# On the rank of two-dimensional simplicial distributions

Cihan<EMAIL_ADDRESS>Department of Mathematics, Bilkent
University, Ankara, Turkey

###### Abstract

Simplicial distributions provide a framework for studying quantum
contextuality, a generalization of Bell’s non-locality. Understanding extremal
simplicial distributions is of fundamental importance with applications to
quantum computing. We introduce a rank formula for twisted simplicial
distributions defined for $2$-dimensional measurement spaces and provide a
systematic approach for describing extremal distributions.

###### Contents

  1. 1 Introduction
  2. 2 Simplicial distributions
    1. 2.1 Two-dimensional simplicial sets
    2. 2.2 Simplicial distributions
    3. 2.3 Twisted distributions
    4. 2.4 Product of distributions
    5. 2.5 Cohomology exact sequence
  3. 3 Distributions on graphs
    1. 3.1 Distributions on graphs
    2. 3.2 Rank of a distribution
    3. 3.3 The rank formula
  4. 4 Examples
    1. 4.1 $N$-cycle scenario
    2. 4.2 Boundary of tetrahedron
    3. 4.3 Mermin torus

##  1 Introduction

The theory of simplicial distributions introduced in [1] is a framework for
describing quantum contextuality, a fundamental feature of quantum theory
generalizing Bell’s non-locality [2, 3]. In this framework, measurements and
outcomes are represented by spaces modeled by combinatorial objects called
simplicial sets [4]. This framework generalizes the theory of non-signaling
distributions formulated in the language of sheaf-theory [5] by elevating sets
of measurements and outcomes to spaces of measurements and outcomes.
Simplicial distributions constitute a polytope whose vertices (extreme
distributions) are of fundamental importance in quantum foundations [6, 7, 8,
9]. Non-contextual distributions are described by a subpolytope whose facets
are given by the Bell inequalities. In this paper, we introduce graph-
theoretic methods to identify the contextual vertices of the polytope of
twisted simplicial distributions.

(a)

(b)

Figure 1: Triangle $\sigma$ with faces $x=d_{2}\sigma$, $y=d_{0}\sigma$ and
$z=d_{1}\sigma$. (a) A twisted simplicial distribution on $\sigma$ with
$\beta(\sigma)=0$. The $d_{0}$-face is given by $p_{y}^{0}=p^{00}+p^{10}$. (b)
A twisted simplicial distribution with $\beta(\sigma)=1$. The $d_{0}$-face is
given by $p_{y}^{0}=p^{01}+p^{11}$.

A simplicial distribution on a scenario consisting of a space $X$ of
measurements and space $Y$ of outcomes is a simplicial set map of the form

$p:X\to D(Y)$

where $D(Y)$ is a simplicial set that models the space of distributions on the
outcome space. In this paper we will restrict to the case where $X$ is a
simplicial set generated by $2$-dimensional simplices
$\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$, and $Y$ is the nerve space of the
additive group $\mathbb{Z}_{2}=\\{0,1\\}$. Topologically $X$ is a
$2$-dimensional space obtained by gluing the $2$-simplices (triangles) along
their faces and possibly collapsing some of them. Our choice of the outcomes
space reflects the restriction that our measurements have binary outcomes.
More concretely, a simplicial distribution $p$ consists of a family
$\\{p_{\sigma_{i}}:\,i=1,2,\cdots,N\\}$ of probability distributions

$\\{p_{\sigma_{i}}^{ab}\in\mathbb{R}_{\geq
0}:\,a,b\in\mathbb{Z}_{2},\,\sum_{a,b}p_{\sigma}^{ab}=1\\}$

satisfying a compatibility condition induced by the face relations. Simplicial
distributions on $X$ define a polytope with finitely many vertices. The face
relations impose conditions in the form of marginals, e.g., for the
$d_{0}$-face of a triangle $\sigma$, we have

$p_{d_{0}\sigma}^{0}=p^{00}_{\sigma}+p^{10}_{\sigma}.$

In this paper, we extend our interest to twisted distributions. For a
$2$-cocycle $\beta:X_{2}\to\mathbb{Z}_{2}$ we write
$\mathsf{sDist}_{\beta}(X)$ for the polytope of $\beta$-twisted simplicial
distributions on $X$. The effect of the twisting appears only in the
$d_{0}$-face, and the marginalization formula above is generalized as follows:

$p_{d_{0}\sigma}^{0}=\left\\{\begin{array}[]{ll}p^{00}_{\sigma}+p^{10}_{\sigma}&\beta(\sigma)=0\\\
p^{11}_{\sigma}+p^{01}_{\sigma}&\beta(\sigma)=1.\end{array}\right.$

Given a twisted simplicial distribution $p$, we define a simplicial subset
$Z_{p}\subset X$ consisting of simplices on which the distribution is
deterministic, i.e., given by a delta-distribution. The rank of $p$ is defined
to be the rank of the matrix consisting of the defining inequalities of the
polytope $\mathsf{sDist}_{\beta}(X)$ that are tight at $p$. Our main result
provides a formula for the rank. For a simplicial set $X$, we will write
$X_{n}^{\circ}$ to denote the subset of non-degenerate simplices. Given a
triangle $\sigma$, the set of $1$-simplices in its boundary is denoted by
$\partial\sigma$.

######  Theorem.

Let $X$ be a simplicial set generated by $2$-simplices
$\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$ such that each $\partial\sigma_{i}$
consists of either three distinct non-degenerate $1$-simplices or two distinct
non-degenerate $1$-simplices and a remaining degenerate $1$-simplex. Consider
a twisted distribution $p\in\mathsf{sDist}_{\beta}(X)$ satisfying the
following conditions:

  * •

for each generating $2$-simplex $\sigma$, $p_{\sigma}^{ab}=0$ for at least one
pair $(a,b)\in\mathbb{Z}_{2}^{2}$, and

  * •

every non-degenerate $1$-simplex of $\bar{X}=X/Z_{p}$ belongs precisely to two
generating $2$-simplices.

Then we have

$\operatorname{rank}(p)=|(Z_{p})_{1}^{\circ}|+|\bar{X}_{2}^{\circ}|-b(\bar{X},\bar{p}).$

The crucial component of the rank formula is $b(X,p)$, a natural number
defined for a twisted simplicial distribution. Our graph-theoretic approach is
based on constructing a signed graph associated with $p$ and $b(X,p)$ is the
number of balanced components in this graph. Balancedness is an important
property for signed graphs. A connected graph is called balanced if the sign
of every circle contained in it, which is simply given by the product of the
signs of the edges in the circle, is positive. In the theorem, we apply this
construction to the quotient space $\bar{X}=X/Z_{p}$ and the twisted
distribution $\bar{p}$ constructed in Proposition 2.20 using a cocycle
obtained by a cohomology long exact sequence. In the last section, we
demonstrate how to apply the rank formula to describe the vertices of
$\mathsf{sDist}(X)$ for the following important scenarios:

  * •

In Section 4.1, we study cycle scenarios whose measurement space consists of a
disk triangulated into $N$ triangles. This scenario is the generalization of
the famous Clauser–Horne–Shimony–Holt (CHSH) scenario [10].

  * •

The scenario whose measurement space is the boundary of a tetrahedron is
considered in Section 4.2. Describing these vertices was a key step in the
topological proof of Fine’s theorem [1], that characterizes non-contextual
distributions on the CHSH scenario.

  * •

The Mermin scenario [11] provides contextual distributions that arise in
quantum theory. In [12], a topological realization is provided where the
underlying measurement space is a torus. The vertices of the polytope of
twisted distributions for different cocycles are studied in [13]. We reproduce
the vertices in Section 4.3.

As displayed in these examples, the rank formula provides a systematic study
of the vertices of the polytope of twisted distributions. We expect this
approach will also be useful in analyzing more complicated polytopes in the
context of classical simulation algorithms for quantum computation [14]. Some
ideas in this direction appear in [13, 15].

The paper is structured as follows. In Section 2, we introduce simplicial sets
focusing on the $2$-dimensional case. We introduce twisted simplicial
distributions in this restricted setting. Our main result of this section,
Proposition 2.20, is proved using ideas from cohomology of simplicial sets and
products of twisted distributions. In Section 3, we develop our graph-
theoretical methods and introduce distributions on graphs of interest. Our
rank formula is proved in Theorem 3.15. Section 4 contains the examples of
practical interest where we put the rank formula in work.

#### Acknowledgments.

This work is supported by the US Air Force Office of Scientific Research under
award number FA9550-21-1-0002 and the Digital Horizon Europe project FoQaCiA,
GA no. 101070558. The author would like to thank Selman Ipek and Walker H.
Stern for their helpful comments on an earlier version of this paper.

##  2 Simplicial distributions

In this section we recall some basic definitions from [1] on simplicial
distributions and introduce a twisted version in the case of $2$-dimensional
simplicial sets based on the exposition in [16]. Our main result is
Proposition 2.20, an important ingredient for the rank formula in Section 3.

