# Characterizations of bilocality and $n$-locality of correlation tensors

Shu Xiao, Huaixin Cao, Zhihua Guo, Kanyuan Han
School of Mathematics and Statistics, Shaanxi Normal University
Xi’an 710119, China

###### Abstract

In the literature, bilocality and $n$-locality of correlation tensors (CTs)
are described by integration local hidden variable models (called C-LHVMs)
rather than by summation LHVMs (called D-LHVMs). Obviously, C-LHVMs are easier
to be constructed than D-LHVMs, while the later are easier to be used than the
former, e.g., in discussing on the topological and geometric properties of the
sets of all bilocal and of all $n$-local CTs. In this context, one may ask
whether the two descriptions are equivalent. In the present work, we first
establish some equivalent characterizations of bilocality of a tripartite CT
${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$, implying that the two descriptions of
bilocality are equivalent. As applications, we prove that all bilocal CTs with
the same size form a compact path-connected set that has many star-convex
subsets. Secondly, we introduce and discuss the bilocality of a tripartite
probability tensor (PT) ${\bf{P}}=\Lbrack P(abc)\Rbrack$, including equivalent
characterizations and properties of bilocal PTs. Lastly, we obtain
corresponding results about $n$-locality of $n+1$-partite CTs
${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ and PTs ${\bf{P}}=\Lbrack
P({\bf{a}}b)\Rbrack$, respectively.

Keywords. bilocality; $n$-locality; correlation tensor; probability tensor;
local hidden variable model.

PACS number(s): 03.65.Ud, 03.67.Mn

##  1 Introduction

As one of important quantum correlations, Bell nonlocality originated from the
Bell’s 1964 paper [1]. He found that when some entangled state is suitably
measured, the probabilities for the outcomes violate an inequality, named the
Bell inequality. This property of quantum states is the so-called Bell
nonlocality and was reviewed by Brunner et al. [2] for the “behaviors”
$P(ab|xy)$ (correlations), a terminology introduced by Tsirelson [3], but not
for quantum states. As an important source in quantum information processing,
Bell nonlocality has been widely discussed, see e.g. [4, 18, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17]. Usually, Bell nonlocality can be checked by
violation of some types Bell inequalities [19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29].

Quantum systems that have never interacted can become nonlocally correlated
through a process called entanglement swapping. To characterize nonlocality in
this context, Branciard et al. [30] introduced local models where quantum
systems that are initially uncorrelated are described by uncorrelated local
variables, leading to stronger tests of nonlocality. More precisely, they
considered the general scenario depicted in Fig. 1.

Figure 1: A general bilocal scenario. A source $\lim_{1}$ sends particles to
Alice and Bob, and a separate source $\lambda_{2}$ sends particles to Charles
and Bob. All parties can perform measurements on their system, labeled $x,y$,
and $z$ for Alice, Bob, and Charles, and they obtain outcomes $a,b$, and $c$,
respectively, with the joint probability $P(abc|xyz)$.

After performing measurements, the correlations between the measurement
outcomes of the three parties are described by the joint probability
distribution $P(abc|xyz)$. Following [30, 31], a joint probability
distribution is said to be bilocal if it can be written in the factorized
form:

$P(abc|xyz)=\iint_{\Lambda_{1}\times\Lambda_{2}}\rho_{1}(\lambda_{1})\rho_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2}){\rm{d}}\lambda_{1}{\rm{d}}\lambda_{2}$
(1.1)

for all possible inputs $x,y,z$ and all outcomes $a,b,c$. We call Eq. (1.1) a
continuous bilocal hidden variable model (C-biLHVM) since hidden variables
$\lambda_{1}$ and $\lambda_{2}$ may be “continuous” ones. Branciard et al.
proved that all bilocal correlations satisfy a quadratic inequality $I\leq
1+E^{2}$ [30, Eq. (10)]. To compare bilocal and nonbilocal correlations in
entanglement-swapping experiments, Branciard et al. [31] extended the analysis
of bilocal correlations initiated in [31] and derived a Bell-type inequality
$\sqrt{I}+\sqrt{J}\leq 1,$ which was proved to be valid for every bilocal
${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$. Gisin et al. [32] proved that all
entangled pure quantum states violate the bilocality inequality. Importantly,
bilocality inequality is related to the 2-locality approach for detecting
quantum correlations in networks, especially, in star networks [33, 34, 36,
37, 38, 39, 35, 40, 41, 42, 43]. For example, Tavakoli et al in [33]
introduced and discussed $n$-locality of a star-network composed by $n+1$
parties (see Fig. 2), where a central node (referred to as Bob, denoted by
$B$) shares a bipartite state $\rho_{A_{i}B_{i}}$ with each $A_{i}$ of $n$
edge nodes (referred to as the Alices, denoted by $A_{1},A_{2},\ldots,A_{n}$).

Figure 2: A star-network with a central node $B$ and $n$ star-nodes
$A_{1},A_{2},\ldots,A_{n}$ where $n=4$.

Usually, both bilocality and $n$-locality are described by integration local
hidden variable models (LHVMs), called C-biLHVMs and C-$n$LHVMs, rather than
by summations LHVMs, called D-biLHVMs and D-$n$LHVMs. In this paper, we will
discuss bilocality and $n$-locality of correlation and probability tensors by
proving equivalences between C-biLHVMs and D-biLHVMs, as well as C-$n$LHVMs
and D-$n$LHVMs. In Sect. 2, we first fix the concept of bilocality of a
tripartite correlation tensor (CT) ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$, and
establish a series of characterizations and many properties of bilocality. In
Sect. 3, we give the concept of bilocality of a probability tensor (PT)
${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ and obtain some equivalent
characterizations and many properties of bilocality of a PT. Sects. 4 and 5
are devoted to the corresponding discussions about $n+1$-partite CTs
${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ and PTs ${\bf{P}}=\Lbrack
P({\bf{a}}b)\Rbrack$, respectively.

##  2 Bilocality of tripartite correlation tensors

In what follows, we use $[n]$ to denote the set $\\{1,2,\ldots,n\\}$. When a
tripartite system is measured by separated three parties $A,B$ and $C$ with
measurements labeled by $x\in[m_{A}],y\in[m_{B}]$ and $z\in[m_{C}]$,
respectively, the joint probability distribution $P(abc|xyz)$ of obtaining
outcomes $a\in[o_{A}],b\in[o_{B}]$ and $c\in[o_{C}]$ forms a tensor
${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over
$\Delta_{3}=[o_{A}]\times[o_{B}]\times[o_{C}]\times[m_{A}]\times[m_{B}]\times[m_{C}]$,
we call it a correlation tensor (CT) [46], just like a matrix. Abstractly, a
tripartite CT over $\Delta_{3}$ is a function
$P:\Delta_{3}\rightarrow{\mathbb{R}}$ such that

$P(abc|xyz)\geq 0(\forall x,y,z,a,b,c){\textrm{\ and\
}}\sum_{a,b,c}{P(abc|xyz)}=1(\forall x,y,z).$

Any function $P:\Delta_{3}\rightarrow{\mathbb{R}}$ is called a correlation-
type tensor (CTT)[46] over $\Delta_{3}$. We use $\mathcal{T}(\Delta_{3})$ and
$\mathcal{CT}(\Delta_{3})$ to denote the sets of all CTTs and CTs over
$\Delta_{3}$, respectively.

For any two elements ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ and
${\bf{Q}}=\Lbrack Q(abc|xyz)\Rbrack$ of $\mathcal{T}(\Delta_{3})$, define

$s{\bf{P}}+t{\bf{Q}}=\Lbrack sP(abc|xyz)+tQ(abc|xyz)\Rbrack,$
$\langle{\bf{P}}|{\bf{Q}}\rangle=\sum_{a,b,c,x,y,z}{P(abc|xyz)}Q(abc|xyz),$

then $\mathcal{T}(\Delta_{3})$ becomes a finite dimensional Hilbert space over
${\mathbb{R}}$. Clearly, the norm-convergence of a sequence in
$\mathcal{T}(\Delta_{3})$ is just the pointwise-convergence and then
$\mathcal{CT}(\Delta_{3})$ becomes a compact convex set in
$\mathcal{T}(\Delta_{3})$.

We fix the concept of the bilocality of a CT over $\Delta_{3}$ according to
[30, 31].

Definition 2.1. A CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over $\Delta_{3}$ is
said to be bilocal if it has a “continuous” bilocal hidden variable model
(C-biLHVM):

$P(abc|xyz)=\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$
(2.1)

for a measure space
$(\Lambda_{1}\times\Lambda_{2},\Omega_{1}\times\Omega_{2},\mu_{1}\times\mu_{2})$
and for all $a,b,c,x,y,z$, where

(a) $q_{1}(\lambda_{1})$ and $P_{A}(a|x,\lambda_{1})(x\in[m_{A}],a\in[o_{A}])$
are $\Omega_{1}$-measurable on $\Lambda_{1}$, $q_{2}(\lambda_{2})$ and
$P_{C}(c|z,\lambda_{2})(z\in[m_{C}],c\in[o_{C}])$ are $\Omega_{2}$-measurable
on $\Lambda_{2}$, and
$P_{B}(b|y,\lambda_{1}\lambda_{2})(y\in[m_{B}],b\in[o_{B}])$ are
$\Omega_{1}\times\Omega_{2}$-measurable on $\Lambda_{1}\times\Lambda_{2}$;

(b)
$q_{i}(\lambda_{i}),P_{A}(a|x,\lambda_{1}),P_{B}(b|y,\lambda_{1}\lambda_{2})$
and $P_{C}(c|z,\lambda_{2})$ are probability distributions (PDs) of
$\lambda_{i},a,b,c,$ respectively.

A CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over $\Delta_{3}$ is said to be non-
bilocal if it is not bilocal. We use
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ to denote the set of all bilocal
CTs over $\Delta_{3}$.

Remark 2.1. By Definition 2.1, when a CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$
over $\Delta_{3}$ is a product of three conditional probability distributions
$P_{A}(a|x),P_{B}(b|y)$ and $P_{C}(c|z)$ of parties $A,B$ and $C$, i.e.,
$P(abc|xyz)=P_{A}(a|x)P_{B}(b|y)P_{C}(c|z)$, we can rewrite it as

$P(abc|xyz)=\sum_{\lambda_{1}\lambda_{2}=1}^{1}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2})$

where $q_{k}(\lambda_{k})=1(k=1,2)$ and

$P_{A}(a|x,\lambda_{1})=P_{A}(a|x),P_{B}(b|y,\lambda_{1}\lambda_{2})=P_{B}(b|y),P_{C}(c|z,\lambda_{2})=P_{C}(c|z),\
\forall\lambda_{k}=1.$

Thus, ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ can be written as (2.1) for the
counting measures $\mu_{k}$ on the set $\Lambda_{k}=\\{1\\}$ and then is
bilocal.

Remark 2.2. From definition, we observe that when a CT ${\bf{P}}=\Lbrack
P(abc|xyz)\Rbrack$ over $\Delta_{3}$ is bilocal, marginal distributions
satisfy:

$P_{AC}(ac|xz)=P_{A}(a|x)P_{C}(c|z),\ \ \forall x,z,a,c.$ (2.2)

By using this property of a bilocal CT, we can find that not all Bell local
CTs over $\Delta_{3}$ are bilocal.

