# Almost Quantum Correlations are Inconsistent with Specker’s Principle

Tomáš Gonda  Perimeter Institute for Theoretical Physics, Waterloo, Ontario,
Canada, N2L 2Y5  Dept. of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario, Canada, N2L 3G1<EMAIL_ADDRESS>Ravi Kunjwal
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L
2Y5  David Schmid  Perimeter Institute for Theoretical Physics, Waterloo,
Ontario, Canada, N2L 2Y5  Dept. of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, Canada, N2L 3G1  Elie Wolfe  Ana Belén Sainz
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L
2Y5

(September 2, 2024)

###### Abstract

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 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 correlations111Strictly speaking, almost quantum correlations have not
been proven to satisfy the principle of Information Causality yet; only
numerical evidence exists for that claim. [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, 11, 12, 13, 14, 15, 16, 17, 18, 19, 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 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, 24, 8].
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 approach222For 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]. by Acín, 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

###### 0Compatible measurements:

[25, 26]
Let $\\{A_{1},\ldots,A_{n}\\}$ be a set of $n$ measurements with the
corresponding outcome sets being $\\{O_{1},\ldots,O_{n}\\}$. We say that the
elements of the set $\\{A_{1},\ldots,A_{n}\\}$ are _compatible_ (or jointly
measurable) if there exists a measurement $A^{\prime}$ with outcome set
$O_{1}\times\ldots\times O_{n}$, such that all the measurements in the set
$\\{A_{1},\ldots,A_{n}\\}$ can be simulated by post-processing the outcomes of
$A^{\prime}$. That is, the statistics of $\\{A_{1},\ldots,A_{n}\\}$ for every
preparation333Although we denote a preparation with the symbol $\rho$, we are
not assuming here that the preparation is a _quantum_ state. $\rho$ can be
reconstructed perfectly by measuring the given preparation $\rho$ with
$A^{\prime}$:

$\displaystyle\operatorname{Pr}\,(a_{i}\\!\mid\\!A_{i},\rho)=\sum\limits_{\alpha\setminus
a_{i}}\operatorname{Pr}\,(\alpha\\!\mid\\!A^{\prime},\rho)\qquad\forall\,\rho,\;\forall\,i\in\\{1,\ldots,n\\},$
(1)

where $\alpha\in O_{1}\times\ldots\times O_{n}$, and $a_{i}\in O_{i}$ 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.

$\displaystyle A_{1}=\left\\{A_{1}^{0},A_{1}^{1}\right\\}$
$\displaystyle:=\left\\{\vbox{\hbox{$\genfrac{}{}{}{1}{1}{2}$}}\left(\mathbbm{1}+\vbox{\hbox{$\genfrac{}{}{}{1}{1}{\sqrt{2}}$}}\,\sigma_{x}\right),\vbox{\hbox{$\genfrac{}{}{}{1}{1}{2}$}}\left(\mathbbm{1}-\vbox{\hbox{$\genfrac{}{}{}{1}{1}{\sqrt{2}}$}}\,\sigma_{x}\right)\right\\},$
(2) $\displaystyle A_{2}=\left\\{A_{2}^{0},A_{2}^{1}\right\\}$
$\displaystyle:=\left\\{\vbox{\hbox{$\genfrac{}{}{}{1}{1}{2}$}}\left(\mathbbm{1}+\vbox{\hbox{$\genfrac{}{}{}{1}{1}{\sqrt{2}}$}}\,\sigma_{z}\right),\vbox{\hbox{$\genfrac{}{}{}{1}{1}{2}$}}\left(\mathbbm{1}-\vbox{\hbox{$\genfrac{}{}{}{1}{1}{\sqrt{2}}$}}\,\sigma_{z}\right)\right\\},$
(3)

where $\sigma_{x}$ and $\sigma_{z}$ are two of the Pauli matrices. The outcome
set for each measurement is $O=\\{0,1\\}$ in this case. If by $\rho$ we denote
the (arbitrary) state of a quantum system, then for any $a_{1},a_{2}\in O$,

$\displaystyle\mathrm{Pr}\left(a_{1}\\!\mid\\!A_{1},\rho\right)=\mathrm{tr}\left[\vbox{\hbox{$\genfrac{}{}{}{1}{1}{2}$}}\left(\mathbbm{1}+\vbox{\hbox{$\genfrac{}{}{}{1}{(-1)^{a_{1}}}{\sqrt{2}}$}}\,\sigma_{x}\right)\rho\right],$
(4)
$\displaystyle\mathrm{Pr}\left(a_{2}\\!\mid\\!A_{2},\rho\right)=\mathrm{tr}\left[\vbox{\hbox{$\genfrac{}{}{}{1}{1}{2}$}}\left(\mathbbm{1}+\vbox{\hbox{$\genfrac{}{}{}{1}{(-1)^{a_{2}}}{\sqrt{2}}$}}\,\sigma_{z}\right)\rho\right].$
(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
$A^{\prime}=\\{N_{a_{1},a_{2}}\\}$, with $N_{a_{1},a_{2}}$ defined for any
$a_{1},a_{2}\in O$ as follows:

$\displaystyle
N_{a_{1},a_{2}}:=\vbox{\hbox{$\genfrac{}{}{}{1}{1}{4}$}}\left(\mathbbm{1}+\frac{(-1)^{a_{1}}\sigma_{x}+(-1)^{a_{2}}\sigma_{z}}{\sqrt{2}}\right)$
(6)

This measurement has the following properties:

$\displaystyle A_{1}^{a_{1}}$ $\displaystyle=\sum_{a_{2}\in O}N_{a_{1},a_{2}}$
$\displaystyle A_{2}^{a_{2}}$ $\displaystyle=\sum_{a_{1}\in O}N_{a_{1},a_{2}}$
(7)

It follows that, for all $\rho$, the probabilities of outcomes of $A_{1}$ and
of $A_{2}$ can be simulated by coarse-graining the probabilities of outcomes
of $A^{\prime}$:

