(H).
One simple consequence of lemma 3.1 is that C(H) is a subset of P. Therefore, it suffices to
show dim(C(H)) = 2
n
−2. Note that the dimension of the classical polytope in an n-Specker
scenario is precisely one less than the number of degrees of freedom in a joint probability
distribution of n binary outcome measurements, i.e. one less than 2
n
−1 [38]. Hence, the
classical polytope has dimension 2
n
−2, as required.
Claim 3.2.2: Every facet of P contains a facet of C(H). Moreover, the facets of P are in
one-to-one correspondence with the inequalities I
n
.
Generically, given a polytope R, a linear inequality A ≥ 0 corresponds to a facet of R if and
only if dim(R) or more vertices of R lie within the hyperplane A = 0. Moreover, if the above
condition is satisfied, the facet of R lies within the hyperplane A = 0.
Accepted in Quantum 2018-08-17, click title to verify 22
Now, to every facet of P, we can associate an inequality from I
n
. Furthermore, each inequality
in I
n
is saturated by a set of exactly 2
n
−2 vertices of C(H), and no two of these sets are the
same. This fact can be seen easily by considering the form of inequalities in equation (15) in
terms of f .
Since 2
n
−2 is also the dimension of C(H) by claim 3.2.1, each inequality in I
n
is associated
with a different facet of C(H). Therefore, every facet of P contains some facet of C(H),
as C(H) ⊆ P. This relation between C(H) and P is depicted in figure 5. Moreover, every
inequality in I
n
corresponds to a different facet of P. The number of facets of P is thus equal
to the number of distinct inequalities in |I
n
|, which is
|I
n
| =
1
2
X
k∈Ω
n
k
!
2
k
2
n−k
= 2
n
2
n−2
= 2
2n−2
, (30)
where Ω denotes the set of odd integers between 1 and n and the factor of
1
/2 takes into
account the fact that I
k,S,s,o
n
= I
k,S,
¯
s,o
n
, so that the corresponding inequality is counted only
once.
R
0
R
Figure 5: Two polytopes, R (gray) and R
0
(blue). R
0
is contained within R, and each
facet of R contains one of R
0
. This schematically depicts the relation between the classical
polytope C(H) (which plays the role of R
0
) and the polytope P (which plays the role of
R) proven in Claim 3.2.2.
Claim 3.2.3: The number of facets of C(H) does not exceed the number of facets of P.
Together with Claim 3.2.2, this implies that C(H) does not have any facets apart from those
contained within facets of P.
The number of vertices of C(H) is N
:
= |O
n
| = 2
n
. These correspond to all the possible
deterministic assignments of probabilities to a joint outcome of the n measurements. As a
matter of fact, no polytope with 2
n
vertices and dimension 2
n
−2 has more than 2
2n−2
facets,
which happens to be the number of facets of P.
In greater generality, McMullen [39] showed that any convex polytope with a given dimension
d and number of vertices N has at most as many faces as the cyclic polytope C(d, N ). Let
us denote the number of facets of C(d, N ) by #
d−1
(C(d, N )). Theorem 4 by Gale [40] asserts
that for even dimensions, d = 2m, this number satisfies
#
2m−1
C(2m, N )
=
N − m
m
!
+
N − m − 1
m − 1
!
. (31)
Accepted in Quantum 2018-08-17, click title to verify 23
Hence, for d = 2
n
− 2 and N = 2
n
, we have the following.
#
d−1
C(d, 2
n
)
=
2
n−1
+ 1
2
n−1
− 1
!
+
2
n−1
2
n−1
− 2
!
(32)
=
1
2
h
2
n−1
+ 1
2
n−1
+ 2
n−1
2
n−1
− 1
i
(33)
= 2
2n−2
(34)
Consequently, the number of facets of C(H) cannot exceed #
d−1
(C(d, 2
n
)) = 2
2n−2
, as
required. It follows that C(H) has no other facets than those it shares with P and thus
C(H) = P.
Lemmas 3.1 and 3.2 provide a complete characterisation of how quantum and almost
quantum behaviours are realized in binary n-Specker scenarios. On the one hand, we have
C(H) ⊆ Q(H) ⊆ Q
1
(H) ⊆ CE(H) as a general hierarchy for all scenarios. On the other hand,
for binary n-Specker scenarios, CE(H) ⊆ P and P = C(H), so that all the inclusions are
actually equalities, concluding the proof of theorem 3.
Hence, binary n-Specker scenarios have the property that every quantum and almost quantum
probabilistic model has a realisation in terms of noncontextual deterministic hidden variable
models. That is, both quantum and almost quantum statistics satisfy the (n−1)-sufficiency
principle for probabilistic models within binary n-Specker scenarios, proving theorem 3.
As a final remark, notice that theorem 3 is not trivial, in the sense that binary n-Specker
scenarios do admit general probabilistic models that violate consistent exclusivity [26].
Accepted in Quantum 2018-08-17, click title to verify 24