Example 2.1. Let $m_{X}=o_{X}=2(X=A,B,C)$ and take

$P_{B}(1|1)=1/2,P_{B}(2|1)=1/2,P_{B}(1|2)=1/2,P_{B}(2|2)=1/2,$
$P^{\prime}_{A}(1|1)=1,P^{\prime}_{A}(2|1)=0,P^{\prime}_{A}(1|2)=1,P^{\prime}_{A}(2|2)=0,$
$P^{\prime\prime}_{A}(1|1)=0,P^{\prime\prime}_{A}(2|1)=1,P^{\prime\prime}_{A}(1|2)=0,P^{\prime\prime}_{A}(2|2)=1,$
$P^{\prime}_{C}(1|1)=1,P^{\prime}_{C}(2|1)=0,P^{\prime}_{C}(1|2)=0,P^{\prime}_{C}(2|2)=1,$
$P^{\prime\prime}_{C}(1|1)=0,P^{\prime\prime}_{C}(2|1)=1,P^{\prime\prime}_{C}(1|2)=1,P^{\prime\prime}_{C}(2|2)=0,$

and define

${\bf{P}}=\frac{1}{2}{\bf{P}}^{\prime}_{A}{\otimes}{\bf{P}}_{B}{\otimes}{\bf{P}}^{\prime}_{C}+\frac{1}{2}{\bf{P}}^{\prime\prime}_{A}{\otimes}{\bf{P}}_{B}{\otimes}{\bf{P}}^{\prime\prime}_{C},$

that is,

$P(abc|xyz)=\frac{1}{2}P^{\prime}_{A}(a|x)P_{B}(b|y)P^{\prime}_{C}(c|z)+\frac{1}{2}P^{\prime\prime}_{A}(a|x)P_{B}(b|y)P^{\prime\prime}_{C}(c|z).$

Clearly, ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ is a Bell local CT. Note that

$|P_{AC}(ac|xz)-P_{A}(a|x)P_{C}(c|z)|=\frac{1}{4}|[P^{\prime}_{A}(a|x)-P^{\prime\prime}_{A}(a|x)][P^{\prime}_{C}(c|z)-P^{\prime\prime}_{C}(c|z)]|\equiv\frac{1}{4},$

we see that

$P_{AC}(ac|xz)\neq P_{A}(a|x)P_{C}(c|z),\ \ \forall x,z,a,c.$

Thus, ${\bf{P}}\notin\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$. Moreover,
from Remark 2.1, we see that
${\bf{P}}^{\prime}_{A}{\otimes}{\bf{P}}_{B}{\otimes}{\bf{P}}^{\prime}_{C}$ and
${\bf{P}}^{\prime\prime}_{A}{\otimes}{\bf{P}}_{B}{\otimes}{\bf{P}}^{\prime\prime}_{C}$
are in $\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$. This shows that the set
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is not convex in the Hilbert
space $\mathcal{T}(\Delta_{3})$.

Generally, the five PDs in a C-biLHVM (2.1) for ${\bf{P}}$ are not necessarily
unique and depend on ${\bf{P}}$. The following proposition ensures that any
two bilocal CTs over $\Delta_{3}$ can be represented by C-biLHVMs with the
same measure space and the same PDs of the same hidden variables.

Proposition 2.1. Let ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ and
${\bf{P}}^{\prime}=\Lbrack P^{\prime}(abc|xyz)\Rbrack$ be any two bilocal CTs
over $\Delta_{3}$. Then there is a product measure space $(S_{1}\times
S_{2},T_{1}\times T_{2},\gamma_{1}\times\gamma_{2})$ and PDs $f_{k}(s_{k})$ of
$s_{k}(k=1,2)$ such that

$\displaystyle P(abc|xyz)$ $\displaystyle=$ $\displaystyle\iint_{S_{1}\times
S_{2}}f_{1}(s_{1})f_{2}(s_{2})P_{A}(a|x,s_{1})$ (2.3) $\displaystyle\times
P_{B}(b|y,s_{1}s_{2})P_{C}(c|z,s_{2}){\rm{d}}\gamma_{1}(s_{1}){\rm{d}}\gamma_{2}(s_{2}),$
$\displaystyle P^{\prime}(abc|xyz)$ $\displaystyle=$
$\displaystyle\iint_{S_{1}\times
S_{2}}f_{1}(s_{1})f_{2}(s_{2})P^{\prime}_{A}(a|x,s_{1})$ (2.4)
$\displaystyle\times
P_{B}^{\prime}(b|y,s_{1}s_{2})P_{C}^{\prime}(c|z,s_{2}){\rm{d}}\gamma_{1}(s_{1}){\rm{d}}\gamma_{2}(s_{2})$

for all $a,b,c,x,y,z.$

Proof. By definition, ${\bf{P}}$ and ${\bf{P}}^{\prime}$ can be represented as

$\displaystyle P(abc|xyz)$ $\displaystyle=$
$\displaystyle\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})$
(2.5) $\displaystyle\times
P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$
$\displaystyle P^{\prime}(abc|xyz)$ $\displaystyle=$
$\displaystyle\iint_{\Lambda_{1}^{\prime}\times\Lambda_{2}^{\prime}}q_{1}^{\prime}(\lambda_{1}^{\prime})q_{2}^{\prime}(\lambda_{2}^{\prime})P_{A}^{\prime}(a|x,\lambda_{1}^{\prime})$
(2.6) $\displaystyle\times
P_{B}^{\prime}(b|y,\lambda_{1}^{\prime}\lambda_{2}^{\prime})P_{C}^{\prime}(c|z,\lambda_{2}^{\prime}){\rm{d}}\mu_{1}^{\prime}(\lambda_{1}^{\prime}){\rm{d}}\mu_{2}^{\prime}(\lambda_{2}^{\prime}),$

for some product measure spaces
$(\Lambda_{1}\times\Lambda_{2},\Omega_{1}\times\Omega_{2},\mu_{1}\times\mu_{2})$
and
$(\Lambda_{1}^{\prime}\times\Lambda_{2}^{\prime},\Omega_{1}^{\prime}\times\Omega_{2}^{\prime},\mu_{1}^{\prime}\times\mu_{2}^{\prime}).$
Putting

$S_{k}=\Lambda_{k}\times\Lambda_{k}^{\prime},T_{k}=\Omega_{k}\times\Omega_{k}^{\prime},\gamma_{k}=\mu_{k}\times\mu_{k}^{\prime},$
$s_{k}=(\lambda_{k},\lambda_{k}^{\prime}),f_{k}(s_{k})=q_{k}(\lambda_{k})q_{k}^{\prime}(\lambda_{k}^{\prime})$

for $k=1,2$, we get a product measure space $(S_{1}\times S_{2},T_{1}\times
T_{2},\gamma_{1}\times\gamma_{2})$, two independent variables $s_{k}$ with
distributions $f_{k}(s_{k})(k=1,2)$. By letting

$P_{A}(a|x,s_{1})=P_{A}(a|x,\lambda_{1}),P_{B}(b|y,s_{1}s_{2})=P_{B}(b|y,\lambda_{1}\lambda_{2}),P_{C}(c|z,s_{2})=P_{C}(c|z,\lambda_{2}),$
$P_{A}^{\prime}(a|x,s_{1})=P_{A}^{\prime}(a|x,\lambda_{1}^{\prime}),P_{B}^{\prime}(b|y,s_{1}s_{2})=P_{B}^{\prime}(b|y,\lambda_{1}^{\prime}\lambda_{2}^{\prime}),P_{C}^{\prime}(c|z,s_{2})=P_{C}^{\prime}(c|z,\lambda_{2}^{\prime}),$

for all $s_{k}=(\lambda_{k},\lambda_{k}^{\prime})$ in $S_{k}$, we obtain from
Eq. (2.5) that

$\displaystyle\iint_{S_{1}\times
S_{2}}f_{1}(s_{1})f_{2}(s_{2})P_{A}(a|x,s_{1})P_{B}(b|y,s_{1}s_{2})P_{C}(c|z,s_{2}){\rm{d}}\gamma_{1}(s_{1}){\rm{d}}\gamma_{2}(s_{2})$
$\displaystyle=$
$\displaystyle\iint_{\Lambda_{1}^{\prime}\times\Lambda_{2}^{\prime}}q_{1}^{\prime}(\lambda_{1}^{\prime})q_{2}^{\prime}(\lambda_{2}^{\prime}){\rm{d}}\mu_{1}^{\prime}(\lambda_{1}^{\prime}){\rm{d}}\mu_{2}^{\prime}(\lambda_{2}^{\prime})$
$\displaystyle\times\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$
$\displaystyle=$ $\displaystyle P(abc|xyz),$

leading to Eq. (2.3). Similarly, Eq. (2.4) follows from Eq. (2.6). The proof
is completed.

Moreover, when a CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over $\Delta_{3}$ has
a C-biLHVM (2.1) given by counting measures $\mu_{i}$ on $\Lambda_{i}$ for all
$i=1,2$, Eq. (2.1) reduces to a discrete biLHVM (D-biLHVM):

$P(abc|xyz)=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2})$
(2.7)

where
$q_{i}(\lambda_{i}),P_{A}(a|x,\lambda_{1}),P_{B}(b|y,\lambda_{1}\lambda_{2})$
and $P_{C}(c|z,\lambda_{2})$ are PDs.

Conversely, if a CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over $\Delta_{3}$ has
a D-biLHVM (2.7), then it has a C-biLHVM (2.1) given by counting measures
$\mu_{i}$ on $\Lambda_{i}$ for all $i=1,2$. To show that the converse is also
valid, we need to establish a decomposition lemma about a row-stochastic
function matrix.

To give our results, some notations are necessary. Put
$N_{X}=(o_{X})^{m_{X}}(X=A,B,C)$, which is the total number of maps from
$[m_{X}]$ into $[o_{X}]$, and let

$\Omega_{A}=\\{J_{1},J_{2},\ldots,J_{N_{A}}\\}=\\{J|J:[m_{A}]\rightarrow[o_{A}]\\},$
$\Omega_{B}=\\{K_{1},K_{2},\ldots,K_{N_{B}}\\}=\\{K|K:[m_{B}]\rightarrow[o_{B}]\\},$
$\Omega_{C}=\\{L_{1},L_{2},\ldots,L_{N_{C}}\\}=\\{L|L:[m_{C}]\rightarrow[o_{C}]\\}.$

It is clear that every $m\times n$ $\\{0,1\\}$-row statistics matrix
$T=[T_{ij}]$ corresponds uniquely to a mapping $F:[m]\rightarrow[n]$ so that
$T_{ij}=\delta_{j,F(i)}$. Thus, the sets of all $\\{0,1\\}$\- row-stochastic
matrices of orders $m_{A}\times o_{A}$, $m_{B}\times o_{B}$, and $m_{C}\times
o_{C}$ can be written as

$RSM^{(0,1)}_{m_{A}\times
o_{A}}=\\{R_{i}:=[\delta_{a,J_{i}(x)}]_{x,a}:i=1,2,\ldots,N_{A}\\},$
$RSM^{(0,1)}_{m_{B}\times
o_{B}}=\\{K_{j}:=[\delta_{b,K_{j}(y)}]_{y,b}:j=1,2,\ldots,N_{B}\\},$
$RSM^{(0,1)}_{m_{C}\times
o_{C}}=\\{L_{k}:=[\delta_{c,L_{k}(z)}]_{z,c}:k=1,2,\ldots,N_{C}\\},$

respectively.