###  2.1 Two-dimensional simplicial sets

A simplicial set $X$ consists of a sequence of sets $X_{n}$ for $n\geq 0$
together with the face maps $d_{i}:X_{n}\to X_{n-1}$ and the degeneracy maps
$s_{j}:X_{n}\to X_{n+1}$ satisfying the simplicial identities [4, 17]. A
simplicial set map $f:X\to Y$ consists of functions $f_{n}:X_{n}\to Y_{n}$ for
$n\geq 0$ compatible with the face and the degeneracy maps. We will write
$f_{\sigma}$ for the simplex $f_{n}(\sigma)\in Y_{n}$ for a given $n$-simplex
$\sigma\in X_{n}$. With this notation compatibility with the simplicial
structure can be expressed as

$d_{i}f_{\sigma}=f_{d_{i}\sigma}\;\;\text{ and
}\;\;s_{j}f_{\sigma}=f_{s_{j}\sigma}.$

A simplex is called non-degenerate if it does not lie in the image of a
degeneracy map. A non-degenerate simplex is called generating if it does not
lie in the image of a face map. For a set $U$ of simplices we write
$U^{\circ}$ to denote the subset of non-degenerate simplices.

In this paper we only consider simplicial sets where each $X_{n}$ is a finite
set. We will also have a restriction on the dimension of the simplicial set. A
$1$-dimensional simplicial set is the same as a directed graph with vertex set
$X_{0}$ and edge set $X_{1}$ (allowing loops and parallel edges). A
$1$-simplex can be treated as an arrow

$v\xrightarrow{x}w$

with source $d_{1}(x)=v$ and target $d_{0}(x)=w$. The degenerate $1$-simplex
$s_{0}(v)$ have both source and target given by $v$. A $2$-dimensional
simplicial set consists of such a directed graph specified by
$(X_{0},X_{1},d_{0},d_{1},s_{0})$ together with a finite set of $2$-simplices
$\sigma_{1},\cdots,\sigma_{N}$ glued to the graph. The gluing is encoded by
specifying the faces $d_{i}(\sigma_{k})\in X_{1}$ for $i=0,1,2$. For example,
the standard $2$-simplex $\Delta^{2}$ has a single generating $2$-simplex
$\sigma$ glued to the directed graph on the three vertices $\\{0,1,2\\}$ with
edges $0\to 1$, $1\to 2$ and $0\to 2$. The face maps are given by

$d_{i}\sigma=\left\\{\begin{array}[]{ll}1\xrightarrow{y}2&i=0\\\
0\xrightarrow{z}2&i=1\\\ 0\xrightarrow{x}1&i=2.\end{array}\right.$

The degenerate edges $0\xrightarrow{s_{0}(0)}0$, $1\xrightarrow{s_{0}(1)}1$
and $2\xrightarrow{s_{0}(2)}2$ are usually omitted. We will write
$\partial\sigma=\\{x,y,z\\}$ for the set of edges in the boundary. Let $f:X\to
Y$ be a simplicial set map where $X$ is generated by the $2$-simplices
$\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$. Such a map assigns a $2$-simplex
$f_{\sigma_{k}}\in Y_{2}$ for each $k=1,\cdots,N$ such that

$d_{i}f_{\sigma_{k}}=d_{j}f_{\sigma_{l}}$

whenever $d_{i}\sigma_{k}=d_{j}\sigma_{l}$. Therefore to study such maps it
suffices to understand the face maps of $Y$ in dimension $2$:

$d_{i}:Y_{2}\to Y_{1},\;\;\;\;i=0,1,2.$

For simplicity when we introduce a simplicial set which will appear only in
the target of a simplicial set we will only specify these face maps.

As an important target space we will be considering the nerve of the additive
group $\mathbb{Z}_{2}=\\{0,1\\}$ of integers modulo $2$. This is a simplicial
set denoted by $N\mathbb{Z}_{2}$ whose $n$-simplices are given by $n$-tuples
of elements in $\mathbb{Z}_{2}$. The face maps in dimension $2$ are given by

$d_{i}(a,b)=\left\\{\begin{array}[]{ll}b&i=0\\\ a+b&i=1\\\
a&i=2.\end{array}\right.$ (1)

######  Proposition 2.1.

Let $X$ be a simplicial set with generating $2$-simplices
$\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$. There is a bijection between
simplicial set maps $s:X\to N\mathbb{Z}_{2}$ and functions
$\varphi:X_{1}\to\mathbb{Z}_{2}$ satisfying

$\varphi(x)+\varphi(y)+\varphi(z)=0\mod 2$

for $x,y,z\in\partial\sigma_{i}$ where $i=1,2,\cdots,N$.

###### Proof.

See [1, Proposition 3.13]. ∎

###  2.2 Simplicial distributions

We will write $D(U)$ for the set of probability distributions on a set $U$. It
consists of functions $p:U\to\mathbb{R}_{\geq 0}$ with finite support, i.e.,
$|\\{u\in U:\,p(u)>0\\}|<\infty$, such that

$\sum_{u\in U}p(u)=1.$

Given a function $f:U\to V$ and a distribution $p\in D(U)$ we define
$f_{*}(p)\in D(V)$ by the formula

$f_{*}(p)(v)=\sum_{u\in f^{-1}(v)}p(u).$

We will write $\delta^{u}$ for the delta-distribution peaked at an element
$u\in U$. Given a simplicial set $Y$ we can construct a new simplicial set
$D(Y)$ whose set of $n$-simplices consists of $D(Y_{n})$. The face and the
degeneracy maps of this simplicial set are given by $d_{i}(p)=(d_{i})_{*}(p)$
and $s_{j}(p)=(s_{j})_{*}(p)$. We will write $\delta:Y\to D(Y)$ for the
simplicial set map that sends a simplex to the delta-distribution peaked at
that simplex.

Our interest is in the space $D(N\mathbb{Z}_{2})$. The set of $n$-simplices
consists of distributions of the form

$p:\mathbb{Z}_{2}^{n}\to\mathbb{R}_{\geq 0}.$

We write $p^{a_{1}\cdots a_{n}}$ for the value $p(a_{1},\cdots,a_{n})$. With
this notation the face maps in dimension $2$ are given by

$(d_{i}p)^{0}=\left\\{\begin{array}[]{ll}p^{00}+p^{10}&i=0\\\
p^{00}+p^{11}&i=1\\\ p^{00}+p^{01}&i=2.\end{array}\right.$ (2)

See Figure (1(a)).

######  Definition 2.2.

A (simplicial) scenario consists of a pair $(X,Y)$ of simplicial sets
representing the space of measurements and outcomes, respectively. A
simplicial distribution on the scenario $(X,Y)$ is a simplicial set map

$p:X\to D(Y)$

We write $\mathsf{sDist}(X,Y)$ for the set of simplicial distributions.

When $Y=N\mathbb{Z}_{2}$ we will simplify the notation and write
$\mathsf{sDist}(X)$ for the set of simplicial distributions on $X$.

######  Proposition 2.3.

Let $X$ be a simplicial set with generating $2$-simplices
$\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$. Then a simplicial distribution
$p:X\to D(N\mathbb{Z}_{2})$ is given by a collection of distributions
$p_{\sigma_{k}}\in D(\mathbb{Z}_{2}^{2})$ satisfying

$d_{i}p_{\sigma_{k}}=d_{j}p_{\sigma_{l}}$ (3)

whenever $d_{i}\sigma_{k}=d_{j}\sigma_{l}$. In particular, $\mathsf{sDist}(X)$
is a (convex) polytope with finitely many vertices.

###### Proof.

A simplicial set map $p:X\to D(N\mathbb{Z}_{2})$ is determined by its values,
i.e., the distributions $p_{\sigma_{i}}\in D(\mathbb{Z}_{2}^{2})$, on the
generating simplices. Note that on the remaining simplices the image is
determined by the simplicial structure maps. The only relations imposed on
$p_{\sigma_{i}}$’s come from the face maps given in Equation (3). Therefore
$\mathsf{sDist}(X)$ is the subspace obtained by intersecting
$[0,1]^{4N}\subset\mathbb{R}^{4N}$ with the linear equations corresponding to
normalization and identifications under the face maps. This subspace is convex
and bounded. Since the set of equations involved is finite there are finitely
many extreme points (vertices). ∎

######  Example 2.4.

The simplest case is when $X=\Delta^{2}$:

$\mathsf{sDist}(\Delta^{2})=\\{(p^{00},p^{01},p^{10},p^{11})\in[0,1]^{4}:\,p^{00}+p^{01}+p^{10}+p^{11}=1\\},$

which defines a polytope in $\mathbb{R}^{3}$, e.g., by retaining the
coordinates $(p^{01},p^{10},p^{11})$.

In general, $X$ is obtained by gluing $N$ of the $\Delta^{2}$’s and
$\mathsf{sDist}(X)$ is a convex polytope in $\mathbb{R}^{3N}$ cut out by the
linear equations (3).

######  Definition 2.5.

A simplicial distribution is called a deterministic distribution if it is of
the form $\delta^{s}:X\xrightarrow{s}Y\xrightarrow{\delta}D(Y)$ where $s:X\to
Y$ is a simplicial set map. We will write $\mathsf{dDist}(X,Y)$ for the set of
deterministic distributions. There is a natural map

$\Theta:D(\mathsf{dDist}(X,Y))\to\mathsf{sDist}(X,Y)$

defined by sending $d=\sum_{s}d(s)\delta^{s}$ to the simplicial distribution
$p$ given by

$p_{\sigma}(\theta)=\sum_{s:s_{\sigma}=\theta}d(s).$

A simplicial distribution $p$ is non-contextual if $p$ is in the image of
$\Theta$. Otherwise, $p$ is called contextual.