$\displaystyle\mathrm{Pr}\left(a_{1}\\!\mid\\!A_{1},\rho\right)=\sum_{a_{2}}\mathrm{tr}\left[N_{a_{1},a_{2}}\,\rho\right]=\sum_{a_{2}}\mathrm{Pr}\left(a_{1},a_{2}\\!\mid\\!A^{\prime},\rho\right),$
(8)
$\displaystyle\mathrm{Pr}\left(a_{2}\\!\mid\\!A_{2},\rho\right)=\sum_{a_{1}}\mathrm{tr}\left[N_{a_{1},a_{2}}\,\rho\right]=\sum_{a_{1}}\mathrm{Pr}\left(a_{1},a_{2}\\!\mid\\!A^{\prime},\rho\right),$
(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
$O_{1}\times\ldots\times O_{n}$. 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
$O_{1}\times\ldots\times O_{n}$ that gives the statistics of each of the
measurements $\\{A_{1},\ldots,A_{n}\\}$, 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 contextuality scenario is defined
by a triplet $(X,O,\mathcal{M})$ and is called a _marginal scenario_ 444Some
works in the literature also refer to these scenarios as _measurement
scenarios_. Here we will stick to the notation used in [8]. 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, $\mathcal{M}$ 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 $\mathcal{M}$. An example of the marginal scenario
usually referred to as _Specker’s triangle_ is presented in figure 1(a).

Given a maximal context $\mathbf{x}\in\mathcal{M}$, let $O^{\mathbf{x}}$
denote the set of all families of outcomes of the measurements in
$\mathbf{x}$. That is, $O^{\mathbf{x}}$ naturally corresponds to the set of
functions $\mathbf{x}\to O$. An assignment of probabilities to measurement
outcomes,
$\\{P_{\mathbf{x}}(\mathbf{a})\,:\,\mathbf{x}\in\mathcal{M},\,\mathbf{a}\in
O^{\mathbf{x}}\\}$, 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
$\mathbf{x},\mathbf{x}^{\prime}\in\mathcal{M}$,
$P_{\mathbf{x}|\mathbf{x}\cap\mathbf{x}^{\prime}}=P_{\mathbf{x}^{\prime}|\mathbf{x}\cap\mathbf{x}^{\prime}}$
holds. Here, $P_{\mathbf{x}|\mathbf{x}\cap\mathbf{x}^{\prime}}$ denotes the
marginal distribution of $P_{\mathbf{x}}$ associated to the measurements in
$\mathbf{x}\cap\mathbf{x}^{\prime}$. 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 $(n,k)$.

###### 0Symmetric marginal scenarios:

An $(n,k)$ _symmetric marginal scenario_ is a triple $(X,O,\mathcal{M})$ with
$|X|=n$ and

$\mathcal{M}=\\{\Omega\subseteq X\,:\,|\Omega|=k\\}$ (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 intersection. If an event
appears in more than one hyperedge, it corresponds to outcomes of implementing
different measurement choices that are operationally equivalent555Two
measurement events are operationally equivalent if their probabilities
coincide for every preparation of the system being probed [6]..

There is a close connection between the two approaches. In particular, given a
marginal scenario $(X,O,\mathcal{M})$, we can define an events-based
hypergraph $H[X,O,\mathcal{M}]$ 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 $H[X,O,\mathcal{M}]$ are defined as
$V(H)=\left\\{(\mathbf{a}|\mathbf{x}):\mathbf{x}\in\mathcal{M},\,\mathbf{a}\in
O^{\mathbf{x}}\right\\}$, where $\mathbf{x}\in\mathcal{M}$ is a maximal
context and $\mathbf{a}\in O^{\mathbf{x}}$ is a family of outcomes of the
measurements in $\mathbf{x}$ as before. The vertices of $H[X,O,\mathcal{M}]$
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.

$A_{2}$$A_{1}$$A_{3}$ (a) Specker’s triangle as a marginal scenario.

$(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
$\mathcal{M}=\\{\\{A_{1},A_{2}\\},\\{A_{1},A_{3}\\},\\{A_{2},A_{3}\\}\\}$. 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 $a_{i},a_{j}\in O$ denote the
outcomes of the measurements $A_{i},A_{j}\in X$ from a context
$\\{A_{i},A_{j}\\}\in\mathcal{M}$. 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.

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 compatible666For a discussion on how to
realise such a scenario within quantum theory, we refer the reader to [26]..
Hence, as a marginal scenario $(X,O,\mathcal{M})$ Specker’s triangle features
measurements $X=\\{A_{1},A_{2},A_{3}\\}$ with outcomes $O=\\{0,1\\}$ and
maximal contexts
$\mathcal{M}=\\{\\{A_{1},A_{2}\\},\\{A_{1},A_{3}\\},\\{A_{2},A_{3}\\}\\}$, as
shown in figure 1(a).

As an events-based hypergraph, Specker’s triangle is shown in figure 1(b). 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
$\\{(a_{i}a_{j}|ij)\\}_{a_{i},a_{j}\in O}$. 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 $p:V\rightarrow[0,1]$ 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, $\sum_{v\in e}p(v)=1$ for all $e\in E(H)$. Since the
specification of a probabilistic model is a tuple of real numbers, we often
treat a probabilistic model as a vector $\vec{p}$ 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
$\mathcal{G}(H)$. There are various subsets of $\mathcal{G}(H)$ which are of
particular interest. Here, we define the subsets corresponding to classical,
quantum, and almost quantum models.

###### 0Classical models:

[8, Def. 4.1.1]
A probabilistic model $p\in\mathcal{G}(H)$ is _classical_ if it can be written
as a convex combination of the deterministic ones. Each deterministic
probabilistic model $p\in\mathcal{G}(H)$ satisfies $p(v)\in\\{0,1\\}$ for all
$v\in V$. The set of all classical models on $H$ is denoted by
$\mathcal{C}(H)$ and forms a polytope, whose vertices are the deterministic
probabilistic models.