Definition 2.2. (1) An $m\times n$ real matrix $B=[b_{ij}]$ is said to be row-
stochastic (RS) if $b_{ij}\geq 0$ for all $i,j$ and
$\sum_{j=1}^{n}b_{ij}(\lambda)=1$ for all $i\in[m].$ Especially, $B=[b_{ij}]$
is said to be $\\{0,1\\}$-RS if for each row index $i$, there is a unique
column index $J(i)$ such that $b_{ij}=\delta_{j,J(i)}$. We use
$\\{R_{k}:k\in[n^{m}]\\}$ to denote the set of all $m\times n$ $\\{0,1\\}$-RS
matrices.

(2) An $m\times n$ function matrix $B(\lambda)=[b_{ij}(\lambda)]$ on a set
$\Lambda$ is said to be RS if for each $\lambda\in\Lambda$,
$b_{ij}(\lambda)\geq 0$ for all $i,j$ and $\sum_{j=1}^{n}b_{ij}(\lambda)=1$
for all $i\in[m].$

Recall that a measurable space (MS) [44] is a pair $(\Lambda,\Omega)$ of a set
$\Lambda$ and a $\sigma$-algebra $\Omega$ of subsets of $\Lambda$. When an MS
$(\Lambda,\Omega)$ is given, the members of $\Omega$ are called
$\Omega$-measurable sets of $\Lambda$ and a function
$f:\Lambda\rightarrow{\mathbb{R}}$ is said to be $\Omega$-measurable
(measurable for short) if the inverse image $f^{-1}(G)$ of every open set $G$
in ${\mathbb{R}}$ is $\Omega$-measurable, i.e., an element of $\Omega$.
Furthermore, a measure $\mu$ on $\Omega$ is a nonnegative extended real-valued
function defined on the set $\Omega$ satisfying the countable additivity:

$A_{i}\in\Omega(i=1,2,\ldots),A_{i}\cap A_{j}=\O(i\neq
j)\Rightarrow\mu\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}\mu(A_{i}).$

Usually, we also assume that $\mu(A)<+\infty$ for some $A\in\Omega$. A measure
space [44] means a triple $(\Lambda,\Omega,\mu)$ of a set $\Lambda$, a
$\sigma$-algebra $\Omega$ on $\Lambda$ and a measure $\mu$ on $\Omega$.

Lemma 2.1. Let $(\Lambda,\Omega)$ be a measurable space and let $m\times n$
function matrix $B(\lambda)=[b_{ij}(\lambda)]$ be $\Omega$-measurable on
$\Lambda$, i.e., $b_{ij}$ are all $\Omega$-measurable functions on $\Lambda$.
Then $B(\lambda)$ is RS if and only if it can be written as:

$B(\lambda)=\sum_{k=1}^{n^{m}}\alpha_{k}(\lambda)R_{k},\
\forall\lambda\in\Lambda,$ (2.8)

where $\alpha_{k}(k=1,2,\ldots,n^{m})$ are all nonnegative and
$\Omega$-measurable functions on $\Lambda$ with
$\sum_{k=1}^{n^{m}}\alpha_{k}(\lambda)=1$ for all $\lambda\in\Lambda$.

The sufficiency is clear and the necessity is proved in Appendix.

Based on this lemma, we have the following theorem, which gives a series of
equivalent characterizations of bilocality of a CT. Especially, it says that a
CT ${\bf{P}}$ over $\Delta_{3}$ has a C-biLHVM if and only if it has a
D-biLHVM. Remarkably, characterization $(ii)$ in Theorem 2.1 was also given in
[31, Eq. (7)] by saying that “it is well known that Alice’s local response
function $P_{A}(a|x,\lambda_{1})$ can (without any loss of generality) be
taken to be deterministic”, based on the Fine’s paper [18]. Indeed, Fine in
[18] did not give a mathematical proof of this conclusion, just a physical
description.

Theorem 2.1. Let ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ be a CT over
$\Delta_{3}$. Then the following statements $(i)$-$(v)$ are equivalent.

(i) ${\bf{P}}$ is bilocal, i.e., it has a C-biLHVM (2.1).

(ii) ${\bf{P}}$ has a D-biLHVM:

$P(abc|xyz)=\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{B}}\sum_{k=1}^{N_{C}}\pi({i,j,k})\delta_{a,J_{i}(x)}\delta_{b,K_{j}(y)}\delta_{c,L_{k}(z)},\
\forall x,a,y,b,z,c,$ (2.9)

where

$\pi({i,j,k})=\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})\alpha_{i}(\lambda_{1})\beta_{j}(\lambda_{1}\lambda_{2})\gamma_{k}(\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2}),$
(2.10)

$q_{1}(\lambda_{1})$ and $q_{2}(\lambda_{2})$ are PDs of $\lambda_{1}$ and
$\lambda_{2}$, respectively, $\alpha_{i}(\lambda_{1})$,
$\beta_{j}(\lambda_{1}\lambda_{2})$ and $\gamma_{k}(\lambda_{2})$ are PDs of
$i,j$ and $k$, respectively, and are measurable w.r.t.
$\lambda_{1},(\lambda_{1}\lambda_{2})$ and $\lambda_{2}$, respectively.

(iii) ${\bf{P}}$ has a D-biLHVM:

$P(abc|xyz)=\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}\pi_{1}(i)\pi_{2}(k)\delta_{a,J_{i}(x)}P_{B}(b|y,i,k)\delta_{c,L_{k}(z)},\
\forall x,a,y,b,z,c,$ (2.11)

where
$q_{i}(\lambda_{i}),P_{A}(a|x,\lambda_{1}),P_{B}(b|y,\lambda_{1}\lambda_{2}),P_{C}(c|z,\lambda_{2})$
are PDs of $\lambda_{i},a,b,c,$ respectively.

(iv) ${\bf{P}}$ is “separable quantum”, i.e., it can be generated by a family

${\mathcal{M}}_{ABC}=\\{M^{xyz}\\}_{(x,y,z)\in[m_{A}]\times[m_{B}]\times[m_{C}]}$
(2.12)

of local POVMs
$M^{xyz}=\\{M_{a|x}{\otimes}N_{b|y}{\otimes}L_{c|z}\\}_{(a,b,c)\in[o_{A}]\times[o_{B}]\times[o_{C}]}$
on a Hilbert space
${\mathcal{H}}_{A}{\otimes}{\mathcal{H}}_{B}{\otimes}{\mathcal{H}}_{C}$
together with a pair $\\{\rho_{AB_{1}},\rho_{B_{2}C}\\}$ of separable states
$\rho_{AB_{1}}$ and $\rho_{B_{2}C}$ of systems
${\mathcal{H}}_{A}{\otimes}{\mathcal{H}}_{B_{1}}$ and
${\mathcal{H}}_{B_{2}}{\otimes}{\mathcal{H}}_{C}$, respectively, in such a way
that

$P(abc|xyz)={\rm{tr}}[(M_{a|x}{{\otimes}}N_{b|y}{\otimes}L_{c|z})(\rho_{AB_{1}}{\otimes}\rho_{B_{2}C})],\
\forall x,a,y,b,z,c.$ (2.13)

(v) ${\bf{P}}$ has a D-biLHVM:

$P(abc|xyz)=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2})$
(2.14)

where
$q_{i}(\lambda_{i}),P_{A}(a|x,\lambda_{1}),P_{B}(b|y,\lambda_{1}\lambda_{2}),P_{C}(c|z,\lambda_{2})$
are PDs of $\lambda_{i},a,b,c,$ respectively.

Proof. $(i)\Rightarrow(ii):$ Let $(i)$ be valid. Then the matrices

$M_{1}(\lambda_{1}):=[P_{A}(a|x,\lambda_{1})]_{x,a},M_{2}(\lambda_{1}\lambda_{2}):=[P_{B}(b|y,\lambda_{1}\lambda_{2})]_{y,b},M_{3}(\lambda_{2})=:[P_{C}(c|z,\lambda_{2})]_{z,c}$

are RS matrices for each parameter $\lambda_{k}\in\Lambda_{k}$. It follows
from Lemma 2.1 that they have the following decompositions:

$M_{1}(\lambda_{1})=\sum_{i=1}^{N_{A}}\alpha_{i}(\lambda_{1})R_{i},$
$M_{2}(\lambda_{1}\lambda_{2})=\sum_{j=1}^{N_{B}}\beta_{j}(\lambda_{1}\lambda_{2})Q_{j},$
$M_{3}(\lambda_{2})=\sum_{k=1}^{N_{C}}\gamma_{k}(\lambda_{2})S_{k};$

equivalently,

$P(a|x,\lambda)=\sum_{i=1}^{N_{A}}\alpha_{i}(\lambda_{1})\delta_{a,J_{i}(x)},$
$P(b|y,\lambda_{1}\lambda_{2})=\sum_{j=1}^{N_{B}}\beta_{j}(\lambda_{1}\lambda_{2})\delta_{b,K_{j}(y)},$
$P(c|z,\lambda_{2})=\sum_{k=1}^{N_{C}}\gamma_{k}(\lambda_{2})\delta_{c,L_{k}(z)},$

where $\alpha_{i}(\lambda_{1})$, $\beta_{j}(\lambda_{1}\lambda_{2})$ and
$\gamma_{k}(\lambda_{2})$ are PDs of $i,j$ and $k$, respectively, and are
measurable w.r.t. $\lambda_{1},(\lambda_{1}\lambda_{2})$ and $\lambda_{2}$,
respectively. Thus, Eq. (2.9) follows from the C-biLHVM (2.1) where
$\pi({i,j,k})$ is given by (2.10). Thus, $(ii)$ is valid.

$(ii)\Rightarrow(iii):$ Let statement $(ii)$ be valid. For all
$(i,k)\in[N_{A}]\times[N_{C}]$, put

$\pi_{1}(i)=\int_{\Lambda_{1}}q_{1}(\lambda_{1})\alpha_{i}(\lambda_{1}){\rm{d}}\mu_{1}(\lambda_{1}),\pi_{2}(k)=\int_{\Lambda_{2}}q_{2}(\lambda_{2})\gamma_{k}(\lambda_{2}){\rm{d}}\mu_{2}(\lambda_{2}),$
$P_{B}(b|y,i,k)=\frac{1}{\pi_{1}(i)\pi_{2}(k)}\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})\alpha_{i}(\lambda_{1})\gamma_{k}(\lambda_{2})P(b|y,\lambda_{1}\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$

for all $y\in[m_{B}],b\in[o_{B}]$ if $\pi_{1}(i)\pi_{2}(k)>0$; otherwise,
define

$P_{B}(b|y,i,k)=\frac{1}{o_{B}},\ \ \forall y\in[m_{B}],\forall b\in[o_{B}].$

Then $\pi_{1}(i)$, $\pi_{2}(k)$ and $P_{B}(b|y,i,k)$ are PDs of $i,j$ and $b$,
respectively, and

$\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})\alpha_{i}(\lambda_{1})\gamma_{k}(\lambda_{2})P(b|y,\lambda_{1}\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})=\pi_{1}(i)\pi_{2}(k)P_{B}(b|y,i,k)$

for all $i,k,y,b$. Note that when $\pi_{1}(i)\pi_{2}(k)=0$, the left-hand side
of above equation is less than or equal to $\pi_{1}(i)\pi_{2}(k)$ and must be
$0$. It follows from Eqs. (2.9) and (2.10) that $\forall x,a,y,b,z,c,$

$\displaystyle P(abc|xyz)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{B}}\sum_{k=1}^{N_{C}}\pi({i,j,k})\delta_{a,J_{i}(x)}\delta_{b,K_{j}(y)}\delta_{c,L_{k}(z)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}\delta_{a,J_{i}(x)}\delta_{c,L_{k}(z)}$
$\displaystyle\times\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})\alpha_{i}(\lambda_{1})\gamma_{k}(\lambda_{2})P(b|y,\lambda_{1}\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}\pi_{1}(i)\pi_{2}(k)\delta_{a,J_{i}(x)}P_{B}(b|y,i,k)\delta_{c,L_{k}(z)}.$

This shows that $(iii)$ is valid.