When $Y=N\mathbb{Z}_{2}$ we will simply write $\mathsf{dDist}(X)$ for the set
of deterministic distributions on $X$.

######  Example 2.6.

Let us consider $X=\Delta^{2}$. By Proposition 2.1 we have

$\mathsf{dDist}(\Delta^{2})\xrightarrow{\cong}\mathbb{Z}_{2}^{2}$

obtained by sending $s:\Delta^{2}\to N\mathbb{Z}_{2}$ to the pair
$(s_{d_{2}\sigma},s_{d_{0}\sigma})$. The corresponding determistic
distribution is given by

$(\delta^{s}_{\sigma})^{ab}=\left\\{\begin{array}[]{ll}1&(a,b)=(s_{d_{2}\sigma},s_{d_{0}\sigma})\\\
0&\text{otherwise.}\end{array}\right.$

For notational convenience we will write $\delta^{ab}$ when $s$ is specified
by $(a,b)$ under the bijection above. Observe that any simplicial distribution
$p$ on the triangle can be written as

$p=\sum_{a,b}p^{ab}\delta^{ab}.$

Therefore every $p$ on $\Delta^{2}$ is non-contextual according to Definition
2.5.

######  Example 2.7.

The simplest contextual example can be obtained by identifying any two edges
of the triangle. Formally this is expressed as a push-out diagram

${\Delta^{1}\sqcup\Delta^{1}}$${\Delta^{2}}$${\Delta^{1}}$${\Delta^{2}/x\sim
z}$

where the top horizontal map sends the generating simplices of $\Delta^{1}$’s
to $x=d_{2}\sigma$ and $z=d_{1}\sigma$. Then we have

$\Theta:D(\\{\delta^{00},\delta^{10}\\})\to\\{(p^{00},p^{01},p^{10},p^{11})\in[0,1]^{4}:\,\sum_{a,b}p^{ab}=1,\;p^{01}=p^{11}\\}$

and a simplicial distribution $p$ is contextual if and only if $p^{11}>0$. The
analysis is similar for the case where the other pairs of edges are
identified.

We can also identify all three edges to obtain $\Delta^{2}/x\sim y\sim z$.
Then

$\Theta:D(\\{\delta^{00}\\})\to\\{(p^{00},p^{01},p^{10},p^{11})\in[0,1]^{4}:\,\sum_{a,b}p^{ab}=1,\;p^{01}=p^{10}=p^{11}\\}$

and again $p$ is contextual if and only if $p^{11}>0$.

######  Example 2.8.

Another topological operation we can perform on a single triangle is to
collapse an edge. This is given by a push-out diagram of the form

${\Delta^{1}}$${\Delta^{2}}$${\Delta[0]}$${\Delta^{2}/z\sim\ast}$

Then we have

$\Theta:D(\\{\delta^{00},\delta^{11}\\})\xrightarrow{\cong}\\{(p^{00},0,0,p^{11})\in[0,1]^{4}:\,p^{00}+p^{11}=1\\}.$

In this case every $p$ is non-contextual. Collapsing two of the edges to
obtain $\Delta^{2}/y\sim z\sim\ast$ we obtain

$\Theta:D(\\{\delta^{00}\\})\xrightarrow{\cong}\\{(1,0,0,0)\\}$

and again every simplicial distribution is non-contextual. The situation is
exactly the same when all three edges are collapsed.

###  2.3 Twisted distributions

We will use the more general notion of simplicial distributions introduced in
[16].

######  Definition 2.9.

A (simplicial) bundle scenario is a simplicial set map $\pi:E\to X$. A
simplicial distribution on the bundle scenario $f$ is a simplicial set map
$p:X\to D(E)$ that makes the following diagram commute

${D(E)}$${X}$${D(X)}$$\scriptstyle{D(\pi)}$$\scriptstyle{p}$$\scriptstyle{\delta}$
(4)

A deterministic distribution on $f$ is a simplicial set map of the form
$\delta^{s}:X\xrightarrow{s}E\xrightarrow{\delta}D(E)$ where $s:X\to E$ is a
section of $\pi$.

The earlier notion given in Definition 2.2 can be recovered by considering
simplicial distributions on the bundle scenario given by the projection map

$\pi:Y\times X\to X$

For twisted distributions we will need the notion of twisted products. Twisted
products model principal bundles [18]. Let us write $H_{n}(X)$ and $H^{n}(X)$
for the $n$-th homology and cohomology groups with coefficients in
$\mathbb{Z}_{2}$, respectively. When $X$ is $2$-dimensional there is a well-
known classification theorem: the set of isomorphism classes of principal
bundles with fiber $N\mathbb{Z}_{2}$ is in bijective correspondence with the
classes in the second cohomology group $H^{2}(X)$. Let us recall the
definition of the $n$-th cohomology group of a simplicial set [4]. Given a
simplicial set $X$ let us write $C^{n}(X)$ for the set
$\mathbb{Z}_{2}^{X_{n}}$ of functions $\alpha:X_{n}\to\mathbb{Z}_{2}$. The
coboundary map $\delta_{n}:C^{n}(X)\to C^{n+1}(X)$ is defined by

$\delta_{n}\alpha(x)=\sum_{i=0}^{n}(-1)^{i}\alpha(d_{i}x).$

Then $H^{n}(X)$ is defined as the quotient of the kernel of $\delta_{n}$ by
the image of $\delta_{n-1}$. A function $\alpha:X_{n}\to\mathbb{Z}_{2}$ is
called a cocycle if it belongs to the kernel of $\delta_{n}$. It is called
normalized if $\alpha(x)=0$ for every degenerate $n$-simplex $x$.

######  Definition 2.10.

Let $\beta:X_{2}\to\mathbb{Z}_{2}$ be a normalized $2$-cocycle. The twisted
product $N\mathbb{Z}_{2}\times_{\beta}X$ is the simplicial set whose set of
$n$-simplices is given by $(N\mathbb{Z}_{2})_{n}\times X_{n}$. The face map in
dimension $2$ is given by

$d_{i}((a,b),\sigma)=\left\\{\begin{array}[]{ll}(b+\beta(\sigma),d_{0}\sigma)&i=0\\\
(a+b,d_{1}\sigma)&i=1\\\ (a,d_{2}\sigma)&i=2.\end{array}\right.$ (5)

Note that the $d_{0}$-face is twisted by the cocycle $\beta$. Projecting onto
the second coordinate gives a principal $N\mathbb{Z}_{2}$-bundle

$\pi:N\mathbb{Z}_{2}\times_{\beta}X\to X$

We are interested in simplicial distributions on such maps.

######  Definition 2.11.

A $\beta$-twisted distribution is a simplicial distribution on
$\pi:N\mathbb{Z}_{2}\times_{\beta}X\to X$. A $\beta$-twisted deterministic
distribution is a deterministic distribution on $\pi$. We will write
$\mathsf{sDist}_{\beta}(X)$ and $\mathsf{dDist}_{\beta}(X)$ for the sets of
$\beta$-twisted simplicial and deterministic distributions on $X$,
respectively.

Let us unravel the definition of a twisted distribution when $X$ is
$2$-dimensional generated by $\sigma_{1},\cdots,\sigma_{N}$. A twisted
simplicial distribution associates a distribution $p_{\sigma_{i}}\in
D(\mathbb{Z}_{2}^{2}\times X_{2})$ with each $2$-simplex $\sigma_{i}$. The
commutativity of diagram (4) implies that the support of the distribution is
contained in $\mathbb{Z}_{2}^{2}\times\\{\sigma_{i}\\}$. Face maps can be
worked out from Equation (5). Only the the $d_{0}$-face is twisted

$(d_{0}p_{\sigma})^{0}=\left\\{\begin{array}[]{ll}p^{00}+p^{10}&\beta(\sigma)=0\\\
p^{11}+p^{01}&\beta(\sigma)=1.\end{array}\right.$ (6)

See Figure (1(b)).

A version of Proposition 2.3 holds for $\mathsf{sDist}_{\beta}(X)$ once the
$d_{0}$-face is twisted in this way. We will provide a more explicit
description of the resulting polytope. We define the correlation function

$c:X_{1}\to\mathbb{R}$ (7)

by $c(x)=p_{x}^{0}-p_{x}^{1}$. The probabilities $p_{\sigma}^{ab}$ for each
$2$-simplex $\sigma\in X_{2}$ can be recovered from the correlation function.

######  Lemma 2.12.

We have

$p_{\sigma}^{ab}=\frac{1}{4}(1+(-1)^{a}c(x_{2})+(-1)^{b+\beta(\sigma)}c(x_{0})+(-1)^{a+b}c(x_{1})).$
(8)

###### Proof.

This formula can be verified using the marginal relations (e.g., $d_{0}$ face
as in Equation (6) and others) and the relation
$p_{x_{i}}^{1}=1-p_{x_{i}}^{0}$. For example, for $(a,b)=(0,0)$ and
$\beta(\sigma)=1$ we have

$\displaystyle\frac{1}{4}(1+c(x_{2})-c(x_{0})+c(x_{1}))$
$\displaystyle=\frac{1}{4}(1+(2p_{x_{2}}-1)-(2p_{x_{0}}-1)+(2p_{x_{1}}-1))$
$\displaystyle=\frac{1}{4}(1+(2(p^{00}_{\sigma}+p^{01}_{\sigma})-1)-(2(p^{11}_{\sigma}+p^{01}_{\sigma})-1)+(2(p^{00}_{\sigma}+p^{11}_{\sigma})-1))$
$\displaystyle=p^{00}_{\sigma}.$

∎

Using this result we can provide a more explicit description for the polytope
of twisted simplicial distributions. For a $m\times d$ matrix $M$ and a column
vector $b$ of size $m$ we will write

$P(M,b)=\\{t\in\mathbb{R}^{d}:\,Mt\geq b\\}$

for the corresponding polytope in $\mathbb{R}^{d}$. Let $\mathbb{1}$ denote a
column vector consisting of $1$’s.