###### 0Quantum models:

[8, Def. 5.1.1]
A probabilistic model $p\in\mathcal{G}(H)$ is _quantum_ if there exists a
Hilbert space $\mathcal{H}$, a single quantum state $\rho$ (density operator
on $\mathcal{H}$) and a projection operator $P_{v}$ on $\mathcal{H}$
associated to every $v\in V$, such that

  1. 1.

$\sum_{v\in e}P_{v}=\mathbbm{1}_{\mathcal{H}}\quad\forall\,e\in E$,

  2. 2.

$p(v)=\mathrm{tr}\left[P_{v}\,\rho\right]\quad\forall\,v\in V$.

The set of all quantum models on $H$ is denoted by $\mathcal{Q}(H)$.

In the specific case of Bell scenarios the set $\mathcal{Q}(H)$ coincides with
the traditional set of quantum correlations [8, Prop. 5.2.1].

###### 0Almost quantum models:

[8, Def. 6.1.2]
A probabilistic model $p\in\mathcal{G}(H)$ is _almost quantum_ if there exists
a Hilbert space $\mathcal{H}$, a single quantum state $\rho$ (density operator
on $\mathcal{H}$) and a projection operator $P_{v}$ on $\mathcal{H}$
associated to every $v\in V$, such that

  1. 1.

$\sum_{v\in e}P_{v}\leq\mathbbm{1}_{\mathcal{H}}\quad\forall\,e\in E$,

  2. 2.

$p(v)=\mathrm{tr}\left[P_{v}\,\rho\right]\quad\forall\,v\in V$.

The set of all almost quantum models on $H$ is denoted by
$\mathcal{Q}_{1}(H)$.

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 scenario777For a discussion on how to define such Bell
hypergraphs we refer the reader to [8, 28]..

A common feature shared by quantum and almost quantum models is that they both
satisfy the consistent exclusivity principle [9, 8]. Explicitly, this means
that all the CE inequalities defined below hold for every almost quantum (and
quantum) probabilistic model.

###### 0CE inequality:

Given a contextuality scenario $H=(V,E)$, an inequality of the form
$\sum_{v\in\mathscr{S}}p(v)\leq 1$ is a _CE inequality_ if and only if
$\mathscr{S}\subseteq V$ is a set of pairwise exclusive events. A set of
events $\mathscr{S}$ is pairwise exclusive if, for each pair
$u,v\in\mathscr{S}$, there is a hyperedge $e\in E$ such that $u,v\in e$.

It is worth mentioning that for scenarios of the type $H[X,O,\mathcal{M}]$
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, $(\mathbf{a}|\mathbf{x})$ and
$(\mathbf{a^{\prime}}|\mathbf{x^{\prime}})$, are exclusive if there is a
measurement in $\mathbf{x}\cap\mathbf{x^{\prime}}$ that assigns different
outcomes in $\mathbf{a}$ and $\mathbf{a^{\prime}}$. Moreover, for these
scenarios the set of empirical models on $(X,O,\mathcal{M})$ is equivalent to
the set of probabilistic models on $H[X,O,\mathcal{M}]$.

###### 0Consistent 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
inequalities888A 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])..

###### 0CE models:

A probabilistic model $p\in\mathcal{G}(H)$ is _consistently exclusive_ if it
satisfies the CE principle. The set of all consistently exclusive models on
$H$ is denoted by $\mathcal{CE}(H)$999The set $\mathcal{CE}(H)$ defined here
corresponds to the set $\mathcal{CE}^{1}(H)$ defined in [8]. 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 stated0:

“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 measurements0:

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. projective) 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 [24, 8]. However, it
is important to recognize that these statistical constraints are not
equivalent 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 “answers” (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 models0:

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.

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 measurements0:

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 $n\geq 3$, as those would merely be
different representations of the same compatibility relations. The qualifier
$n\geq 3$ 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 models0:

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 $n\geq 3$.

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 $n\geq 4$, 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. ∎

Specker’s principle$\Longleftrightarrow$Pairwise sufficiencyfor
measurements$\Longleftrightarrow$All-but-one sufficiencyfor
measurements$\mathrel{\mbox{=}}\joinrel\Rightarrow$$\mathrel{\mbox{=}}\joinrel\Rightarrow$$\mathrel{\mbox{=}}\joinrel\Rightarrow$Consistent
exclusivityPairwise sufficiencyfor probabilistic
models$\mathrel{\mbox{=}}\joinrel\Rightarrow$All-but-one sufficiencyfor
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 $\mathcal{Q}(H)$ for an events-based hypergraph
$H$ obtained from an $(n,2)$ symmetric marginal scenario coincides with
$\mathcal{C}(H)$. Here, we demonstrate that there are probabilistic models in
$\mathcal{Q}_{1}(H)$ for $H$ obtained from an $(4,2)$ symmetric marginal
scenario (see figure 3), which are outside of $\mathcal{C}(H)$. 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

$\displaystyle I_{\mathrm{pent}}=$ $\displaystyle-\langle
X_{1}X_{2}\rangle-\langle X_{1}X_{3}\rangle+\langle X_{1}X_{4}\rangle+\langle
X_{1}\rangle-\langle X_{2}X_{3}\rangle$ (11) $\displaystyle+\langle
X_{2}X_{4}\rangle+\langle X_{2}\rangle+\langle X_{3}X_{4}\rangle+\langle
X_{3}\rangle-\langle X_{4}\rangle\leq 2\,,$

where $\langle X_{i}X_{j}\rangle=\sum_{a_{i},a_{j}\in
O}(-1)^{a_{i}+a_{j}}p(a_{i}a_{j}|ij)$ and $\langle X_{j}\rangle=\sum_{a_{j}\in
O}(-1)^{a_{j}}p(a_{j}|j)$. 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_{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
$\mathcal{M}=\\{\\{A_{1},A_{2}\\},\\{A_{1},A_{3}\\},\\{A_{1},A_{4}\\},\\{A_{2},A_{3}\\},\\{A_{2},A_{4}\\},\\{A_{3},A_{4}\\}\\}$.