$(iii)\Rightarrow(iv):$ Let $(iii)$ be valid. By putting

${\mathcal{H}}_{A}={\mathcal{H}}_{B_{1}}=\mathbb{C}^{N_{A}},{\mathcal{H}}_{C}={\mathcal{H}}_{B_{2}}=\mathbb{C}^{N_{C}},{\mathcal{H}}_{B}={\mathcal{H}}_{B_{1}}{\otimes}{\mathcal{H}}_{B_{2}}=\mathbb{C}^{N_{A}}{\otimes}\mathbb{C}^{N_{C}},$

taking orthonormal bases $\\{|e_{i}\rangle\\}_{i=1}^{N_{A}}$ and
$\\{|f_{k}\rangle\\}_{k=1}^{N_{C}}$ for ${\mathcal{H}}_{A}$ and
${\mathcal{H}}_{C}$, respectively, defining POVMs:

$M_{a|x}=\sum_{i=1}^{N_{A}}\delta_{a,J_{i}(x)}|e_{i}\rangle\langle
e_{i}|,L_{c|z}=\sum_{k=1}^{N_{C}}\delta_{c,L_{k}(z)}|f_{k}\rangle\langle
f_{k}|,$
$N_{b|y}=\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}P_{B}(b|y,i,k)|e_{i}\rangle\langle
e_{i}|{\otimes}|f_{k}\rangle\langle f_{k}|,$

and constructing separable states:

$\rho_{AB_{1}}=\sum_{s=1}^{N_{A}}\pi_{1}(s)|e_{s}\rangle\langle
e_{s}|{\otimes}|e_{s}\rangle\langle
e_{s}|,\rho_{B_{2}C}=\sum_{t=1}^{N_{C}}\pi_{2}(t)|f_{t}\rangle\langle
f_{t}|{\otimes}|f_{t}\rangle\langle f_{t}|,$ (2.15)

we get

$\rho_{AB_{1}}{\otimes}\rho_{B_{2}C}=\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}\pi_{1}(i)\pi_{2}(k)|e_{i}\rangle\langle
e_{i}|{\otimes}(|e_{i}\rangle\langle e_{i}|{\otimes}|f_{k}\rangle\langle
f_{k}|){\otimes}|f_{k}\rangle\langle f_{k}|,$

and then obtain Eq. (2.13) from Eq. (2.11).

$(iv)\Rightarrow(v):$ Let $(iv)$ be valid. Since $\rho_{AB_{1}}$ and
$\rho_{B_{2}C}$ are separable states of systems $AB_{1}$ and $B_{2}C$, they
have the following decompositions:

$\rho_{AB_{1}}=\sum_{\lambda_{1}=1}^{d_{1}}q_{1}(\lambda_{1})|e_{\lambda_{1}}\rangle\langle
e_{\lambda_{1}}|{\otimes}|f_{\lambda_{1}}\rangle\langle f_{\lambda_{1}}|,$
$\rho_{B_{2}C}=\sum_{\lambda_{2}=1}^{d_{2}}q_{2}(\lambda_{2})|g_{\lambda_{2}}\rangle\langle
g_{\lambda_{2}}|{\otimes}|h_{{\lambda_{2}}}\rangle\langle h_{{\lambda_{2}}}|,$

where $q_{i}(\lambda_{i})$ is a PD of $\lambda_{i}$,
$\\{|e_{\lambda_{1}}\rangle\\}_{{\lambda_{1}}=1}^{d_{1}}$,
$\\{|f_{\lambda_{1}}\rangle\\}_{{\lambda_{1}}=1}^{d_{1}}$,
$\\{|g_{\lambda_{2}}\rangle\\}_{\lambda_{2}=1}^{d_{2}}$ and
$\\{|h_{{\lambda_{2}}}\rangle\\}_{\lambda_{2}=1}^{d_{2}}$ are sets of pure
states of ${\mathcal{H}}_{A}$, ${\mathcal{H}}_{B_{1}}$,
${\mathcal{H}}_{B_{2}}$ and ${\mathcal{H}}_{C}$, respectively. Thus,

$\rho_{AB_{1}}{\otimes}\rho_{B_{2}C}=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})|e_{\lambda_{1}}\rangle\langle
e_{\lambda_{1}}|{\otimes}|f_{\lambda_{1}}\rangle\langle
f_{\lambda_{1}}|{\otimes}|g_{\lambda_{2}}\rangle\langle
g_{\lambda_{2}}|{\otimes}|h_{\lambda_{2}}\rangle\langle h_{\lambda_{2}}|,$

and so Eq. (2.13) yields Eq. (2.14) by taking
$P_{B}(b|y,\lambda_{1}\lambda_{2})=\langle
f_{\lambda_{1}}g_{\lambda_{2}}|N_{b|y}|f_{\lambda_{1}}g_{\lambda_{2}}\rangle,$
and

$P_{A}(a|x,\lambda_{1})=\langle
e_{\lambda_{1}}|M_{a|x}|e_{\lambda_{1}}\rangle,P_{C}(c|z,\lambda_{2})=\langle
h_{\lambda_{2}}|L_{c|z}|h_{\lambda_{2}}\rangle.$

$(v)\Rightarrow(i):$ When Eq. (2.14) holds, Eq. (2.1) is valid for
$\Lambda_{i}=[d_{i}]$, $\Omega_{i}=2^{\Lambda_{i}}$ (power set of
$\Lambda_{i}$) and the counting measures $\mu_{i}$ on $\Lambda_{i}(i=1,2)$.
Hence, ${\bf{P}}$ is bilocal. The proof is completed.

Proposition 2.2. A CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over $\Delta_{3}$
is bilocal if and only if it can be written as

$P(abc|xyz)=\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{B}}\sum_{k=1}^{N_{C}}\pi_{1}(i)\pi_{2}(k)p(j|i,k)\delta_{a,J_{i}(x)}\delta_{b,K_{j}(y)}\delta_{c,L_{k}(z)}$
(2.16)

for all $x,y,z,a,b,c,$ where $\pi_{1}(i),\pi_{2}(k)$ and $p(j|i,k)$ are PDs of
$i,k$ and $j$, respectively.

Proof. When (2.16) is valid, letting

$P(b|y,i,k)=\sum_{j=1}^{N_{B}}p(j|i,k)\delta_{b,K_{j}(y)}$ (2.17)

yields Eq. (2.11). It follows from Theorem 2.1 that
${\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$. Conversely, when
${\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$, Theorem 2.1 yields
that Eq. (2.11) holds. By considering measurable space
$([N_{A}]\times[N_{C}],P([N_{A}]\times[N_{C}]))$ and then using Lemma 2.1 for
the matrix $M(i,k)=[P(b|y,i,k)]$ with $(y,b)$-entry $P(b|y,i,k)$, we conclude
the composition (2.17) of $P(b|y,i,k)$ in (2.11). Thus, (2.11) becomes (2.16).
The proof is completed.

Note that [46, Theorem 5.1] every ${\bf{P}}=\Lbrack
P(abc|xyz)\Rbrack\in\mathcal{CT}^{\textrm{Bell-local}}(\Delta_{3})$ if and
only if it can be written as

$P(abc|xyz)=\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{B}}\sum_{k=1}^{N_{C}}q_{ijk}\delta_{a,J_{i}(x)}\delta_{b,K_{j}(y)}\delta_{c,L_{k}(z)},\
\forall x,a,y,b,z,c,$ (2.18)

where $\\{q_{ijk}\\}$ is a PD of $i,j,k$. Thus, from the decomposition (2.16)
of a bilocal CT, we see that every bilocal CT is Bell local, but not the
inverse. Based on decomposition (2.18), it is easy to check that the set
$\mathcal{CT}^{\textrm{Bell-local}}(\Delta_{3})$ is a convex compact set, and
so

$\mathcal{CT}^{\textrm{Bell-
local}}(\Delta_{3})\supset{\rm{conv}}\left(\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})\right).$

Remark 2.1 yields that
${\bf{D}}_{ijk}:=\Lbrack\delta_{a,J_{i}(x)}\delta_{b,K_{j}(y)}\delta_{c,L_{k}(z)}\Rbrack\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$
for all $i,j,k.$ Thus, Eq. (2.18) implies that

$\mathcal{CT}^{\textrm{Bell-
local}}(\Delta_{3})={\rm{conv}}\left(\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})\right),$
(2.19)

which was pointed out in [31, II.B].

Since the hidden variables $\lambda_{1}$ and $\lambda_{2}$ in a C-biLHVM (2.1)
or a D-triLHVM (2.14) are assumed to be independent, the sets
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is not necessarily convex
(Remark 2.2). Branciard et al. claimed in [31, Appendix A.1] that the bilocal
set is connected. Next, we give a detail proof of the last conclusion.

Corollary 2.1.(Path-connectedness)[31, Appendix A] The set
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is path-connected in the Hilbert
space $\mathcal{T}(\Delta_{3})$. See Fig. 3.

Figure 3: Path-connectness of the set
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$

Proof. Let ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ and ${\bf{Q}}=\Lbrack
Q(abc|xyz)\Rbrack$ be any two elements of
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$. Then Theorem 2.1 implies that
${\bf{P}}$ and ${\bf{Q}}$ have D-biLHVMs:

$P(abc|xyz)=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}p_{1}(\lambda_{1})p_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2})$
(2.20)

where $p_{i}(\lambda_{i}),P_{A}(a|x,\lambda_{1}),$
$P_{B}(b|y,\lambda_{1}\lambda_{2}),P_{C}(c|z,\lambda_{2})$ are PDs of
$\lambda_{i},a,b,c,$ respectively., and

$Q(abc|xyz)=\sum_{\xi_{1}=1}^{d_{1}^{\prime}}\sum_{\xi_{2}=1}^{d_{2}^{\prime}}q_{1}(\xi_{1})q_{2}(\xi_{2})Q_{A}(a|x,\xi_{1})Q_{B}(b|y,\xi_{1}\xi_{2})Q_{C}(c|z,\xi_{2})$
(2.21)

where
$q_{i}(\xi_{i}),Q_{A}(a|x,\xi_{1}),Q_{B}(b|y,\xi_{1}\xi_{2}),Q_{C}(c|z,\xi_{2})$
are PDs of $\xi_{i},a,b,c,$ respectively.