######  Proposition 2.13.

Let $d=|X_{1}|$ and $m=|X_{2}\times\mathbb{Z}_{2}^{2}|$. Then

$\mathsf{sDist}_{\beta}(X)=P(M,-\mathbb{1})$

where $M$ is the $m\times d$ matrix defined by

$M_{(\sigma,ab),x}=\left\\{\begin{array}[]{ll}(-1)^{b+\beta(\sigma)}&x=d_{0}\sigma\\\
(-1)^{a+b}&x=d_{1}\sigma\\\ (-1)^{a}&x=d_{2}\sigma.\end{array}\right.$

###### Proof.

This follows directly from Lemma 2.12. ∎

######  Proposition 2.14.

We have the following properties.

  1. 1.

The set $\mathsf{dDist}_{\beta}(X)$ is non-empty if and only if $[\beta]=0$.

  2. 2.

There is a bijection

$\mathsf{sDist}_{\beta}(X)\xrightarrow{\cong}\mathsf{sDist}_{\alpha}(X)$

when $[\beta]=[\alpha]$.

###### Proof.

Part (1): The cohomology class $[\beta]$ vanishes if and only if the principal
bundle $N\mathbb{Z}_{2}\times_{\beta}X\to X$ trivial. The latter holds if and
only if it admits a section. Part (2): The cohomology classes coincide if and
only the corresponding principal bundles are isomorphic (as principal
bundles). This gives a commutative diagram where the top arrow is an
isomorphism

${D(N\mathbb{Z}_{2}\times_{\beta}X)}$${D(N\mathbb{Z}_{2}\times_{\alpha}X)}$${D(X)}$$\scriptstyle{\cong}$

Therefore the resulting sets of simplicial distributions are in bijective
correspondence. ∎

######  Example 2.15.

Let $D_{N}$ be a simplicial set obtained by gluing $N$ many $2$-simplices to
obtain a disk, e.g., as in Figure (2). For a more general definition see [19,
Definition 3.1]. Since the resulting space is contractible we have
$H_{2}(D_{N})=0$. Therefore for any normalized cocycle
$\beta:(D_{N})_{2}\to\mathbb{Z}_{2}$ we have $[\beta]=0$. By Proposition 2.14
we have $\mathsf{dDist}_{\beta}(D_{N})\neq\emptyset$ and

$\mathsf{sDist}_{\beta}(D_{N})\cong\mathsf{sDist}(D_{N}).$

Then the gluing lemma of [1] implies that every
$p\in\mathsf{sDist}_{\beta}(D_{N})$ is non-contextual.

Figure 2:

For a simplicial subset $Z\subset X$ and a simplicial distribution on $X$ we
write $p|_{Z}$ for the composite $Z\xrightarrow{i}X\xrightarrow{p}D(E)$ where
$i$ is the inclusion map. The following simple observation will be very useful
in analyzing the vertices of the polytope of twisted distributions.

######  Lemma 2.16.

([13]) Let $Z\subset X$ be a simplicial subset with a single generating
$2$-simplex $\sigma$. If two of the faces, say $x,y\in\partial\sigma$, satisfy
$p_{x}=\delta^{a}$ and $p_{y}=\delta^{b}$ for some $a,b\in\mathbb{Z}_{2}$ then
$p|_{Z}$ is a deterministic distribution.

###### Proof.

When the distribution is deterministic on two of the edges this forces three
of the four parameters in $\mathsf{sDist}_{\beta}(\Delta^{2})\subset[0,1]^{4}$
to be zero. Together with the normalization condition we obtain a unique
solution, i.e., a deterministic distribution. ∎

######  Example 2.17.

The measurement space $\tilde{C}_{4}$ of the Clauser–Horne–Shimony–Holt (CHSH)
scenario consists of four triangles
$\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}$ glued as in Figure (3(a)). This
is a special case of cycle scenarios we will examine in Section 4.1. It is
well-known that the polytope $\mathsf{sDist}(\tilde{C}_{4})$ has two kinds of
vertices consisting of the deterministic distributions and contextual
distributions known as the Popescu–Rohrlich (PR) boxes [7]; see also [19,
Example 3]. PR boxes are characterized by the property that the restriction
$p|_{\partial\tilde{C}_{4}}$ to the boundary, which consists of four edges
$x_{1},x_{2},x_{3},x_{4}$, is a deterministic distribution $\delta^{s}$ such
that

$\sum_{i=1}^{4}a_{i}=1\mod 2$

where $\delta^{s}_{x_{i}}=\delta^{a_{i}}$ for $i=1,2,\cdots,N$.

(a)

(b)

Figure 3: (a) The measurement space of the CHSH scenario. (b) PR box with
$a_{1}+a_{2}+a_{3}+a_{4}=1\mod 2$. Green color indicates the deterministic
edges.

###  2.4 Product of distributions

In [20] a product is introduced for simplicial distributions. We can extend
this product to $2$-dimensional twisted distributions. The convolution product
of $p,q\in D(\mathbb{Z}_{2}^{n})$ is the distribution $p\ast q$ defined by

$p\ast q(c)=\sum_{a+b=c}p(a)q(b)$

where the sum runs over $a,b\in\mathbb{Z}_{2}^{n}$ such that $a+b=c$. Using
the convolution product we define a map

$\mathsf{sDist}_{\alpha}(X)\times\mathsf{sDist}_{\beta}(X)\to\mathsf{sDist}_{\alpha+\beta}(X)$

by sending $(p,q)$ to the distribution $p\cdot q$ given by

$(p\cdot q)_{\sigma}=p_{\sigma}\ast q_{\sigma}.$

######  Lemma 2.18.

The product $p\cdot q$ is an $(\alpha+\beta)$-twisted distribution.

###### Proof.

We have

$\displaystyle((p\cdot q)_{d_{0}\sigma})^{0}$
$\displaystyle=p_{d_{0}\sigma}^{0}q_{d_{0}\sigma}^{0}+p_{d_{0}\sigma}^{1}q_{d_{0}\sigma}^{1}$
$\displaystyle=\sum_{a}p_{\sigma}^{a(\alpha(\sigma))}\sum_{b}q_{\sigma}^{b(\beta(\sigma))}+\sum_{a}p_{\sigma}^{a(\alpha(\sigma)+1)}\sum_{b}q_{\sigma}^{b(\beta(\sigma)+1)}$
$\displaystyle=(p\cdot q)^{0(\alpha(\sigma)+\beta(\sigma))}+(p\cdot
q)^{1(\alpha(\sigma)+\beta(\sigma))}$ $\displaystyle=(d_{0}(p\cdot
q)_{\sigma})^{0}.$

Similarly one can verify that $p\cdot q$ is compatible with the remaining
simplicial structure maps. Commutativity of diagram (4) follows from the
observation that the support of $p\cdot q$ is contained in
$\mathbb{Z}_{2}^{2}\times\\{\sigma_{i}\\}$ by definition of the product. ∎

It is instructive to consider the action of $\mathsf{dDist}_{\alpha}(X)$ on
$\mathsf{sDist}_{\beta}(X)$ induced by this product. Note that because of part
(1) of Proposition 2.14 we assume $[\alpha]=0$, or for computational
simplicity $\alpha=0$. Then we have

$(\delta^{ab}\cdot q)^{cd}_{\sigma}=q_{\sigma}^{(c+a)(b+d)}$ (9)

where $\delta^{ab}\in\mathsf{dDist}(X)$ and $q\in\mathsf{sDist}_{\beta}(X)$.

###  2.5 Cohomology exact sequence

Let $Z\subset X$ be a simplicial subset. Given a normalized $2$-cocycle
$\beta:X_{2}\to\mathbb{Z}_{2}$ we will write $\beta|_{Z}$ for the pull-back
$i^{*}\beta$ along the inclusion map $i:Z\to X$. We assume $[\beta|_{Z}]=0$ so
that there exists a normalized $1$-cochain $s:Z_{1}\to\mathbb{Z}_{2}$ such
that $\beta|_{Z}=\delta s$. We will write

$\mathsf{sDist}_{\beta}(X,s)=\\{p\in\mathsf{sDist}_{\beta}(X):\,p|_{Z}=\delta^{s}\\}.$

Consider the cofiber sequence

$Z\xrightarrow{i}X\xrightarrow{q}X/Z$

and the associated cohomology long exact sequence

$H^{1}(X/Z)\to H^{1}(X)\to H^{1}(Z)\xrightarrow{\zeta}H^{2}(X/Z)$

Let $\zeta(s)$ denote the $2$-cocycle obtained by the snake lemma [21], i.e.,
first extending $s$ to $X$ and applying the coboundary:

  * •

Let $\tilde{s}:X_{1}\to\mathbb{Z}_{d}$ denote the $2$-cochain defined by

$\tilde{s}(x)=\left\\{\begin{array}[]{ll}s(x)&x\in Z_{1}\\\ 0&x\in
X_{1}-Z_{1}.\end{array}\right.$

  * •

By applying the coboundary map $\delta:C_{1}(X)\to C_{2}(X)$ we obtain

$\delta\tilde{s}(\sigma)=\tilde{s}(d_{0}\sigma)-\tilde{s}(d_{1}\sigma)+\tilde{s}(d_{2}\sigma)$

where $\sigma\in X_{2}$.