A linear program optimisation over the maximum values of equation (11) for
general probabilistic models yields a value of $I_{\mathrm{pent}}^{*}=6$.
Similarly, optimising equation (11) via a semidefinite program [8] yields a
maximum value of $I_{\mathrm{pent}}^{\mathrm{AQ}}=2.5$ 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 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
effects101010In 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. [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 $\mathcal{C}(H)\neq\mathcal{Q}_{1}(H)$ 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 $\mathcal{Q}_{1}(H)$ and
$\mathcal{C}(H)$ 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 $n\geq 2$ and dichotomic measurements, then the sets
$\mathcal{C}(H)$, $\mathcal{Q}(H)$, $\mathcal{Q}_{1}(H)$ and $\mathcal{CE}(H)$
all coincide.

The proof of theorem 3 is presented in appendix A. This theorem tells us that
the sets $\mathcal{Q}_{1}(H)$ and $\mathcal{C}(H)$ coincide whenever $H$
corresponds to an $(n,n-1)$ symmetric marginal scenario with binary outcome
measurements, in agreement with the conjecture above.

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 [29, 9, 24, 34, 35, 36], in the sense that they are
uniquely identified111111Almost 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. 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 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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##  Appendix 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 1(b), a generalized event $g=(a_{1}|1)$ can be defined
as either the coarse graining of
$\\{(a_{1},a_{2}|1,2),(a_{1},\bar{a}_{2}|1,2)\\}$ or that of
$\\{(a_{1},a_{3}|1,3),(a_{1},\bar{a}_{3}|1,3)\\}$, since the measurement
$A_{1}$ is compatible with both $A_{2}$ and $A_{3}$ and the no-disturbance
condition holds. Each of these sets
$\\{(a_{1},a_{j}|ij),(a_{1},\bar{a}_{j}|ij)\\}$ 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 $g,g^{\prime}$ are said to be exclusive if
events in a refinement of $g$ are pairwise exclusive with events in a
refinement of $g^{\prime}$. Equivalently, one can say that the union of any
refinements of $g$ and $g^{\prime}$ 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 $\mathcal{M}$ 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 1(a) depicts a $3$-Specker scenario and
figure 4 a $4$-Specker scenario.

$A_{1}$$A_{2}$$A_{3}$$A_{4}$ Figure 4: The marginal scenario representation of
the $\mathbf{4}$-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
$\mathcal{M}=\\{\\{A_{1},A_{2},A_{3}\\},\\{A_{1},A_{2},A_{4}\\},\\{A_{1},A_{3},A_{4}\\},\\{A_{2},A_{3},A_{4}\\}\\}$.

From the very definition of the scenario it immediately follows that every
quantum empirical model in $(X,O,\mathcal{M})$ 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
$X=\\{1,2,\dots,n\\}$. The events in $H\coloneqq H[X,O,\mathcal{M}]$ are
denoted by $(a_{i_{1}},\dots,a_{i_{n-1}})$, where
$\\{i_{1},\dots,i_{n-1}\\}\in\mathcal{M}$ and $a_{i_{k}}\in O$ is the outcome
of measurement $i_{k}$. By $(a_{i_{1}},\dots,a_{i_{k}})$, we denote a
generalized event with the remaining $n-k-1$ measurements in the context
$\\{i_{1},\dots,i_{n-1}\\}$ disregarded. A probabilistic model121212For
simplicity we will denote by $p(a_{i_{1}},\dots,a_{i_{k}})$ the conditional
probability distribution $p(a_{i_{1}},\dots,a_{i_{k}}\mid i_{1},\dots,i_{k})$.
Similarly, the events $(a_{i_{1}},\dots,a_{i_{n-1}}\mid i_{1},\dots,i_{n-1})$
in $H[X,O,\mathcal{M}]$ are simply denoted by $(a_{i_{1}},\dots,a_{i_{n-1}})$.
$\vec{p}$ is then a family of probabilities indexed by the contexts and
outcomes of the measurements in the respective context

$\vec{p}=\bigl{\\{}p(a_{i_{1}},\dots,a_{i_{n-1}})\,:\,\text{$\Omega\coloneqq\\{i_{1},\dots,i_{n-1}\\}\in\mathcal{M}$,
$(a_{i_{1}},\dots,a_{i_{n-1}})\in O^{\Omega}$}\bigr{\\}}.$ (12)

In this notation, the no-disturbance condition guarantees that the marginal
distribution ${p(a_{i_{1}},\dots,a_{i_{k}})}$ 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 $n\geq 2$. Then the sets
$\mathcal{C}(H)$, $\mathcal{Q}(H)$, $\mathcal{Q}_{1}(H)$ and $\mathcal{CE}(H)$
all coincide.

First of all notice that there are inclusions
$\mathcal{C}(H)\subseteq\mathcal{Q}(H)\subseteq\mathcal{Q}_{1}(H)\subseteq\mathcal{CE}(H)$,
which hold in general for any contextuality scenario H [8]. Therefore, in
order to prove theorem 3, we only need to show
$\mathcal{CE}(H)\subseteq\mathcal{C}(H)$. This can be broken down into three
parts. Firstly, we define a finite set of linear inequality constraints
$\mathcal{I}_{n}$ on $\mathcal{G}(H)$ and the corresponding polytope
$\mathcal{P}$, which satisfies them. Then we show in lemma 3.1 that all these
inequalities are actually CE inequalities, so that
$\mathcal{CE}(H)\subseteq\mathcal{P}$ holds. The proof concludes with lemma
3.2, which establishes the equality of $\mathcal{C}(H)$ and $\mathcal{P}$.