Put $I(abc|xyz)\equiv\frac{1}{o_{A}o_{B}o_{C}}$, then ${\bf{I}}:=\Lbrack
I(abc|xyz)\Rbrack\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3}).$ For every
$t\in[0,1/2]$, set

$P^{t}_{1}(a|x,\lambda_{1})=(1-2t)P_{1}(a|x,\lambda_{1})+2t\frac{1}{o_{A}};$
$P^{t}_{2}(b|y,\lambda_{1}\lambda_{2})=(1-2t)P_{2}(b|y,\lambda_{1}\lambda_{2})+2t\frac{1}{o_{B}};$
$P^{t}_{3}(c|z,\lambda_{2})=(1-2t)P_{3}(c|z,\lambda_{2})+2t\frac{1}{o_{C}},$

which are clearly PDs of $a,b$ and $c$, respectively. Putting

$\displaystyle
P^{t}(abc|xyz)=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}p_{1}(\lambda_{1})p_{2}(\lambda_{2})P^{t}_{1}(a|x,\lambda_{1})P^{t}_{2}(b|y,\lambda_{1}\lambda_{2})P^{t}_{3}(c|z,\lambda_{2})$

yields a bilocal CT ${f}(t):=\Lbrack P^{t}(abc|xyz)\Rbrack$ over $\Delta_{3}$
for all $t\in[0,1/2]$ with ${f}(0)={\bf{P}}$ and ${f}(1/2)={\bf{I}}$.
Obviously, the map $t\mapsto{f}(t)$ from $[0,1/2]$ into
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is continuous.

Similarly, for every $t\in[1/2,1]$, set

$Q^{t}_{1}(a|x,\xi_{1})=(2t-1)Q_{1}(a|x,\xi_{1})+2(1-t)\frac{1}{o_{A}};$
$Q^{t}_{2}(b|y,\xi_{1}\xi_{2})=(2t-1)Q_{2}(b|y,\xi_{1}\xi_{2})+2(1-t)\frac{1}{o_{B}};$
$Q^{t}_{3}(c|z,\xi_{2})=(2t-1)Q_{3}(c|z,\xi_{2})+2(1-t)\frac{1}{o_{C}},$

which are clearly PDs of $a,b$ and $c$, respectively. Put

$\displaystyle
Q^{t}(abc|xyz)=\sum_{\xi_{1}=1}^{d_{1}^{\prime}}\sum_{\xi_{2}=1}^{d_{2}^{\prime}}q_{1}(\xi_{1})q_{2}(\xi_{2})Q^{t}_{1}(a|x,\xi_{1})Q^{t}_{2}(b|y,\xi_{1}\xi_{2})Q^{t}_{3}(c|z,\xi_{2}),$

then ${g}(t):=\Lbrack Q^{t}(abc|xyz)\Rbrack$ is a bilocal CT over $\Delta_{3}$
for all $t\in[1/2,1]$ with ${g}(1/2)={\bf{I}}$ and ${g}(1)={\bf{Q}}$.
Obviously, the map $t\mapsto{g}(t)$ from $[1/2,1]$ into
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is continuous. Thus, the
function $f:[0,1]\rightarrow\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$
defined by

$p(t)=\left\\{\begin{array}[]{cc}{f}(t),&t\in[0,1/2];\\\
{g}(t),&t\in(1/2,1],\end{array}\right.$

is continuous everywhere and then induces a path $p$ in
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$, connecting ${\bf{P}}$ and
${\bf{Q}}$. This shows that $\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is
path-connected. The proof is completed.

Next, let us discuss the star-convexity of the set of all bilocal CTs over
$\Delta_{3}$ such that Alice has a fixed distribution. Recall that a subset
$D$ of a vector space $V$ is said to star-convex if it has an element $s$,
called a sun of $S$, such that $(1-t)s+tD\subset D$ for all $t\in[0,1].$

To do this, take a CT ${\bf{E}}=\Lbrack E(a|x)\Rbrack$ over
$[o_{A}]\times[m_{A}]$, we let

$\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})=\left\\{{\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3}):\
{\bf{P}}_{A}={\bf{E}}\right\\}.$

The proof of the following corollary is suggested by an argument in [31,
Appendix A.2].

Corollary 2.2.(Star-convexity)[31, Appendix A.2] The set
$\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})$ is star-convex the
Hilbert space $\mathcal{T}(\Delta_{3})$. See Fig. 4.

Figure 4: Star-convexity of the set
$\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})$.

Proof. Put $P_{0}(abc|xyz)=E(a|x)\times\frac{1}{o_{B}}\times\frac{1}{o_{C}}$,
then ${\bf{P}}_{0}:=\Lbrack P_{0}(abc|xyz)\Rbrack$ is a bilocal CT over
$\Delta_{3}$ (Remark 2.1). Next we show that ${\bf{P}}_{0}$ is a sun of
$\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})$, i.e.,

$(1-t){\bf{P}}_{0}+t\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})\subset\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3}),\
\forall t\in[0,1].$ (2.22)

Let ${\bf{P}}=\Lbrack
P(abc|xyz)\Rbrack\in\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})$.
We see from Theorem 2.1 that ${\bf{P}}$ has a D-biLHVM:

$P(abc|xyz)=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}p_{1}(\lambda_{1})p_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2}).$
(2.23)

The condition $P_{A}(a|x)=E(a|x)$ becomes

$\sum_{\lambda_{1}=1}^{d_{1}}p_{1}(\lambda_{1})P_{A}(a|x,\lambda_{1})=E(a|x),\
\forall x\in[m_{A}],a\in[o_{A}].$ (2.24)

For every $t\in[0,1]$, set

$f(\lambda_{2},s)=\left\\{\begin{array}[]{cc}p_{2}(\lambda_{2})(1-t),&s=0;\\\
p_{2}(\lambda_{2})t,&s=1,\end{array}\right.$
$P_{B}(b|y,\lambda_{1},(\lambda_{2},s))=\left\\{\begin{array}[]{cc}\frac{1}{o_{B}},&s=0;\\\
P_{B}(b|y,\lambda_{1}\lambda_{2}),&s=1,\end{array}\right.$
$P_{C}(c|z,(\lambda_{2},s))=\left\\{\begin{array}[]{cc}\frac{1}{o_{C}},&s=0;\\\
P_{C}(c|z,\lambda_{2}),&s=1,\end{array}\right.$

which are PDs of $(\lambda_{2},s),b$ and $c$, respectively. For all
$x,y,z,a,b,z$, we see from (2.24) that

$\displaystyle\sum_{\lambda_{1}\in[d_{1}]}\sum_{(\lambda_{2},s)\in[d_{2}]\times\\{0,1\\}}p_{1}(\lambda_{1})f(\lambda_{2},s)P_{A}(a|x,\lambda_{1})$
$\displaystyle\times
P_{B}(b|y,\lambda_{1}(\lambda_{2},s))P_{C}(c|z,\lambda_{2}(\lambda_{2},s))$
$\displaystyle=$
$\displaystyle\sum_{\lambda_{1}\in[d_{1}]}\sum_{\lambda_{2}\in[d_{2}]}p_{1}(\lambda_{1})p_{2}(\lambda_{2})(1-t)P_{A}(a|x,\lambda_{1})\frac{1}{o_{B}}\frac{1}{o_{C}}$
$\displaystyle+\sum_{\lambda_{1}\in[d_{1}]}\sum_{\lambda_{2}\in[d_{2}]}p_{1}(\lambda_{1})p_{2}(\lambda_{2})tP_{A}(a|x,\lambda_{1})P_{B}(b|y,\lambda_{1}\lambda_{2})P_{C}(c|z,\lambda_{2})$
$\displaystyle=$ $\displaystyle(1-t)P_{0}(abc|xyz)+tP(abc|xyz).$

This shows that ${\bf{Q}}:=(1-t){\bf{P}}_{0}+t{\bf{P}}$ is bilocal with
${\bf{Q}}_{A}={\bf{E}}$ and then an element of
${\bf{P}}\in\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})$. This
shows that (2.22) holds and so
$\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3})$ is star-convex with
a sun ${\bf{P}}_{0}$. The proof is completed.

Similarly, for any fixed CT ${\bf{F}}=\Lbrack F(c|z)\Rbrack$ over
$[o_{C}]\times[m_{C}]$, the set

$\mathcal{CT}_{C-{\bf{F}}}^{\textrm{bilocal}}(\Delta_{3})=\left\\{{\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3}):\
{\bf{P}}_{C}={\bf{F}}\right\\}$

is also star-convex.

Another application of Theorem 2.1 is to prove the compactness of
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$.

Corollary 2.3.(Compactness) The set
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is compact in the Hilbert space
$\mathcal{T}(\Delta_{3})$.

Proof. Let $\textbf{P}\in\mathcal{CT}(\Delta_{3})$,
$\\{{\bf{P}}_{n}\\}_{n=1}^{\infty}\subset\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$
with $\textbf{P}_{n}\rightarrow\textbf{P}$ as $n\rightarrow\infty$, i.e.,
$P_{n}(abc|xyz)\rightarrow P(abc|xyz)$ as $n\rightarrow\infty$ for all
$(a,b,c,x,y,z)\in\Delta_{3}$. According to Theorem 2.1, we may assume that

$P_{n}(abc|xyz)=\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}\pi^{(n)}_{1}(i)\pi^{(n)}_{2}(k)\delta_{a,J_{i}(x)}P^{(n)}_{B}(b|y,i,k)\delta_{c,L_{k}(z)}$
(2.25)

for all $x,a,y,b,z,c,$ where, for each $n=1,2,\ldots,$

$\\{\pi^{(n)}_{1}(i)\\}_{i\in[N_{A}]},\
\\{\pi^{(n)}_{2}(k)\\}_{k\in[N_{C}]},{\rm{\ and\
}}\\{P^{(n)}_{B}(b|y,i,k)\\}_{b\in[o_{B}]}(\forall y,i,k)$

are PDs of $i,k,b$, respectively. By taking subsequences if necessary, we may
assume that

$\lim_{n\rightarrow\infty}\pi^{(n)}_{1}(i)=\pi_{1}(i)(\forall i),\
\lim_{n\rightarrow\infty}\pi^{(n)}_{2}(k)=\pi_{2}(k)(\forall k),$

and

$\lim_{n\rightarrow\infty}P^{(n)}_{B}(b|y,i,k)=P_{B}(b|y,i,k)(\forall
y,b,i,k).$

Obviously, $\\{\pi_{1}(i)\\}_{i\in[N_{A}]}$ and
$\\{\pi_{2}(k)\\}_{k\in[N_{C}]}$, and
$\\{P_{B}(b|y,i,k)\\}_{b\in[o_{B}]}(\forall y,i,k)$ are PDs of $i,k,b$.
Letting $n\rightarrow\infty$ in Eq. (2.25) yields that

$P(abc|xyz)=\sum_{i=1}^{N_{A}}\sum_{k=1}^{N_{C}}\pi_{1}(i)\pi_{2}(k)\delta_{a,J_{i}(x)}P_{B}(b|y,i,k)\delta_{c,L_{k}(z)},\
\forall x,a,y,b,z,c.$

Using Theorem 2.1 again implies that
${\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$. Since
$\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ is also bounded, it is compact.
The proof is completed.

Similar to Definition 2.1 we can introduce and discuss bilocality of a
tripartite CT ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over $\Delta_{3}$ defined
by

$\displaystyle P(abc|xyz)$ $\displaystyle=$
$\displaystyle\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1}\lambda_{2})$
(2.26) $\displaystyle\times
P_{B}(b|y,\lambda_{1})P_{C}(c|z,\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$

or

$\displaystyle P(abc|xyz)$ $\displaystyle=$
$\displaystyle\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|x,\lambda_{1})$
(2.27) $\displaystyle\times
P_{B}(b|y,\lambda_{2})P_{C}(c|z,\lambda_{1}\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2}).$

The last type here of bilocality is just the special case of $n$-locality for
$n=2$, i.e., the $2$-local case, but not the first one.