  * •

Since $\delta\tilde{s}|_{Z}=0$ it comes from a $2$-cochain, which will be
denoted by $\zeta(s):(X/Z)_{2}\to\mathbb{Z}_{2}$.

See also [1, Section 5]

######  Lemma 2.19.

The deterministic distribution $\delta^{\tilde{s}}$ on $X$ defined by

$(\delta^{\tilde{s}}_{\sigma})^{ab}=\left\\{\begin{array}[]{ll}1&(a,b)=(\tilde{s}(d_{2}\sigma),\tilde{s}(d_{0}\sigma))\\\
0&\text{otherwise}\end{array}\right.$

is a $\delta\tilde{s}$-twisted distribution.

###### Proof.

This is direct verification. ∎

######  Proposition 2.20.

We have a commutative diagram

${D(\mathsf{dDist}_{\beta}(X,s))}$${\mathsf{sDist}_{\beta}(X,s)}$${D(\mathsf{dDist}_{\beta+\zeta(s)}(X/Z))}$${\mathsf{sDist}_{\beta+\zeta(s)}(X/Z)}$$\scriptstyle{\Theta}$$\scriptstyle{D(\phi)}$$\scriptstyle{\cong}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\scriptstyle{\Theta}$

where $\phi(p)=\delta^{\tilde{s}}\cdot p$. Both vertical arrows are
isomorphisms.

###### Proof.

Let $p\in\mathsf{sDist}_{\beta}(X)$ be such that $p|_{Z}=\delta^{s}$ for some
$s$ satisfying $\delta s=\beta|_{Z}$. By Lemma 2.18 and 2.19 the distribution
$\delta^{\tilde{s}}\cdot p$ is $(\beta+\zeta(s))$-twisted and

$(\delta^{\tilde{s}}\cdot p)|_{Z}=\delta^{\tilde{s}}|_{Z}\cdot
p|_{Z}=\delta^{s}\cdot\delta^{s}=\delta^{0}$

where $\delta^{0}:Z\to D(N\mathbb{Z}_{2})$ is the constant map (with image the
unique vertex). Note that in the last equation we used Equation (9). Therefore
$\delta^{\tilde{s}}\cdot p$ factors through the quotient $X/Z$. The inverse of
$\phi$ is given by the composite

$\phi^{-1}:\mathsf{sDist}_{\beta+\zeta(s)}(X/Z)\xrightarrow{q^{*}}\mathsf{sDist}_{\beta+d\tilde{s}}(X)\xrightarrow{\delta^{\tilde{s}}\cdot}\mathsf{sDist}_{\beta}(X)$

The maps $\phi$ and $\phi^{-1}$ restrict to a bijection between
$\mathsf{dDist}_{\beta}(X,s)$ and $\mathsf{dDist}_{\beta+\zeta(s)}(X/Z)$. ∎

######  Corollary 2.21.

If $\phi(p)$ is non-contextual then $p$ is non-contextual.

(a)

(b)

Figure 4: Twisted simplicial distributions on a triangle where $x$ is
identified with $z$ and $y$ is collapsed: (a) $\beta(\sigma)=0$ and (b)
$\beta(\sigma)=1$. The resulting polytope can be identified with the unit
interval $[0,1]$. Here $\bar{t}=1-t$.

######  Example 2.22.

Let us consider $\mathsf{sDist}(X)$ where $X=\Delta^{2}/x\sim z$; see Example
2.7. We take $Z$ to be the simplicial subset given by the edge $y$. If the
restriction $p|_{Z}$ is deterministic then it is either $\delta^{0}$ or
$\delta^{1}$. Then by Proposition 2.20 we have an isomorphism

$\mathsf{sDist}(X,s)\xrightarrow{\cong}\mathsf{sDist}_{\beta}(X/y\sim\ast)$

Note that $\beta(\sigma)=a$ if $p_{y}^{a}=\delta^{a}$. See Figure (4).

##  3 Distributions on graphs

Our goal in this section is to describe twisted distributions as distributions
on graphs. This is achieved by associating a graph with a simplicial set.
Throughout the paper we only consider simple graphs, i.e., graphs with no
loops and parallel edges. To land in simple graphs we impose some conditions
on the simplicial sets that represent the measurements. We will restrict to a
measurement space $X$ given by a simplicial set

  * •

generated by the $2$-simplices $\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$, and

  * •

each set $\partial\sigma_{k}$ of edges in the boundary consists of either
three distinct non-degenerate edges $\\{x_{0},x_{1},x_{2}\\}$ or two distinct
non-degenerate edges $\\{x_{i},x_{j}\\}$, where $i>j$, and a remaining
degenerate edge.

Therefore we have the situation depicted in Figure (5).

Figure 5:

###  3.1 Distributions on graphs

We begin by constructing a bipartite graph associated to the simplicial set
$X$. A bipartite graph $\Gamma$ consists of a vertex set $V(\Gamma)$,
partitioned into two sets $V^{0}(\Gamma)\sqcup V^{1}(\Gamma)$, and an edge set
$E(\Gamma)$ connecting the vertices from these two sets. We will also consider
graphs with a sign given by a function $\gamma:E(\Gamma)\to\\{\pm 1\\}$.

######  Definition 3.1.

Given a finite $2$-dimensional simplicial set $X$ let $\Gamma(X)$ denote the
bipartite graph with

  * •

vertex set $V=X_{1}^{\circ}\sqcup X_{2}^{\circ}$,

  * •

edge set $E$ consisting of $\\{x,\sigma\\}$ where
$x\in\partial\sigma^{\circ}$.

Next, we enlarge this graph by including the set of outcomes into the picture.
The idea is to replace each vertex corresponding to $\sigma$ by four vertices
labeled by $s_{\sigma}^{ab}$ where $a,b\in\mathbb{Z}_{2}$. The new vertices
are also connected to the same vertices corresponding to the $1$-simplices in
$\partial\sigma^{\circ}$. These edges of the graph are also assigned a sign
$\pm 1$ indicating the outcomes. See Figure (6) for the local picture over a
triangle with two kinds of boundaries, one with $|\partial\sigma^{\circ}|=3$
and $2$.

######  Definition 3.2.

Given a normalized $2$-cocycle $\beta:X_{2}\to\mathbb{Z}_{2}$ we define a
signed bipartite graph $\Gamma_{\beta}(X)$ with

  * •

vertex set $X_{1}^{\circ}\sqcup(X_{2}^{\circ}\times\mathbb{Z}_{2}^{2})$,

  * •

edge set consisting of $\\{x,s_{\sigma}^{ab}\\}$ where
$x\in\partial\sigma^{\circ}$,

  * •

and sign given by

$\gamma(x,s_{\sigma}^{ab})=(-1)^{s_{\sigma}(x)}$ (10)

where $s_{\sigma}:\partial\sigma\to\mathbb{Z}_{2}$ is defined by

$s_{\sigma}(x)=\left\\{\begin{array}[]{ll}(-1)^{b+\beta(\sigma)}&x=d_{0}\sigma\\\
(-1)^{a+b}&x=d_{1}\sigma\\\ (-1)^{a}&x=d_{2}\sigma.\end{array}\right.$

Next, we introduce the notion of a distribution on the bipartite graph
associated to the pair $(X,\beta)$. The goal is to capture the notion of
twisted simplicial distributions as distributions on graphs. The next
definition is motivated by Proposition 2.13.

######  Definition 3.3.

A distribution on $\Gamma_{\beta}(X)$ is a function
$p:X_{2}^{\circ}\times\mathbb{Z}_{2}^{2}\to\mathbb{R}_{\geq 0}$ such that

$\sum_{a,b}p(s_{\sigma}^{ab})=1$

for every $\sigma\in X_{2}^{\circ}$ and

$\sum_{a,b}\gamma(x,s_{\sigma}^{ab})p(s_{\sigma}^{ab})=\sum_{a,b}\gamma(x,s_{\sigma^{\prime}}^{ab})p(s_{\sigma^{\prime}}^{ab})$
(11)

for every $\sigma,\sigma^{\prime}\in X_{2}^{\circ}$ such that
$x\in\partial\sigma^{\circ}\cap\partial\sigma^{\prime\circ}$. In addition, for
simplices $\sigma\in X_{2}^{\circ}$ whose boundary contains a single
degenerate edge $x^{\prime}$ we also require that

$\left\\{\begin{array}[]{ll}p(s_{\sigma}^{(0+\beta(\sigma))(0+\beta(\sigma))})+p(s_{\sigma}^{(1+\beta(\sigma))(0+\beta(\sigma))})=1&x^{\prime}=d_{0}\sigma\\\
p(s_{\sigma}^{00})+p(s_{\sigma}^{11})=1&x^{\prime}=d_{1}\sigma\\\
p(s_{\sigma}^{00})+p(s_{\sigma}^{01})=1&x^{\prime}=d_{2}\sigma.\end{array}\right.$
(12)

We write $\mathsf{Dist}(\Gamma_{\beta}(X))$ for the set of distributions on
the graph $\Gamma_{\beta}(X)$.