Consider a set of measurements $S\subseteq X$ with $k\coloneqq|S|$. Let
$\mathbf{s}=\\{a_{r}:r\in S\\}\in O^{S}$ be a collection of outcomes of the
$k$ measurements $S$. Similarly $\mathbf{o}=\\{a_{r}:r\in X\setminus S\\}\in
O^{X\setminus S}$ is a collection of outcomes of the $n-k$ measurements
$X\setminus S$. We define the set $\mathcal{I}_{n}$ in terms of linear
functionals $\mathcal{I}_{n}^{k,S,\mathbf{s},\mathbf{o}}$ 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.

$\mathcal{I}_{n}^{k,S,\mathbf{s},\mathbf{o}}\coloneqq
p(\mathbf{o})+\sum_{j=1}^{k-1}(-1)^{j}\sum_{\\{i_{1},\dots,i_{j}\\}\subseteq
S}p(a_{i_{1}},\dots,a_{i_{j}},\mathbf{o}),$ (13)

with $p(\mathbf{o})=1$ whenever $k=n$. We use the notation
$\mathcal{I}_{n}^{k}$ for the set of corresponding inequalities with a given
$k$.

$\displaystyle\mathcal{I}_{n}^{k}\coloneqq\left\\{\mathcal{I}_{n}^{k,S,\mathbf{s},\mathbf{o}}\geq
0\,:\,\text{$\mathbf{s}\in O^{S}$, $\mathbf{o}\in O^{X\setminus S}$,
$S\subseteq X$, $|S|=k$}\right\\}$ (14)

We define the full set of inequalities, $\mathcal{I}_{n}$, as the union of all
$\mathcal{I}_{n}^{k}$ 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 $\vec{p}$, which
specifies a probability distribution for each proper subset of $X$, there is a
function $f\colon O^{X}\to\mathbb{R}$ that recovers $\vec{p}$ by
marginalization131313The marginal of $f$ over $\Omega\subseteq X$ is computed
by summing the values of $f$ over $O^{\Omega}$, which yields a function
$O^{X\setminus\Omega}\to\mathbb{R}$. [7]. That is, by marginalizing $f$ over
any measurement $i$, we obtain the original probabilistic model for the
context $X\setminus\\{i\\}$ Even though this function is not unique, we can
write $\mathcal{I}_{n}^{k,S,\mathbf{s},\mathbf{o}}$ in terms of $f$ uniquely.
In particular, we have

$\mathcal{I}_{n}^{k,S,\mathbf{s},\mathbf{o}}=f(\mathbf{s},\mathbf{o})+f(\bar{\mathbf{s}},\mathbf{o}).$
(15)

Notice that one can choose $f$ to be a probability distribution over $O^{X}$
if and only if $\vec{p}$ 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 $\mathcal{I}_{3}$ are

$\displaystyle\mathcal{I}_{3}^{1}$ $\displaystyle=\left\\{p(a_{i},a_{j})\geq
0\,:\,\text{$\\{i,j\\}\in\\{1,2,3\\}$, $(a_{i},a_{j})\in
O^{\\{i,j\\}}$}\right\\},$ (16) $\displaystyle\mathcal{I}_{3}^{3}$
$\displaystyle=\left\\{1-p(a_{1})-p(a_{2})-p(a_{3})+p(a_{1},a_{2})+p(a_{1},a_{3})+p(a_{2},a_{3})\geq
0\,:\,a_{i}\in O^{\\{i\\}}\right\\}.$ (17)

The first ones are just the positivity constraints, which are always CE
inequalities. Likewise, inequalities in $\mathcal{I}_{3}^{3}$ can be written
in a form that is manifestly CE, as

$\mathcal{I}_{3}^{3}=\left\\{1\geq
p(a_{1},\bar{a}_{2})+p(a_{2},\bar{a}_{3})+p(\bar{a}_{1},a_{3})\,:\,a_{i}\in
O^{\\{i\\}}\right\\},$ (18)

where $\bar{a}_{i}\coloneqq a_{i}+1\pmod{2}$. A generalization of the fact
that $\mathcal{I}_{3}$ consists of CE inequalities to all $\mathcal{I}_{n}$ is
the content of the next lemma.

######  Lemma 3.1.

Every element of $\mathcal{I}_{n}$ 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${}_{\text{m}}$).

For all $m$,

  1. (a)

every element of $\mathcal{I}_{m}^{k}$ is a CE inequality for any odd $k$
between 1 and $m-1$.

Moreover, if $m$ is odd, then given any inequality141414Note that when $k$ is
equal to $n$ in $\mathcal{I}_{n}^{k,S,\textbf{s},\textbf{o}}$, $S$ has to be
$X$ itself and o contains no outcomes.
$\mathcal{I}_{m}^{m,X,\textbf{s},\emptyset}\geq 0$ in $\mathcal{I}_{m}^{m}$,
there exists a set $\mathcal{D}_{m}^{\textbf{s}}$ of generalized events (which
by a slight abuse of notation we refer to simply as events), such that the
following properties hold:

  1. (b)

One can express $\mathcal{I}_{m}^{m,X,\textbf{s},\emptyset}$ as

$\mathcal{I}_{m}^{m,X,\textbf{s},\emptyset}=1-\sum_{\alpha\in\mathcal{D}_{m}^{\textbf{s}}}p(\alpha).$
(19)

  2. (c)

Each of the events in $\mathcal{D}_{m}^{\textbf{s}}$ involves outcomes of at
most $m-1$ measurements among $X$.

  3. (d)

The set $\mathcal{D}_{m}^{\textbf{s}}$ consists of pairwise exclusive events.

  4. (e)

Each event $g$ in the set $\mathcal{D}_{m}^{\textbf{s}}$ 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 $\bar{\textbf{s}}$.

Notice that the statement that all elements of $\mathcal{I}_{m}$ 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${}_{\text{2}}$ and
IH${}_{\text{3}}$ both hold (base case), and that IH${}_{\text{m-2}}$ with
IH${}_{\text{m-1}}$ together imply IH${}_{\text{m}}$ (inductive step).