##  3 Bilocality of probability tensors

The usual Bell nonlocality of a quantum state or a quantum network is the
property that is exhibited by performing a set of non-compatible local POVM
measurements. Renou et al. [43] pointed out that quantum nonlocality in a
quantum network can be demonstrated without the need of having various input
settings, but only by considering the joint statistics of fixed local
measurement outputs. For example, for a tripartite network, it suffices to
consider the tripartite probability distribution $\\{P(a,b,c)\\}$ of local
measurement outcomes $a,b,c$, where $a\in[o_{A}],b\in[o_{B}]$ and
$c\in[o_{C}]$. These probabilities form a tensor ${\bf{P}}=\Lbrack
P(a,b,c)\Rbrack$ over $O_{3}=[o_{A}]\times[o_{B}]\times[o_{C}]$, we call it a
probability tensor (PT)[45]. Note that every PT ${\bf{P}}=\Lbrack
P(a,b,c)\Rbrack$ over $O_{3}$ can be written as

$P(a,b,c)=\sum_{\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in
O_{3}}q(\lambda)P_{A}(a|\lambda)P_{B}(b|\lambda)P_{C}(c|\lambda)(\forall
a,b,c),$

where

$q(\lambda)=P(\lambda_{1},\lambda_{2},\lambda_{3}),P_{A}(a|\lambda)=\delta_{a,\lambda_{1}},P_{B}(b|\lambda)=\delta_{b,\lambda_{2}},P_{C}(c|\lambda)=\delta_{c,\lambda_{3}},$

which a PDs of $\lambda,a,b$ and $c$. Thus, every PT ${\bf{P}}$ is always Bell
local.

Definition 3.1. A PT ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over $O_{3}$ is said
to bilocal if it admits a C-biLHVM:

$P(a,b,c)=\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|\lambda_{1})P_{B}(b|\lambda_{1}\lambda_{2})P_{C}(c|\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2})$
(3.1)

for a measure space
$(\Lambda_{1}\times\Lambda_{2},\Omega_{1}\times\Omega_{2},\mu_{1}\times\mu_{2})$,
where

(a) $q_{1}(\lambda_{1})$ and $P_{A}(a|\lambda_{1})(a\in[o_{A}])$ are
nonnegative measurable on $\Lambda_{1}$, $q_{2}(\lambda_{2})$ and
$P_{C}(c|\lambda_{2})(c\in[o_{C}])$ are nonnegative measurable on
$\Lambda_{2}$, and $P_{B}(b|\lambda_{1}\lambda_{2})(b\in[o_{B}])$ are
nonnegative measurable on $\Lambda_{1}\times\Lambda_{2}$;

(b) $q_{i}(\lambda_{i}),P_{A}(a|\lambda_{1}),P_{B}(b|\lambda_{1}\lambda_{2})$
and $P_{C}(c|\lambda_{2})$ are PDs of $\lambda_{i},a,b,c,$ respectively.

We call Eq. (3.1) a C-biLHVM of ${\bf{P}}$ and denote by
$\mathcal{PT}^{\textrm{bilocal}}(O_{3})$ the set of all bilocal PTs over
$O_{3}$.

For example, the PT ${\bf{p}}=\Lbrack\pi({i,j,k})\Rbrack$ given by Eq. (2.10)
is bilocal.

Remark 3.1. From Definition 3.1, we observe that when a PT ${\bf{P}}=\Lbrack
P(a,b,c)\Rbrack$ over $O_{3}$ is bilocal, its marginal distributions satisfy:

$P_{AC}(a,c)=P_{A}(a)P_{C}(c),\ \ \forall a,c.$ (3.2)

By using this property of a bilocal PT, we can find that not all PTs over
$O_{3}$ are bilocal.

Example 3.1. Let $o_{X}=2(X=A,B,C)$ and define a PT ${\bf{P}}=\Lbrack
P(a,b,c)\Rbrack$ by

$P(1,1,1)=P(2,2,2)=\frac{1}{2},P(a,b,c)=0{\rm{\ if\ }}(a,b,c)\neq(1,1,1){\rm{\
or\ }}(a,b,c)\neq(2,2,2).$

Then

$P_{AC}(1,1)=P_{AC}(2,2)=1/2,P_{AC}(a,c)=0{\rm{\ if\ }}(a,c)\neq(1,1){\rm{\
or\ }}(a,c)\neq(2,2),$ $P_{A}(1)=P_{A}(2)=1/2,P_{C}(1)=P_{C}(2)=1/2,$

while

$P_{AC}(1,1)=\frac{1}{2}\neq\frac{1}{4}=P_{A}(1)P_{C}(1).$

Thus, ${\bf{P}}\notin\mathcal{PT}^{\textrm{bilocal}}(O_{3})$.

Clearly, a PT ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over $O_{3}$ is bilocal if
and only if the CT ${\bf{P}}_{1}:=\Lbrack P(abc|111)\Rbrack$ over
$\Delta_{3}=[o_{A}]\times[o_{B}]\times[o_{C}]\times[1]\times[1]\times[1]$ is
C-bilocal, in this case $N_{A}=o_{A},N_{B}=o_{B},N_{C}=o_{C}$. Thus, as a
special case of Theorem 2.1, we have the following.

Theorem 3.1. For a PT ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over $O_{3}$, the
following statements $(i)$-$(iv)$ are equivalent.

(i) ${\bf{P}}$ is bilocal, i.e., it has a C-biLHVM (3.1).

(ii) ${\bf{P}}$ has a D-biLHVM:

$P(a,b,c)=\sum_{i=1}^{o_{A}}\sum_{k=1}^{o_{C}}\pi_{1}(i)\pi_{2}(k)\delta_{a,i}P_{B}(b|i,k)\delta_{c,k},\
\forall a,b,c,$ (3.3)

where $\pi_{1}(i)$, $\pi_{2}(k)$, and $P_{B}(b|i,k)$ are PDs of $i,k,$ and
$b$, respectively.

(iii) ${\bf{P}}$ is “separable quantum”, i.e., it can be generated by a local
POVM
$M=\\{M_{a}{\otimes}N_{b}{\otimes}L_{c}\\}_{(a,b,c)\in[o_{A}]\times[o_{B}]\times[o_{C}]}$
on the Hilbert space
${\mathcal{H}}_{A}\otimes({\mathcal{H}}_{B_{1}}{\otimes}{\mathcal{H}}_{B_{2}}){\otimes}{\mathcal{H}}_{C}$
together with a pair $\\{\rho_{AB_{1}},\rho_{B_{2}C}\\}$ of separable states
$\rho_{AB_{1}}$ and $\rho_{B_{2}C}$ of systems
${\mathcal{H}}_{A}\otimes{\mathcal{H}}_{B_{1}}$ and
${\mathcal{H}}_{B_{2}}{\otimes}{\mathcal{H}}_{C}$, respectively, in such a way
that

$P(a,b,c)={\rm{tr}}[(M_{a}{\otimes}N_{b}{\otimes}L_{c})(\rho_{AB_{1}}{\otimes}\rho_{B_{2}C})],\
\forall a,b,c.$ (3.4)

(iv) ${\bf{P}}$ has a D-biLHVM:

$P(a,b,c)=\sum_{\lambda_{1}=1}^{d_{1}}\sum_{\lambda_{2}=1}^{d_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|\lambda_{1})P_{B}(b|\lambda_{1}\lambda_{2})P_{C}(c|\lambda_{2})$
(3.5)

where $q_{i}(\lambda_{i}),P_{A}(a|\lambda_{1})$,
$P_{B}(b|\lambda_{1}\lambda_{2})$, and $P_{C}(c|\lambda_{2})$ are PDs of
$\lambda_{i},a,b,$ and $c$, respectively.

Motivated by [43], we call a PT ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over
$O_{3}$ to be trilocal if it can be written as

$\displaystyle
P(a,b,c)=\sum_{\lambda_{1}=1}^{n_{1}}\sum_{\lambda_{2}=1}^{n_{2}}\sum_{\lambda_{3}=1}^{n_{3}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})q_{3}(\lambda_{3})P_{A}(a|\lambda_{3}\lambda_{1})P_{B}(b|\lambda_{1}\lambda_{2})P_{C}(c|\lambda_{2}\lambda_{3})$
(3.6)

for all $a\in[o_{A}],b\in[o_{B}],c\in[o_{C}]$, where

$\\{q_{k}(\lambda_{k})\\}_{\lambda_{k}\in[n_{k}]},\\{P_{A}(a|\lambda_{3}\lambda_{1})\\}_{a\in[o_{A}]},\\{P_{B}(b|\lambda_{1}\lambda_{2})\\}_{b\in[o_{B}]},\\{P_{C}(c|\lambda_{2}\lambda_{3})\\}_{c\in[o_{C}]}$

are PDs. Such a PT $P$ is also called to be classical in the $C_{3}$ scenario,
according to [45, Definition 2.12]. Let ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$
over $O_{3}$ be a bilocal PT. Then it has a D-biLHVM (3.5), which implies
(3.6) by putting

$\lambda_{3}=1,q_{3}(\lambda_{3})=1,P_{A}(a|\lambda_{3}\lambda_{1})=P_{A}(a|\lambda_{1}),P_{C}(c|\lambda_{2}\lambda_{3})=P_{C}(c|\lambda_{2}).$

Hence, it is trilocal. It was proved in [43] that there exists a quantum
nontrilocal PT ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over
$O_{3}=[2]\times[4]\times[2]$. Thus, there exists a quantum PT that is non
bilocal.

As special cases of Corollaries 2.1, 2.2 and 2.3, we have the following
conclusions.

Corollary 3.1. The set $\mathcal{PT}^{\textrm{bilocal}}(O_{3})$ is path-
connected w.r.t. the pointwise-convergence topology on $O_{3}$.

By taking PDs ${\bf{E}}=\\{E(a)\\}_{a\in[o_{A}]}$ and
${\bf{F}}=\\{F(c)\\}_{c\in[o_{C}]}$, letting

$\mathcal{PT}_{A-{\bf{E}}}^{\textrm{bilocal}}(O_{3})=\left\\{{\bf{P}}\in\mathcal{PT}^{\textrm{bilocal}}(O_{3}):\
{\bf{P}}_{A}={\bf{E}}\right\\},$
$\mathcal{PT}_{C-{\bf{F}}}^{\textrm{bilocal}}(O_{3})=\left\\{{\bf{P}}\in\mathcal{PT}^{\textrm{bilocal}}(O_{3}):\
{\bf{P}}_{C}={\bf{F}}\right\\},$

we obtain have the following.

Corollary 3.2. The sets $\mathcal{PT}_{A-{\bf{E}}}^{\textrm{bilocal}}(O_{3})$
and $\mathcal{PT}_{C-{\bf{F}}}^{\textrm{bilocal}}(O_{3})$ and are star-convex.

Corollary 3.3. The set $\mathcal{PT}^{\textrm{bilocal}}(O_{3})$ is a compact
subset of the set $\mathcal{PT}(O_{3})$ of all PTs over $O_{3}$ w.r.t. the
pointwise-convergence topology on $O_{3}$.