(a)

(b)

Figure 6: Signs on the edges induced by $s_{\sigma}^{ab}$ are indicated by
blue and red color for $+1$ and $-1$, respectively. The probabilities are
given by $p^{ab}=p(s_{\sigma}^{ab})$. In this example $\beta(\sigma)=0$.

Equation (12) has the following interpretation. As in simplicial distributions
we want $p_{x^{\prime}}=\delta^{0}$ for each degenerate edge $x^{\prime}$.
This condition is imposed via this equation separately since the degenerate
edges of $X$ does not appear in the graph $\Gamma_{\beta}(X)$. See Figure (6)
for the local picture over a simplex. Definition 3.3 is tailored so that the
following identification can be done.

######  Proposition 3.4.

Sending a twisted distribution $p\in\mathsf{sDist}_{\beta}(X)$ to the
distribution on the graph $\Gamma_{\beta}(X)$ defined by

$p(s_{\sigma}^{ab})=p_{\sigma}^{ab}$

gives a bijection of convex sets

$\mathsf{sDist}_{\beta}(X)\xrightarrow{\cong}\mathsf{Dist}(\Gamma_{\beta}(X))$
(13)

###### Proof.

Follows from Proposition 2.13. ∎

We will identify twisted simplicial distributions and distributions on the
associated enlarged graph.

###  3.2 Rank of a distribution

######  Definition 3.5.

Given a distribution $p\in\mathsf{Dist}(\Gamma_{\beta}(X))$ we consider the
induced signed subgraph ${\Gamma_{\beta}(X,p)}\subset\Gamma_{\beta}(X)$
determined by the vertex set

$X_{1}^{\circ}\sqcup\\{s_{\sigma}^{ab}\in
X_{2}^{\circ}\times\mathbb{Z}_{2}^{2}:\,p(s_{\sigma}^{ab})=0\\}.$

The edge set is determined by the vertices, that is, $\\{x,s_{\sigma}^{ab}\\}$
is an edge if $p(s_{\sigma}^{ab})=0$ and $x\in\partial\sigma^{\circ}$.

Let $A({\Gamma_{\beta}(X,p)})$ denote the adjacency matrix of the signed
graph. Its rows and columns are indexed by the vertices of the graph and its
entries are $0,\pm 1$ indicating whether there exists an edge connecting the
pair of vertices (taking into account the sign). Since the graph is bipartite
the adjacency matrix has the form

$A(\Gamma_{b})=\left(\begin{matrix}\mathbb{0}&B({\Gamma_{\beta}(X,p)})\\\
B({\Gamma_{\beta}(X,p)})^{T}&\mathbb{0}\end{matrix}\right)$

where

$B({\Gamma_{\beta}(X,p)})_{s_{\sigma}^{ab},x}=\left\\{\begin{array}[]{ll}\gamma(x,s_{\sigma}^{ab})&\\{x,s_{\sigma}^{ab}\\}\in
E(\Gamma_{\beta}(X,p))\\\ 0&\text{otherwise.}\end{array}\right.$

######  Definition 3.6.

The rank of $p\in\mathsf{Dist}(\Gamma_{\beta}(X))$ is defined to be the rank
of the matrix $B({\Gamma_{\beta}(X,p)})$. The rank of a twisted distribution
$p\in\mathsf{sDist}_{\beta}(X)$ is the rank of the associated distribution on
the graph.

######  Corollary 3.7.

A twisted distribution
$p\in\mathsf{sDist}_{\beta}(X)\subset{\mathbb{R}^{|X_{1}^{\circ}|}}$ is a
vertex if and only if $\operatorname{rank}(p)={|X_{1}^{\circ}|}$.

###### Proof.

This observation follows from the general theory of polytopes; see [22,
Theorem 18.1]. ∎

Note that the polytope $\mathsf{sDist}_{\beta}(X)$ is not full-dimensional in
$\mathbb{R}^{|X_{1}^{\circ}|}$ as its dimension is given by

$d(X)=|X_{1}^{\circ}|-\sum_{\sigma\in
X_{2}^{\circ}}|\partial\sigma-\partial\sigma^{\circ}|.$

This dimension count follows from Example 2.4 and 2.8.

######  Definition 3.8.

Let $p\in\mathsf{sDist}_{\beta}(X)$ be a twisted distribution. A non-
degenerate $1$-simplex $x$ is called a deterministic edge if
$p_{x}=\delta^{a}$ for some $a\in\mathbb{Z}_{2}$. Similarly a non-degenerate
$2$-simplex $\sigma$ is called a deterministic triangle if
$p_{\sigma}=\delta^{ab}$ for some $(a,b)\in\mathbb{Z}_{2}^{2}$. We will write
$Z_{p}\subset X$ for the simplicial subset generated by the deterministic
edges and triangles with respect to $p$.

Recall the bijection in Proposition 2.20:

$\phi:\mathsf{sDist}_{\beta}(X,s)\to\mathsf{sDist}_{\bar{\beta}}(\bar{X})$

where $\bar{X}=X/Z_{p}$ and $\bar{\beta}=\beta+\zeta(s)$. In the next result
we show that the rank of $\bar{p}=\phi(p)$ is related to the rank of $p$.

######  Lemma 3.9.

For $p\in\mathsf{sDist}_{\beta}(X)$ we have

$\operatorname{rank}(p)=\operatorname{rank}({\bar{p}})+|(Z_{p}^{\circ})_{1}|.$
(14)

###### Proof.

For $\sigma\in X_{2}^{\circ}$ let us write
${\Gamma_{\beta}(X,p)}|_{\sigma}\subset{\Gamma_{\beta}(X,p)}$ for the induced
subgraph on the vertices
$\partial\sigma^{\circ}\sqcup(\\{\sigma\\}\times\mathbb{Z}_{2}^{2})$. We will
take the quotient in steps, that is, one deterministic simplex at a time. By
Lemma 2.16 each $\sigma\in(Z_{p})_{2}^{\circ}$ satisfies
$|\partial\sigma^{\circ}|=2$ or $3$. Therefore we have

$X=X^{(0)}\to X^{(1)}\to\cdots\to X^{(i)}\to X^{(i+1)}\to\cdots\to
X^{(t-1)}\to X^{(t)}=X/Z_{p}$

where $X^{(i+1)}$ is obtained from $X^{(i)}$ by killing a non-degenerate
deterministic edge or a non-degenerate deterministic triangle. Let us consider
these two types of quotients:

  1. 1.

Killing a non-degenerate deterministic edge $x$ which does not belong to a
deterministic triangle: Let $\sigma$ be a non-degenerate triangle whose
boundary contains $x$. Since $p_{\sigma}$ is not deterministic
$|\partial\sigma^{\circ}|=3$ (by Lemma 2.16). We have

$B({\Gamma_{\beta}(X,p)}|_{\sigma})=\begin{bmatrix}(-1)^{s(x)}&1&(-1)^{s(x)+\beta(\sigma)}\\\
(-1)^{s(x)}&-1&(-1)^{s(x)+\beta(\sigma)+1}\end{bmatrix}\sim\begin{bmatrix}0&1&(-1)^{s(x)+\beta(\sigma)}\\\
1&0&0\end{bmatrix}$

where $\sim$ indicates a sequence of elementary row operations. On the other
hand, we have

$B({\Gamma_{\bar{\beta}}(\bar{X},\bar{p})}|_{\sigma})=\begin{bmatrix}1&(-1)^{\bar{\beta}(\sigma)}\\\
-1&(-1)^{\bar{\beta}(\sigma)+1}.\end{bmatrix}=\begin{bmatrix}1&(-1)^{s(x)+\beta(\sigma)}\\\
-1&(-1)^{s(x)+\beta(\sigma)+1}\end{bmatrix}\sim\begin{bmatrix}1&(-1)^{s(x)+\beta(\sigma)}\\\
0&0\end{bmatrix}$

where $\bar{\beta}(\sigma)=\beta(\sigma)+\zeta(s)(\sigma)$. Therefore we have
$\operatorname{rank}(B({\Gamma_{\beta}(X,p)}|_{\sigma}))=\operatorname{rank}(B({\Gamma_{\bar{\beta}}(\bar{X},\bar{p})}|_{\sigma}))+1$.
That is, the rank at $\sigma$ has two parts. First part comes from the new
distribution on the quotient space and the second part from the deterministic
edge. Therefore Equation (14) holds.

  2. 2.

Killing a non-degenerate deterministic triangle $\sigma$: Since $p_{\sigma}$
is deterministic $B({\Gamma_{\beta}(X,p)}|_{\sigma})$ is a $3\times 3$-matrix
of rank $3$. This time all the contribution comes from the deterministic edges
and thus Equation (14) holds.