######  Base case.

The fact that IH${}_{\text{2}}$ holds follows because $\mathcal{I}_{2}$
consists of positivity constraints only. Since 3 is odd, we need to check all
five properties in order to establish IH${}_{\text{3}}$. Let’s analyze them
one by one.

  1. BC(a)

As we have seen in example 1, $\mathcal{I}_{3}^{1}$ consists of positivity
constraints, which are CE inequalities.

  2. BC(b)

Following the discussion in example 1, we can define

$\mathcal{D}_{3}^{(a_{1},a_{2},a_{3})}\coloneqq\\{(a_{1},\bar{a}_{2}),(a_{2},\bar{a}_{3}),(\bar{a}_{1},a_{3})\\},$
(20)

whence equation (19) holds for $m=3$ and $\textbf{s}=(a_{1},a_{2},a_{3})$.

  3. BC(c)

This property is immediate from the definition of
$\mathcal{D}_{3}^{(a_{1},a_{2},a_{3})}$ above.

  4. BC(d)

For every pair of events in $\mathcal{D}_{3}^{(a_{1},a_{2},a_{3})}$, there is
a measurement with different outcome for each of the two events in the pair.
Therefore, the events in $\mathcal{D}_{3}^{(a_{1},a_{2},a_{3})}$ are pairwise
exclusive.

  5. BC(e)

Each event in $\mathcal{D}_{3}^{(a_{1},a_{2},a_{3})}$ is of the form
$g=(a_{i},\bar{a}_{j})$, for some $i$ and $j$ in $\\{1,2,3\\}$. Hence,
measurement $i$ assigns different outcomes to $g$ and $\bar{\textbf{s}}$,
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${}_{\text{m-2}}$ and IH${}_{\text{m-1}}$ implies IH${}_{\text{m}}$. Let’s
check that the properties (a) through (e) are satisfied if we assume
IH${}_{\text{m-2}}$ and IH${}_{\text{m-1}}$ in turn.

  1. IS(a)

Consider an arbitrary inequality
$\mathcal{I}_{m}^{k,S,\textbf{s},\textbf{o}}\geq 0$ from
$\mathcal{I}_{m}^{k}$, where $k$ is odd and smaller than $m$. Since o is non-
empty, we can choose a measurement $i\in X\setminus S$ and one of its outcomes
$a_{i}\in\textbf{o}$. If we define
$\textbf{o}^{\prime}\coloneqq\textbf{o}\setminus\\{a_{i}\\}$, then the
inequality $\mathcal{I}_{m-1}^{k,S,\textbf{s},\textbf{o}^{\prime}}\geq 0$ is
an element of $\mathcal{I}_{m-1}^{k}$. Notice that we have restricted $X$ to
$X\setminus\\{i\\}$, but $S$ can remain unchanged, since we have chosen $i$ in
a way such that $S\subseteq X\setminus\\{i\\}$.

It follows from IH${}_{\text{m-1}}$ that
$\mathcal{I}_{m-1}^{k,S,\textbf{s},\textbf{o}^{\prime}}\geq 0$ is a CE
inequality. Moreover, appending an outcome $a_{i}$ of one extra measurement
$i$ to a set of pairwise exclusive events doesn’t affect their pairwise
exclusivity. Therefore, $\mathcal{I}_{m}^{k,S,\textbf{s},\textbf{o}}\geq 0$ is
also a CE inequality, which completes the proof of the implication
IH${}_{\text{m-1}}$$\implies$IH${}_{\text{m}}$(a).

In order to address properties (b) through (e), we first present a candidate
for $\mathcal{D}_{m}^{\textbf{s}_{m}}$, given an arbitrary inequality
$\mathcal{I}_{m}^{m,X_{m},\textbf{s}_{m},\emptyset}\geq 0$ in
$\mathcal{I}_{m}^{m}$. It is defined recursively, by constructing
$\mathcal{D}_{m}^{\textbf{s}_{m}}$ from
$\mathcal{D}_{m-2}^{\textbf{s}_{m-2}}$, which corresponds to the inequality
$\mathcal{I}_{m-2}^{m-2,X_{m-2},\textbf{s}_{m-2},\emptyset}\geq 0$ in
$\mathcal{I}_{m-2}^{m-2}$. The labels for the measurements and events in the
two relevant scenarios, $(X_{m},O,\mathcal{M}_{m})$ and
$(X_{m-2},O,\mathcal{M}_{m-2})$, are

$\displaystyle X_{m}$ $\displaystyle=\\{1,\dots,m\\}$
$\displaystyle\textbf{s}_{m}$ $\displaystyle=\left(a_{1},\dots,a_{m}\right)$
(21) $\displaystyle X_{m-2}$ $\displaystyle=\\{3,\dots,m\\}$
$\displaystyle\textbf{s}_{m-2}$
$\displaystyle=\left(a_{3},\dots,a_{m}\right),$ (22)

so that $\textbf{s}_{m-2}$ is just $\textbf{s}_{m}$ with the (outcomes of the)
first two measurements omitted. With these defined, we can proceed to define
$\mathcal{D}_{m}^{\textbf{s}_{m}}$ as

$\mathcal{D}_{m}^{\textbf{s}_{m}}\coloneqq\bigl{\\{}(\bar{a}_{1},a_{2}),(\bar{a}_{2},\textbf{s}_{m-2}),(a_{1},\bar{\textbf{s}}_{m-2})\bigr{\\}}\cup\Bigl{[}\bigl{\\{}(a_{1},a_{2}),(a_{1},\bar{a}_{2}),(\bar{a}_{1},\bar{a}_{2})\bigr{\\}}\times\mathcal{D}_{m-2}^{\textbf{s}_{m-2}}\Bigr{]}.$
(23)

In the above expression, the Cartesian product has the effect of concatenating
the corresponding strings of measurement outcomes. Now we can move on to
proving that $\mathcal{D}_{m}^{\textbf{s}_{m}}$ satisfies properties (b)
through (e).