Similar to Definition 3.1 we can introduce and discuss bilocality of a
tripartite PT ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over $O_{3}$ described by

$P(a,b,c)=\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|\lambda_{1}\lambda_{2})P_{B}(b|\lambda_{1})P_{C}(c|\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2}),$
(3.7)

or

$P(a,b,c)=\iint_{\Lambda_{1}\times\Lambda_{2}}q_{1}(\lambda_{1})q_{2}(\lambda_{2})P_{A}(a|\lambda_{1})P_{B}(b|\lambda_{2})P_{C}(c|\lambda_{1}\lambda_{2}){\rm{d}}\mu_{1}(\lambda_{1}){\rm{d}}\mu_{2}(\lambda_{2}).$
(3.8)

##  4 $n$-Locality of $n+1$-partite CTs

Consider a star-network measurement scenario Fig. 5, where Bob and the $i$-th
Alice $A_{i}$ perform simultaneously POVM measurements at the central node $B$
and $i$-th star-node $A_{i}$, labeled by $y$ and $x_{i}$, and get their
outcomes $b$ and $a_{i}$, respectively, where $n\geq 2$.

Figure 5: Star-network measurement scenario under the $n$-locality assumption
where $n=4$.

The conditional probabilities $P(a_{1},\ldots,a_{n},b|x_{1},\ldots,x_{n},y)$
of obtaining results $a_{1},a_{2},\ldots,a_{n},b$ satisfy

$P({\bf{a}}b|{\bf{x}}y)\geq 0(\forall{\bf{a}},b,{\bf{x}},y),\
\sum_{{\bf{a}},b}P({\bf{a}}b|{\bf{x}}y)=1(\forall{\bf{x}},y),$ (4.1)

where $n\geq 2$, and

${\bf{a}}=(a_{1},\ldots,a_{n})\equiv a_{1}\ldots
a_{n}\in[o_{1}]\times\ldots\times[o_{n}],$
${\bf{x}}=(x_{1},\ldots,x_{n})\equiv x_{1}\ldots
x_{n}\in[m_{1}]\times\ldots\times[m_{n}],$

and then form an $n+1$-partite CT [46]:

${\bf{P}}=\Lbrack P(a_{1},\ldots,a_{n},b|x_{1},\ldots,x_{n},y)\Rbrack:=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ (4.2)

over

$\Delta_{n+1}=\left(\prod_{i=1}^{n}[o_{i}]\right)\times[o_{B}]\times\left(\prod_{i=1}^{n}[m_{i}]\right)\times[m_{B}].$
(4.3)

As a generalization of Section 2, this section is devoted to the discussion
about $n$-locality of CTs. To do this, we use $\mathcal{CT}(\Delta_{n+1})$ to
denote the set of all CTs over $\Delta_{n+1}$ (functions $P$ with condition
(4.1)) and put $N_{i}=(o_{i})^{m_{i}}(i=1,2,\ldots,n),N_{B}=(o_{B})^{m_{B}}$.
To consider operations of CTs, we use $\mathcal{T}(\Delta_{n+1})$ to denote
the set of all tensors (i.e., real functions on $\Delta_{n+1}$) over
$\Delta_{n+1}$. For any two elements ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ and ${\bf{Q}}=\Lbrack
Q({\bf{a}}b|{\bf{x}}y)\Rbrack$ of $\mathcal{T}(\Delta_{n+1})$, define

$s{\bf{P}}+t{\bf{Q}}=\Lbrack
sP({\bf{a}}b|{\bf{x}}y)+tQ({\bf{a}}b|{\bf{x}}y)\Rbrack,\ \ \forall
s,t\in{\mathbb{R}},$
$\langle{\bf{P}}|{\bf{Q}}\rangle=\sum_{{\bf{a}},b,{\bf{x}},y}P({\bf{a}}b|{\bf{x}}y)Q({\bf{a}}b|{\bf{x}}y).$

Then $\mathcal{T}(\Delta_{n+1})$ becomes a real Hilbert space. The elements of
$\mathcal{T}(\Delta_{n+1})$ were called correlation-type tensors in [46].
Clearly, the norm-topology of $\mathcal{T}(\Delta_{n+1})$ is just the
pointwise convergent topology on $\Delta_{n+1}$. Thus,
$\mathcal{CT}(\Delta_{n+1})$ forms a compact convex subset of
$\mathcal{T}(\Delta_{n+1})$.

According to [33, 34, 36], we fix the concept of $n$-locality of a CT over
$\Delta_{n+1}$ as follows.

Definition 4.1. An $n+1$-partite CT ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over $\Delta_{n+1}$ is said to $n$-local if it
admits a continuous $n$-LHVM (C-$n$LHVM):

$P({\bf{a}}b|{\bf{x}}y)=\int_{\Lambda}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|x_{i},\lambda_{i})\times
P_{B}(b|y,\lambda){\rm{d}}\mu(\lambda)$ (4.4)

for some product measure space

$(\Lambda,\Omega,\mu)=\left(\Lambda_{1}\times\ldots\times\Lambda_{n},\Omega_{1}\times\ldots\times\Omega_{n},\mu_{1}\times\ldots\times\mu_{n}\right),$

where
${\lambda}=(\lambda_{1}\lambda_{2},\ldots,\lambda_{n})\equiv\lambda_{1}\lambda_{2}\ldots\lambda_{n}\in\Lambda_{1}\times\ldots\times\Lambda_{n}$,
and

(a) $q_{i}(\lambda_{i})$ and
$P_{i}(a_{i}|x_{i},\lambda_{i})(x_{i}\in[m_{i}],a_{i}\in[o_{i}])$ are
nonnegative $\Omega_{i}$-measurable functions on $\Lambda_{i}$, and
$P_{B}(b|y,\lambda)(y\in[m_{B}],b\in[o_{B}])$ are nonnegative
$\Omega$-measurable functions on $\Lambda$;

(b) $q_{i}(\lambda_{i}),P_{i}(a_{i}|x_{i},\lambda_{i})$ and
$P_{B}(b|y,\lambda)$ are PDs of $\lambda_{i},a_{i}$ and $b$, respectively, for
all $i\in[n]$, $x_{i}\in[m_{i}],y\in[m_{B}]$ and all $\lambda\in\Lambda.$

Moreover, ${\bf{P}}$ is said to non-$n$-local if it is not $n$-local.

Remark 4.1. Mathematically, $n$-local CTs ${\bf{P}}$ are just ones whose
entries $P({\bf{a}}b|{\bf{x}}y)$ can be factorized as a weighted average of
the product of $n$ local conditional distributions
$P_{i}(a_{i}|x_{i},\lambda_{i})$ with parameters $\lambda_{i}(i=1,2,\ldots,n)$
and a conditional distribution $P_{B}(b|y,\lambda)$ with $n$ parameters
$\lambda_{1},\ldots,\lambda_{n}$. For example, when ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over $\Delta_{n+1}$ is a product CT, i.e.,

$P({\bf{a}}b|{\bf{x}}y)=P_{1}(a_{1}|x_{1})\ldots
P_{n}(a_{n}|x_{n})P_{B}(b|y),$

it can be written as the form of (4.4) for the counting measures $\mu_{i}$ on
$\Lambda_{i}=\\{1\\}$ and

$q_{i}(\lambda_{i})=1,P_{i}(a_{i}|x_{i},\lambda_{i})=P_{i}(a_{i}|x_{i})(i=1,2,\ldots,n),P_{B}(b|y,\lambda)=P_{B}(b|y).$

Thus, every product CT over $\Delta_{n+1}$ is $n$-local, but not the converse.

Remark 4.2. If an $n+1$-partite CT ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over $\Delta_{n+1}$ admits a discrete $n$-LHVM
(D-$n$LHVM):

$P({\bf{a}}b|{\bf{x}}y)=\sum_{\lambda_{1}=1}^{d_{1}}\cdots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|x_{i},\lambda_{i})\times
P_{B}(b|y,\lambda)$ (4.5)

for all ${\bf{x}},{\bf{a}},y,b,$ where

$\\{q_{i}(\lambda_{i})\\}_{\lambda_{i}=1}^{n_{i}},\\{P_{i}(a_{i}|x_{i},\lambda_{i})\\}_{a_{i}=1}^{o_{i}}{\rm{\
and\ }}\\{P_{B}(b|y,\lambda)\\}_{b=1}^{o_{B}}$

are PDs, then Eq. (4.4) holds for the counting measures $\mu_{i}$ on
$\Lambda_{i}=[d_{i}](i=1,2,\ldots,n)$. This shows that if ${\bf{P}}$ has a
D-$n$LHVM (4.5), then it has a C-$n$LHVM (4.4). Indeed, the converse is also
valid, see the following theorem.

Remark 4.3. Let ${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ be $n$-local.
Then it has a C-$n$LHVM (4.4) and is then nonsignaling[46] with the marginal
distribution on system $A_{i}A_{j}B(i<j)$:

$\displaystyle P_{A_{i}A_{j}B}(a_{i}a_{j}b|x_{i}x_{j}y)$ $\displaystyle=$
$\displaystyle\iint_{\Lambda_{i}\times\Lambda_{j}}q_{i}(\lambda_{i})q_{j}(\lambda_{j})P_{i}(a_{i}|x_{i},\lambda_{i})$
$\displaystyle\times
P_{j}(a_{j}|x_{j},\lambda_{j})P_{B}(b|y,\lambda_{i}\lambda_{j}){\rm{d}}\mu_{i}(\lambda_{i}){\rm{d}}\mu_{j}(\lambda_{j}).$

Thus, ${\bf{P}}_{A_{i}A_{j}B}=\Lbrack
P_{A_{i}A_{j}B}(a_{i}a_{j}b|x_{i}x_{j}y)\Rbrack$ is a bilocal tripartite CT in
the sense of (2.27) for all $1\leq i<j\leq n$. Hence, if there exists an index
$(i_{0},j_{0})$ with $1\leq i_{0}<j_{0}\leq n$ such that the tripartite CT
${\bf{P}}_{A_{i_{0}}A_{j_{0}}B}$ is not bilocal according to definition
(2.27), then ${\bf{P}}$ must be non-$n$-local. Furthermore, by tracing out
part $B$, we get the marginal distribution of ${\bf{P}}$ on system
$A_{1}\cdots A_{n}$:

$P_{A_{1}\cdots A_{n}}({\bf{a}}|{\bf{x}})=P_{A_{1}}(a_{1}|x_{1})\cdots
P_{A_{n}}(a_{n}|x_{n}),\ \forall{\bf{a}},{\bf{x}},$

that is, ${\bf{P}}_{A_{1}\cdots
A_{n}}={\bf{P}}_{A_{1}}{\otimes}\cdots{\otimes}{\bf{P}}_{A_{n}},$ a complete
product $n$-partite CT. Moreover, when trace out any Alice, say $A_{1}$, we
get

Next, we use $\\{J^{(i)}_{k_{i}}:k_{i}\in[N_{i}]\\}$ to denote the set of all
mappings from $[m_{i}]$ into $[o_{i}]$ for all $i=1,2,\ldots,n$ and use
$\\{K_{j}:j\in[N_{B}]\\}$ to denote the set of all mappings from $[m_{B}]$
into $[o_{B}]$.

Theorem 4.1. For a CT ${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over
$\Delta_{n+1}$, the following statements $(i)$-$(v)$ are equivalent.

(i) ${\bf{P}}$ is $n$-local.