When a deterministic edge is killed, either as in (1) or part of a
deterministic triangle as in (2), the corresponding column in
$B({\Gamma_{\beta}(X,p)})$ can be removed. Once all such columns are removed
the remaining matrix has the same rank as
$B({\Gamma_{\bar{\beta}}(\bar{X},\bar{p})})$ by the local computations at (1)
and (2). ∎

By this lemma we can assume that $p$ does not give rise to any deterministic
simplex (edge or triangle). Otherwise, we can always take a quotient by the
simplicial subset $Z_{p}$ and use the formula in Lemma 3.9. If
$\partial\sigma$ contains a degenerate edge $x^{\prime}$ then
$p(s_{\sigma}^{ab})=0$ for precisely two of the pairs
$(a,b)\in\mathbb{Z}_{2}^{2}$. Note that these pairs are of the form $(a,b)$
and $(a+1,b+1)$. We remove one of these vertices for each such simplex keeping
only $s_{\sigma}^{ab}$ with $a=0$.

######  Definition 3.10.

Let $p\in\mathsf{sDist}_{\beta}(X)$ be a twisted distribution with no
deterministic simplices (edges or triangles). Let ${\Gamma^{0}_{\beta}(X,p)}$
denote the induced signed subgraph of ${\Gamma_{\beta}(X,p)}$ where only the
vertex $s_{\sigma}^{0b}$ is kept when $\sigma$ has a degenerate $1$-simplex in
its boundary together with the sign defined by

$\gamma_{p}(x,\sigma)=\gamma(x,s_{\sigma}^{ab})$ (15)

where $s_{\sigma}^{ab}$ is the unique vertex with $p(s_{\sigma}^{ab})=0$ if
$|\partial\sigma^{\circ}|=3$, or $s_{\sigma}^{0b}$ if
$|\partial\sigma^{\circ}|=2$.

The sign $\gamma_{p}$ satisfies

$\prod_{x\in\partial\sigma^{\circ}}\gamma_{p}(x,\sigma)=\left\\{\begin{array}[]{ll}(-1)^{\beta(\sigma)}&|\partial\sigma^{\circ}|=3\\\
(-1)^{a+\beta(\sigma)+1}&|\partial\sigma^{\circ}|=2,\,x^{\prime}=d_{1}\sigma\\\
(-1)^{a+1}&|\partial\sigma^{\circ}|=2,\,x^{\prime}=d_{0}\sigma,\text{ or
}d_{2}\sigma\end{array}\right.$ (16)

where $x^{\prime}$ is the unique degenerate edge in the case
$|\partial\sigma^{\circ}|=2$.

######  Proposition 3.11.

Let $p\in\mathsf{sDist}_{\beta}(X)$ be a twisted distribution with no
deterministic simplices. Then

$\operatorname{rank}(p)=\operatorname{rank}(B({\Gamma^{0}_{\beta}(X,p)})).$

The choice we made in the definition of the graph does not affect the rank.

(a)

(b)

Figure 7: (a) The graph $\Gamma_{\bar{\beta}}(\bar{X},\bar{p})$ where $p$ is a
PR box. (b) The subgraph graph $\Gamma_{\bar{\beta}}^{0}(\bar{X},\bar{p})$.

###  3.3 The rank formula

We recall some basic definitions about signed graphs following [23]. Let
$\Sigma=(V,E)$ be a simple (undirected) graph. A path is a sequence of edges
$e_{1}e_{2}\cdots e_{k}$ with no repetition such that $e_{i-1}$ and $e_{i}$
has a common vertex. If $v$ and $w$ are the vertices of $e_{1}$ and $e_{k}$
not common to $e_{2}$ and $e_{k-1}$, respectively, then we say the path is
from $v$ to $w$. A path is called closed if $v=w$. A circle is the graph
determined by a closed path. If $\Sigma$ comes with a sign $\gamma:E\to\\{\pm
1\\}$ then a circle $C$ given by the sequence $e_{1}e_{2}\cdots e_{k}$ has
sign

$\gamma(C)=\gamma(e_{1})\gamma(e_{2})\cdots\gamma(e_{k}).$

If $\gamma(C)=1$ ($\gamma(C)=-1$) the circle is called positive (negative).

######  Definition 3.12.

A signed graph $\Sigma$ is called balanced if every circle in it is positive.
We will write $b(\Sigma)$ for the number of components of the graph that are
balanced.

A bidirected graph comes with a choice of sign $\eta(v,e)=\pm 1$ for each $v$
incident to an edge $e$. Given a signed graph $(\Sigma,\gamma)$ we can define
a bidirected graph $(\Sigma,\eta)$ such that

$\gamma(e)=-\eta(v,e)\eta(w,e)$

for every edge $e=\\{v,w\\}$. This bidirected graph is not unique. An
incidence matrix of $(\Sigma,\gamma)$ is a $|V|\times|E|$-matrix $H(\Sigma)$
such that

$H(\Sigma)_{v,e}=\left\\{\begin{array}[]{ll}\eta(v,e)&v\in e\\\
0&\text{otherwise.}\end{array}\right.$

######  Theorem 3.13.

$H(\Sigma)$ has rank $|V|-b(\Sigma)$.

###### Proof.

See [23, Theorem IV.1]. ∎

We will apply this rank result to bipartite graphs.

######  Definition 3.14.

Let $\Sigma$ be a signed bipartite graph with vertex set
$V(\Sigma)=V^{0}(\Sigma)\sqcup V^{1}(\Sigma)$. Suppose that every vertex in
$V^{0}(\Sigma)$ is incident to exactly two edges. Define a new bidirected
graph $(\hat{\Sigma},\eta)$ with

  * •

vertex set $V^{0}(\Sigma)$,

  * •

edge set $V^{1}(\Sigma)$, and

  * •

bidirection defined by

$\eta(v,e)=\gamma(v,e)$

where $v\in V^{0}(\Sigma)$ and $e\in V^{1}(\Sigma)$ are connected by an edge
in $\Sigma$.

The vertices incident to an edge in $V^{1}(\Sigma)$ are those vertices in
$V^{0}(\Sigma)$ connected to it by an edge in $\Sigma$. Effectively, to obtain
$\hat{\Sigma}$ we merge any two edges incident to a vertex in $V^{0}(\Sigma)$
into a single edge. Then using Theorem 3.13 we have

$\operatorname{rank}(B(\Sigma))=\operatorname{rank}(H(\hat{\Sigma}))=|V^{0}(\Sigma)|-b(\hat{\Sigma}).$
(17)

Note that the rank does not depend on the choice of $\eta$ since a different
choice would amount to multiplying the corresponding row with $-1$. If
$\Gamma_{\beta}^{0}(X,p)$ satisfies the assumption of Definition 3.14 we can
apply the construction $\Sigma\mapsto\hat{\Sigma}$ to the graph
$\Gamma_{\beta}^{0}(X,p)$. We define

$b(X,p)=b(\hat{\Gamma}_{\beta}^{0}(X,p)).$ (18)

######  Theorem 3.15.

Let $X$ be a simplicial set generated by $2$-simplices
$\sigma_{1},\sigma_{2},\cdots,\sigma_{N}$ such that each $\partial\sigma_{i}$
consists of either three distinct non-degenerate $1$-simplices or two distinct
non-degenerate $1$-simplices and a remaining degenerate $1$-simplex. Consider
a twisted distribution $p\in\mathsf{sDist}_{\beta}(X)$ satisfying the
following conditions:

  * •

for each generating $2$-simplex $\sigma$, $p_{\sigma}^{ab}=0$ for at least one
pair $(a,b)\in\mathbb{Z}_{2}^{2}$, and

  * •

every non-degenerate $1$-simplex of $\bar{X}=X/Z_{p}$ belongs precisely to two
generating $2$-simplices.

Then we have

$\operatorname{rank}(p)=|(Z_{p})_{1}^{\circ}|+|\bar{X}_{2}^{\circ}|-b(\bar{X},\bar{p}).$

###### Proof.

By Lemma 3.9, Proposition 3.11, and Equation (17) we have

$\operatorname{rank}(p)=|(Z_{p})_{1}^{\circ}|+\operatorname{rank}(\Gamma_{\bar{\beta}}^{0}(\bar{X},\bar{p}))=|(Z_{p})_{1}^{\circ}|+|X_{2}^{\circ}|-b(X,p).$

The assumption on $p$ implies that the number of the vertices of
$\hat{\Gamma}_{\bar{\beta}}^{0}(\bar{X},\bar{p})$ is given by
$|X_{2}^{\circ}|$. ∎

Note that if the condition $p_{\sigma}^{ab}=0$ fails for at least one pair of
outcomes for every non-degenerate simplex then to compute the rank we can
restrict to the simplicial subset for which this condition holds.

##  4 Examples

The rank formula in Theorem 3.15 is very useful in finding the vertices of the
polytope of twisted distributions. We can partition
$\mathsf{sDist}_{\beta}(X)$ by fixing a simplicial subset $Z\subset X$ and
considering those twisted distributions $p$ with the property that $Z_{p}=Z$.
Our approach will be to combine this observation with the following result.

######  Lemma 4.1.

Let $p\in\mathsf{sDist}_{\beta}(X)$ be a vertex. Then

$|(Z_{p})_{1}^{\circ}|{-|(Z_{p})_{2}^{\circ}|}\geq|X_{1}^{\circ}|-|X_{2}^{\circ}|.$

###### Proof.

By Corollary 3.7 $p$ is a vertex if and only if
$\operatorname{rank}(p)=|X_{1}^{\circ}|$. We have
$\operatorname{rank}(\bar{p})\leq|X_{2}^{\circ}|-{|(Z_{p})_{2}^{\circ}|}$
since $Z_{p}$ is $1$-dimensional. Then the result follows from Lemma 3.9. ∎

Figure 8:

The rank formula contains the number of balanced components of the associated
graph. For the computation of the signs of the circle graphs we will use the
following formula. See Figure (8).