  1. IS(b)

Recall the expression (15), which corresponds to

$\mathcal{I}_{m}^{m,X_{m},\mathbf{s}_{m},\emptyset}=f(\mathbf{s}_{m})+f(\bar{\mathbf{s}}_{m})$
(24)

in our case. If we define $\mathcal{B}_{m}^{\textbf{s}_{m}}\coloneqq
O^{m}\setminus\\{\textbf{s}_{m},\bar{\textbf{s}}_{m}\\}$, then we
have151515This is a consequence of the function $f$ being normalized, which
follows from the normalization of the probabilistic model $p$ obtained from
$f$ by marginalization.

$\mathcal{I}_{m}^{m,X_{m},\mathbf{s}_{m},\emptyset}=1-\sum_{\mathbf{\alpha}\in\mathcal{B}_{m}^{\textbf{s}_{m}}}f(\mathbf{\alpha}).$
(25)

In order to match equation (25) with equation (19), we identify the sum over
$\mathcal{B}_{m}^{\textbf{s}_{m}}$ with a sum over
$\mathcal{D}_{m}^{\textbf{s}_{m}}$ via the following:

$\displaystyle\scalebox{0.75}{\mbox{$\displaystyle\sum_{\mathbf{\alpha}\in\mathcal{B}^{\mathbf{s}_{m}}_{m}}$}}f(\mathbf{\alpha})$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\sum_{\mathbf{\alpha}\in
O^{m-2}}$}}\big{[}f(a_{1},\bar{a}_{2},\mathbf{\alpha})+f(\bar{a}_{1},a_{2},\mathbf{\alpha})\big{]}+\scalebox{0.75}{\mbox{$\displaystyle\sum_{{\mathbf{\alpha}\in\mathcal{B}^{\mathbf{s}_{m-2}}_{m-2}}}$}}\big{[}f(a_{1},a_{2},\mathbf{\alpha})+f(\bar{a}_{1},\bar{a}_{2},\mathbf{\alpha})\big{]}$
(26)
$\displaystyle\qquad+f(\bar{a}_{1},\bar{a}_{2},a_{3},\dots,a_{m})+f(a_{1},a_{2},\bar{a}_{3},\dots,\bar{a}_{m})$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\sum\limits_{\mathbf{\alpha}\in\mathcal{B}^{\mathbf{s}_{m-2}}_{m-2}}$}}\left[f(a_{1},a_{2},\mathbf{\alpha})+f(a_{1},\bar{a}_{2},\mathbf{\alpha})+f(\bar{a}_{1},\bar{a}_{2},\mathbf{\alpha})\right]$
(27)
$\displaystyle\qquad+f(\bar{a}_{1},a_{2})+f(\bar{a}_{2},a_{3},\dots,a_{m})+f(a_{1},\bar{a}_{3},\dots,\bar{a}_{m})$
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\sum_{\mathbf{\beta}\in\mathcal{D}_{m}^{\mathbf{s}_{m}}}$}}f(\mathbf{\beta})$
(28)
$\displaystyle=\scalebox{0.75}{\mbox{$\displaystyle\sum_{\mathbf{\beta}\in\mathcal{D}_{m}^{\mathbf{s}_{m}}}$}}p(\mathbf{\beta})$
(29)

Equation (26) is just an expansion of the sum over
$\mathcal{B}_{m}^{\textbf{s}_{m}}$ to sums over
$\mathcal{B}_{m-2}^{\textbf{s}_{m-2}}$. In equation (27) we rearrange the
terms and perform marginalization. Equation (28) uses the fact that the set of
events being summed over is precisely $\mathcal{D}_{m}^{\textbf{s}_{m}}$.
Moreover, we use property (b) for $\mathcal{D}_{m-2}^{\textbf{s}_{m-2}}$,
which is a part of IH${}_{\text{m-2}}$. Finally, equation (29) replaces $f$
with $p$. This is possible, because $\mathcal{D}_{m}^{\textbf{s}_{m}}$
satisfies property (c), which is justified in the following step.

  2. IS(c)

Since $\mathcal{D}_{m-2}^{\textbf{s}_{m-2}}$ involves outcomes of no more than
$m-3$ measurements, it is easy to verify that
$\mathcal{D}_{m}^{\textbf{s}_{m}}$ refers to events with at most $m-1$
measurement outcomes.

  3. IS(d)

The fact that $\mathcal{D}_{m-2}^{\textbf{s}_{m-2}}$ satisfies both properties
(d) and (e) implies that $\mathcal{D}_{m}^{\textbf{s}_{m}}$ defined by
equation (23) consists of pairwise exclusive generalized events. One can take
a step further and take $\mathcal{D}_{m-2}^{\textbf{s}_{m-2}}$ 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
$\mathcal{D}_{m}^{\textbf{s}_{m}}$ is a set of pairwise exclusive events in
$H[X_{m},O,\mathcal{M}_{m}]$.

  4. IS(e)

Direct inspection shows that every event in $\mathcal{D}_{m}^{\textbf{s}_{m}}$
comprises a measurement that assigns a different outcome to $\textbf{s}_{m}$,
and one that assigns a different outcome to $\bar{\textbf{s}}_{m}$. Hence,
$\mathcal{D}_{m}^{\textbf{s}_{m}}$ satisfies property (e) as well.

Thus the proof that IH${}_{\text{m-2}}$ and IH${}_{\text{m-1}}$ imply
IH${}_{\text{m}}$ 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
$\mathcal{CE}(H)\subseteq\mathcal{P}$. The rest of the proof of theorem 3
consists of demonstrating $\mathcal{P}=\mathcal{C}(H)$.

######  Lemma 3.2.

The set of probabilistic models satisfying all inequalities in
$\mathcal{I}_{n}$ coincides with the polytope of classical probabilistic
models, i.e. $\mathcal{P}=\mathcal{C}(H)$.

###### 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
$\mathcal{P}$ contains a facet of $\mathcal{C}(H)$. Finally, we prove that
$\mathcal{C}(H)$ does not have any facets _other_ than those contained within
facets of $\mathcal{P}$. These facts together imply
$\mathcal{P}=\mathcal{C}(H)$.

###### Claim 3.2.1:

The dimension of both $\mathcal{C}(H)$ and $\mathcal{P}$ is $2^{n}-2$.

Notice that every element of the probabilistic model $p\in\mathcal{G}(H)$ can
be written as a linear combination of terms that refer to outcome(s) $0$ only.
There are $\sum_{j=1}^{n-1}{{n}\choose{j}}=2^{n}-2$ such terms, which gives an
upper bound on the dimensions of $\mathcal{G}(H)$, $\mathcal{P}$ and
$\mathcal{C}(H)$.

One simple consequence of lemma 3.1 is that $\mathcal{C}(H)$ is a subset of
$\mathcal{P}$. Therefore, it suffices to show
$\mathrm{dim}(\mathcal{C}(H))=2^{n}-2$. 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 $\mathcal{P}$ contains a facet of $\mathcal{C}(H)$. Moreover,
the facets of $\mathcal{P}$ are in one-to-one correspondence with the
inequalities $\mathcal{I}_{n}$.

Generically, given a polytope $\mathcal{R}$, a linear inequality
$\mathcal{A}\geq 0$ corresponds to a facet of $\mathcal{R}$ if and only if
$\mathrm{dim(\mathcal{R})}$ or more vertices of $\mathcal{R}$ lie within the
hyperplane $\mathcal{A}=0$. Moreover, if the above condition is satisfied, the
facet of $\mathcal{R}$ lies within the hyperplane $\mathcal{A}=0$.

Now, to every facet of $\mathcal{P}$, we can associate an inequality from
$\mathcal{I}_{n}$. Furthermore, each inequality in $\mathcal{I}_{n}$ is
saturated by a set of exactly $2^{n}-2$ vertices of $\mathcal{C}(H)$, 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 $\mathcal{C}(H)$ by claim 3.2.1, each
inequality in $\mathcal{I}_{n}$ is associated with a different facet of
$\mathcal{C}(H)$. Therefore, every facet of $\mathcal{P}$ contains some facet
of $\mathcal{C}(H)$, as $\mathcal{C}(H)\subseteq\mathcal{P}$. This relation
between $\mathcal{C}(H)$ and $\mathcal{P}$ is depicted in figure 5. Moreover,
every inequality in $\mathcal{I}_{n}$ corresponds to a different facet of
$\mathcal{P}$. The number of facets of $\mathcal{P}$ is thus equal to the
number of distinct inequalities in $|\mathcal{I}_{n}|$, which is

$|\mathcal{I}_{n}|=\frac{1}{2}\sum_{k\in\Omega}{{n}\choose{k}}\,2^{k}\,2^{n-k}=2^{n}\,2^{n-2}=2^{2n-2},$
(30)

where $\Omega$ denotes the set of odd integers between 1 and $n$ and the
factor of ${\scriptstyle\nicefrac{{1}}{{2}}}$ takes into account the fact that
$\mathcal{I}_{n}^{k,S,\mathbf{s},\mathbf{o}}=\mathcal{I}_{n}^{k,S,\bar{\mathbf{s}},\mathbf{o}}$,
so that the corresponding inequality is counted only once.

$\mathcal{R}^{\prime}$$\mathcal{R}$ Figure 5: Two polytopes, $\mathcal{R}$
(gray) and $\mathcal{R}^{\prime}$ (blue). $\mathcal{R}^{\prime}$ is contained
within $\mathcal{R}$, and each facet of $\mathcal{R}$ contains one of
$\mathcal{R}^{\prime}$. This schematically depicts the relation between the
classical polytope $\mathcal{C}(H)$ (which plays the role of
$\mathcal{R}^{\prime}$) and the polytope $\mathcal{P}$ (which plays the role
of $\mathcal{R}$) proven in Claim 3.2.2.

###### Claim 3.2.3:

The number of facets of $\mathcal{C}(H)$ does not exceed the number of facets
of $\mathcal{P}$. Together with Claim 3.2.2, this implies that
$\mathcal{C}(H)$ does not have any facets apart from those contained within
facets of $\mathcal{P}$.

The number of vertices of $\mathcal{C}(H)$ is $N\coloneqq|O^{n}|=2^{n}$. 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 $\mathcal{P}$.

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

$\\#_{2m-1}\big{(}C(2m,N)\big{)}={N-m\choose m}+{N-m-1\choose m-1}.$ (31)

Hence, for $d=2^{n}-2$ and $N=2^{n}$, we have the following.

$\displaystyle\\#_{d-1}\big{(}C(d,2^{n})\big{)}$
$\displaystyle={2^{n-1}+1\choose 2^{n-1}-1}+{2^{n-1}\choose 2^{n-1}-2}$ (32)
$\displaystyle=\frac{1}{2}\left[\bigl{(}2^{n-1}+1\bigr{)}2^{n-1}+2^{n-1}\bigl{(}2^{n-1}-1\bigr{)}\right]$
(33) $\displaystyle=2^{2n-2}$ (34)

Consequently, the number of facets of $\mathcal{C}(H)$ cannot exceed
$\\#_{d-1}(C(d,2^{n}))=2^{2n-2}$, as required. It follows that
$\mathcal{C}(H)$ has no other facets than those it shares with $\mathcal{P}$
and thus $\mathcal{C}(H)=\mathcal{P}$. ∎

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
$\mathcal{C}(H)\subseteq\mathcal{Q}(H)\subseteq\mathcal{Q}_{1}(H)\subseteq\mathcal{CE}(H)$
as a general hierarchy for all scenarios. On the other hand, for binary
$n$-Specker scenarios, $\mathcal{CE}(H)\subseteq\mathcal{P}$ and
$\mathcal{P}=\mathcal{C}(H)$, 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].