######  Lemma 4.2.

Let $p\in\mathsf{sDist}_{\beta}(X)$ and consider $\gamma_{p}$ defined in
Equation (15). For a non-degenerate $2$-simplex $\sigma$ whose boundary
$\partial\sigma=\\{x,y,z\\}$ contains a deterministic edge, say $z$, with
$p_{z}=\delta^{a}$ we have

$\gamma_{p}(x,\sigma)\gamma_{p}(y,\sigma)=(-1)^{a+\beta(\sigma)+1}.$

###### Proof.

The distribution $p_{\sigma}\in D(\mathbb{Z}_{2}^{2})$ is zero if and only if
$(c,d)$ satisfies $c+d=a+\beta(\sigma)+1$. We have $\gamma(x,\sigma)=(-1)^{c}$
and $\gamma(y,\sigma)=(-1^{d})$. Hence the formula follows. ∎

###  4.1 $N$-cycle scenario

For $N\geq 2$, let $\tilde{C}_{N}$ denote the measurement space of the
$N$-cycle scenario defined as the following simplicial set:

  * •

Generating $2$-simplices: $\sigma_{1},\cdots,\sigma_{N}$.

  * •

Identifying relations:

$d_{i^{\prime}_{1}}\sigma_{1}=d_{i_{2}}\sigma_{2},\;d_{i_{2}^{\prime}}\sigma_{2}=d_{i_{3}}\sigma_{3}\;\cdots\;d_{i^{\prime}_{N}}\sigma_{N}=d_{i_{1}}\sigma_{1}$

where $i_{k}\neq i_{k}^{\prime}\in{\\{0,1\\}}$ for $1\leq k\leq N$.

The case $N=4$ where the boundary is oriented in the counter-clockwise
direction is depicted in Figure (3(a)). See also [19, Definition 4.1]. The
edges on the boundary will be denoted by $x_{1},x_{2},\cdots,x_{N}$.

(a)

(b)

Figure 9: (a) Deterministic interior edge. (b) Collapsed scenario.

Let $p\in\mathsf{sDist}_{\beta}(\tilde{C}_{N})$ be a vertex. Let us consider
the possibilities for $Z_{p}$.

  1. 1.

$Z_{p}$ has a deterministic edge connecting a boundary vertex to the central
vertex: In this case the quotient $\tilde{C}_{N}/Z_{p}$ can be identified with
a wedge sum of $D_{K_{i}}$’s of Example 2.15 where $K_{i}\leq N$. We know that
every twisted distribution on $D_{K_{i}}$ is non-contextual. Then Corollary
2.21 implies that $p$ is non-contextual, hence a deterministic distribution.
The situation is depicted in Figure (9).

  2. 2.

All deterministic edges of $Z_{p}$ are on the boundary: Note that by the
previous case we know that $p$ would be a deterministic distribution if
$Z_{p}$ contained a deterministic triangle. Assuming that $Z_{p}$ does not
contain a deterministic triangle, Lemma 4.1 gives

$|(Z_{p})_{1}^{\circ}|\geq|X_{1}^{\circ}|-|X_{2}^{\circ}|=2N-N=N.$

This forces $Z_{p}$ to be the boundary $\partial\tilde{C}_{N}$ of the disk.
Assume that $p|_{\partial\tilde{C}_{N}}=\delta^{s}$. Then we have

$\delta^{s}_{x_{i}}=\delta^{a_{i}},\;\;\;i=1,2,\cdots,N$

for some $a_{i}\in\mathbb{Z}_{2}$. The graph
$\hat{\Gamma}_{\bar{\beta}}^{0}(\bar{X},\bar{p})$ is given by a circle $C$
with edges $e_{1},e_{2},\cdots,e_{N}$ as in Figure (10(a)). The sign of the
circle is given by

$\gamma_{\bar{p}}(C)=\prod_{i=1}^{N}\gamma_{\bar{p}}(e_{i})=(-1)^{N}\prod_{i=1}^{N}(-1)^{a_{i}+\beta(\sigma_{i})+1}=(-1)^{N}\prod_{i=1}^{N}(-1)^{\bar{\beta}(\sigma_{i})+1}=(-1)^{[\bar{\beta}]}$

and by the rank formula we obtain

$\operatorname{rank}(p)=\left\\{\begin{array}[]{ll}2N&[\bar{\beta}]=1\\\
2N-1&[\bar{\beta}]=0.\end{array}\right.$ (19)

Therefore $p$ is a vertex if and only if $[\bar{\beta}]=1$, that is,
$[\beta]=1+\sum_{i=1}^{N}a_{i}$.

As a special case let us take $[\beta]=0$. In this case $p$ is a vertex if and
only if $\sum_{i=1}^{N}a_{i}=1$. When $N=4$ this reproduces the PR boxes in
Example 2.17.

(a)

(b)

Figure 10: (a) The graph
$\hat{\Gamma}_{\bar{\beta}}^{0}(\tilde{C}_{4}/\partial\tilde{C}_{4})$. (b)
When the boundary is collapsed the resulting space is a sphere.

###  4.2 Boundary of tetrahedron

Let $\partial\Delta^{3}$ denote the simplicial set given by the boundary of a
tetrahedron with generating $2$-simplices
$\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}$. Let
$p\in\mathsf{sDist}_{\beta}(\partial\Delta^{3})$ be a vertex. Then Lemma 4.1
gives

$|(Z_{p})_{1}^{\circ}|-|(Z_{p})_{2}^{\circ}|\geq 4-2=2$

and we have $|(Z_{p})_{2}^{\circ}|=0,1$ or $4$. The last case gives a
deterministic distribution if $[\beta]=0$. Let us consider the remaining
cases.

  1. 1.

$|(Z_{p})_{2}^{\circ}|=0$: In this case
${\hat{\Gamma}_{\bar{\beta}}^{0}(\bar{X},\bar{p})}$ is a circle $C$ with edges
$e_{1},e_{2},e_{3},e_{4}$; see Figure (11(a)). Its sign is given by

$\gamma_{\bar{p}}(C)=\prod_{i=1}^{4}\gamma_{\bar{p}}(e_{i})=(-1)^{a+\beta(\sigma_{1})+1}(-1)^{b+\beta(\sigma_{2})+1}(-1)^{a+\beta(\sigma_{3})+1}(-1)^{b+\beta(\sigma_{4})+1}=(-1)^{[\beta]}$

and by the rank formula

$\operatorname{rank}(p)=\left\\{\begin{array}[]{ll}6&[\beta]=1\\\
5&[\beta]=0.\end{array}\right.$ (20)

Therefore $p$ is a vertex if and only if $[\beta]=1$.

  2. 2.

$|(Z_{p})_{2}^{\circ}|=1$: This case (Figure (11(b))) reduces to the cycle
scenario $\tilde{C}_{3}$ where $p|_{\partial\tilde{C}_{3}}$ is deterministic
with $a+b+c=0$. Therefore by Equation (19), $p$ is a vertex if and only if
$[\beta]=1$.

In the case $[\beta]=0$ we observe that all the vertices are deterministic,
which provides an alternative approach to the computation in [1, Proposition
4.12].

(a)

(b)

Figure 11:

###  4.3 Mermin torus

Let $T$ denote the Mermin torus introduced in [13]. Let
$p\in\mathsf{sDist}_{\beta}(T)$ be a vertex. The distribution $p$ is
deterministic if and only if $[\beta]=0$. Let us assume $p$ is not
deterministic. Lemma 4.1 gives

$|(Z_{p})_{1}^{\circ}|-|(Z_{p})_{2}^{\circ}|\geq 3$

which can only be satisfies if $Z_{p}$ consists of three generating
$1$-simplices as in Figure (12(a)) or two generating triangles as in Figure
(12(b)). In both cases $\hat{\Gamma}_{\bar{\beta}}^{0}(\bar{X},\bar{p})$ is a
circle $C$. We compute the sign of this circle for each case:

  1. 1.

In Figure (12(a)) we have

$\gamma_{\bar{p}}(C)=(-1)^{a+\beta(\sigma_{1})+1}(-1)^{b+\beta(\sigma_{2})+1}(-1)^{c+\beta(\sigma_{3})+1}(-1)^{a+\beta(\sigma_{4})+1}(-1)^{b+\beta(\sigma_{5})+1}(-1)^{c+\beta(\sigma_{6})+1}=(-1)^{[\beta]}.$

Therefore $p$ is a vertex if and only if $[\beta]=1$.

  2. 2.

In Figure (12(b)) we have

$\gamma_{\bar{p}}(C)=(-1)^{a+\beta(\sigma_{1})+1}(-1)^{b+\beta(\sigma_{3})+1}(-1)^{a+b+c+\beta(\sigma_{2})+\beta(\sigma_{5})+\beta(\sigma_{4})+1}(-1)^{c+\beta(\sigma_{6})+1}=(-1)^{[\beta]}$

and $p$ is a vertex if and only if $[\beta]=1$.

Combining these two observations gives the main result of [13].
$\mathsf{sDist}_{\beta}(T)$ has only deterministic vertices if $[\beta]=0$. On
the other hand, if $[\beta]=1$ then there are two kinds of vertices as given
in Figure (12).

(a)

(b)

Figure 12:

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