(ii) ${\bf{P}}$ has the form:

$P({\bf{a}}b|{\bf{x}}y)=\sum_{k_{1}=1}^{N_{1}}\cdots\sum_{k_{n}=1}^{N_{n}}\sum_{j=1}^{N_{B}}\pi({k_{1},\ldots,k_{n},j})\times\prod_{i=1}^{n}\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}\times\delta_{b,K_{j}(y)}$
(4.6)

for all ${\bf{x}},{\bf{a}},y,b,$ where

$\pi({k_{1},\ldots,k_{n},j})=\int_{\Lambda}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}\alpha_{k_{i}}(\lambda_{i})\times\beta_{j}(\lambda){\rm{d}}\mu(\lambda),$
(4.7)

$q_{i}(\lambda_{i})$ is PD of $\lambda_{i}$, $\alpha_{k_{i}}(\lambda_{i})$ and
$\beta_{j}(\lambda)$ are PDs of $k_{i}$ and $j$, respectively, and are
measurable w.r.t. $\lambda_{i}$ and $\lambda$, respectively.

(iii) ${\bf{P}}$ admits a D-$n$LHVM:

$P({\bf{a}}b|{\bf{x}}y)=\sum_{k_{1}=1}^{N_{1}}\cdots\sum_{k_{n}=1}^{N_{n}}\prod_{i=1}^{n}\pi_{i}(k_{i})\times\prod_{i=1}^{n}\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}\times
P_{B}(b|y,k_{1},\ldots,k_{n})$ (4.8)

where $\pi_{i}(k_{i})$ and $P_{B}(b|y,k_{1},\ldots,k_{n})$ are PDs of $k_{i}$
and $b$, respectively.

(iv) ${\bf{P}}$ is “separable quantum”, i.e., it can be generated by a family
of local POVMs

$M^{xy}=\\{M_{a_{1}|x_{1}}{\otimes}\ldots{\otimes}M_{a_{n}|x_{n}}{\otimes}N_{b|y}\\}_{(a_{1},\ldots,a_{n},b)\in[o_{1}]\times\ldots\times[o_{n}]\times[o_{B}]}$

on a Hilbert space
$\left({\otimes}_{i=1}^{n}{\mathcal{H}}_{A_{i}}\right)\otimes{\mathcal{H}}_{B}$
(${\mathcal{H}}_{B}={\otimes}_{i=1}^{n}{\mathcal{H}}_{B_{i}}$) together with
$n$ separable states $\rho_{A_{i}B_{i}}$ of systems
${\mathcal{H}}_{A_{i}}\otimes{\mathcal{H}}_{B_{i}}$, in such a way that

$P({\bf{a}}b|{\bf{x}}y)={\rm{tr}}[(({\otimes}_{i=1}^{n}M_{a_{i}|x_{i}}){\otimes}N_{b|y})\rho_{n}]$
(4.9)

for all ${\bf{x}},{\bf{a}},y,b$, where

$\rho_{n}=\mathcal{T}(\rho_{A_{1}B_{1}}{\otimes}\ldots{\otimes}\rho_{A_{n}B_{n}})\mathcal{T}^{\dagger},$
(4.10)

and $\mathcal{T}$ denotes the canonical unitary isomorphism from
${\otimes}_{i=1}^{n}({\mathcal{H}}_{A_{i}}\otimes{\mathcal{H}}_{B_{i}})$ onto
$\left({\otimes}_{i=1}^{n}{\mathcal{H}}_{A_{i}}\right)\otimes\left({\otimes}_{i=1}^{n}{\mathcal{H}}_{B_{i}}\right)$,
referring to Fig. 2 for the case where $n=4$.

(v) P admits a D-$n$LHVM (4.5).

Proof. $(i)\Rightarrow(ii):$ Let $(i)$ be valid. Then the conditions $(a)$ and
$(b)$ in Definition 4.1 imply that

$M_{i}(\lambda_{i})=[P_{A}(a_{i}|x_{i},\lambda_{i})]_{x_{i},a_{i}}(i=1,2,\ldots,n),M(\lambda)=[P_{B}(b|y,\lambda)]_{y,b}$

are measurable RS function matrices on $\Lambda_{i}$ and $\Lambda$,
respectively. It follows from Lemma 2.1 that they have the following
decompositions:

$\displaystyle
M_{i}(\lambda_{i})=\sum_{k_{i}=1}^{N_{i}}\alpha_{k_{i}}(\lambda_{i})[\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}],\
M(\lambda)=\sum_{j=1}^{N_{B}}\beta_{j}(\lambda)[\delta_{b,K_{j}(y)}],$ (4.11)

where $\alpha_{k_{i}}(\lambda_{i})$ and $\beta_{j}(\lambda)$ are PDs of
$k_{i}$ and $j$, and are measurable w.r.t. $\lambda_{i}$ and $\lambda$,
respectively, $\\{R^{(i)}_{k_{i}}\\}_{k_{i}=1}^{N_{i}}$ and
$\\{Q_{j}\\}_{j=1}^{N_{B}}$ denote the sets of all $\\{0,1\\}$-row-stochastic
matrices of $m_{i}\times o_{i}$ and $m_{B}\times o_{B}$, respectively. Thus,

$P_{i}(a_{i}|x_{i},\lambda_{i})=\sum_{k_{i}=1}^{N_{i}}\alpha_{k_{i}}(\lambda_{i})\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})},P_{B}(b|y,\lambda)=\sum_{j=1}^{N_{B}}\beta_{j}(\lambda)\delta_{b,K_{j}(y)},$

and Eqs. (4.6) and (4.7) are then obtained from the C-LHVM (4.4).

$(ii)\Rightarrow(iii):$ Let statement $(ii)$ be valid. For all
$(i,k_{i})\in[n]\times[N_{i}]$, put

$\pi_{i}(k_{i})=\int_{\Lambda_{i}}q_{i}(\lambda_{i})\alpha_{k_{i}}(\lambda_{i}){\rm{d}}\mu_{i}(\lambda_{i}),f(b,y,\lambda)=\sum_{j=1}^{N_{B}}\delta_{b,K_{j}(y)}\beta_{j}(\lambda),$
$P_{B}(b|y,k_{1},\ldots,k_{n})=\frac{1}{\pi_{1}({k_{1}})\ldots\pi_{n}(k_{n})}\int_{\Lambda}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}\alpha_{k_{i}}(\lambda_{i})\times
f(b,y,\lambda){\rm{d}}\mu(\lambda)$

for all $y\in[m_{B}],b\in[o_{B}]$ if $\pi_{1}({k_{1}})\ldots\pi_{n}(k_{n})>0$;
otherwise, define

$P_{B}(b|y,k_{1},\ldots,k_{n})=\frac{1}{o_{B}},\ \ \forall y\in[m_{B}],\forall
b\in[o_{B}].$

Then $\\{\pi_{i}(k_{i})\\}_{k_{i}\in[N_{A}]}$ and
$\\{P_{B}(b|y,k_{1},\ldots,k_{n})\\}_{b\in[o_{B}]}$ are PDs, and

$\int_{\Lambda}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}\alpha_{k_{i}}(\lambda_{i})\times
f(b,y,\lambda){\rm{d}}\mu(\lambda)=\prod_{i=1}^{n}\pi_{i}(k_{i})\times
P_{B}(b|y,k_{1},\ldots,k_{n})$

for all $k_{i}\in[N_{i}],y\in[m_{B}],b\in[o_{B}]$. Then Eq. (4.6) leads to Eq.
(4.8) and so $(iii)$ is valid.

$(iii)\Rightarrow(iv):$ Let $(iii)$ be valid. By putting

${\mathcal{H}}_{A_{i}}={\mathcal{H}}_{B_{i}}=\mathbb{C}^{N_{i}},{\mathcal{H}}_{B}={\mathcal{H}}_{B_{1}}{\otimes}{\mathcal{H}}_{B_{2}}{\otimes}\ldots{\otimes}{\mathcal{H}}_{B_{n}},$

taking orthonormal bases $\\{|e^{(i)}_{k_{i}}\rangle\\}_{k_{i}=1}^{N_{i}}$ for
${\mathcal{H}}_{A_{i}}={\mathcal{H}}_{B_{i}}$, defining POVMs:

$M^{xy}=\\{M_{a_{1}|x_{1}}{\otimes}\ldots{\otimes}M_{a_{n}|x_{n}}{\otimes}N_{b|y}\\}_{(a_{1},\ldots,a_{n},b)\in[o_{1}]\times\ldots\times[o_{n}]\times[o_{B}]}$

on a Hilbert space
${\otimes}_{i=1}^{n}{\mathcal{H}}_{A_{i}}\otimes{\mathcal{H}}_{B}$ with

$M_{a_{i}|x_{i}}=\sum_{k_{i}=1}^{N_{i}}\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}|e^{(i)}_{k_{i}}\rangle\langle
e^{(i)}_{k_{i}}|,$
$N_{b|y}=\sum_{k_{1},\ldots,k_{n}}P_{B}(b|y,k_{1},\ldots,k_{n}){\otimes}_{i=1}^{n}|e^{(i)}_{k_{i}}\rangle\langle
e^{(i)}_{k_{i}}|,$

and constructing separable states:

$\rho_{A_{i}B_{i}}=\sum_{k_{i}=1}^{N_{i}}\pi_{i}(k_{i})|e^{(i)}_{k_{i}}\rangle_{A_{i}}\langle
e^{(i)}_{k_{i}}|{\otimes}|e^{(i)}_{k_{i}}\rangle_{B_{i}}\langle
e^{(i)}_{k_{i}}|,$ (4.12)

we obtain Eq. (4.9) from Eq. (4.8).

$(iv)\Rightarrow(v):$ Let $(iv)$ be valid. Since $\rho_{A_{i}B_{i}}$ is a
separable state of system $A_{i}B_{i}$, it can be written as

$\rho_{A_{i}B_{i}}=\sum_{\lambda_{i}=1}^{d_{i}}q_{i}(\lambda_{i})|e^{(i)}_{\lambda_{i}}\rangle\langle
e^{(i)}_{\lambda_{i}}|{\otimes}|f^{(i)}_{\lambda_{i}}\rangle\langle
f^{(i)}_{\lambda_{i}}|,$

where $\\{q_{i}(\lambda_{i})\\}_{\lambda_{i}\in[d_{i}]}(i\in[n])$ are PDs,
$\\{|e^{(i)}_{\lambda_{i}}\rangle\\}_{\lambda_{i}=1}^{d_{i}}$ and
$\\{|f^{(i)}_{\lambda_{i}}\rangle\\}_{\lambda_{i}=1}^{d_{i}}$ are pure states
of ${\mathcal{H}}_{A_{i}}$ and ${\mathcal{H}}_{B_{i}}$, respectively. Thus,

$\displaystyle\rho_{n}$ $\displaystyle:=$
$\displaystyle\mathcal{T}(\rho_{A_{1}B_{1}}{\otimes}\ldots{\otimes}\rho_{A_{n}B_{n}})\mathcal{T}^{\dagger}$
$\displaystyle=$
$\displaystyle\sum_{\lambda_{1},\ldots,\lambda_{n}}\prod_{i=1}^{n}q_{i}(\lambda_{i})\left({\otimes}_{i=1}^{n}|e^{(i)}_{\lambda_{i}}\rangle\langle
e^{(i)}_{\lambda_{i}}|\right){\otimes}\left({\otimes}_{i=1}^{n}|f^{(i)}_{\lambda_{i}}\rangle\langle
f^{(i)}_{\lambda_{i}}|\right),$

and so

$\displaystyle P({\bf{a}}b|{\bf{x}}y)$ $\displaystyle=$
$\displaystyle{\rm{tr}}[(({\otimes}_{i=1}^{n}M_{a_{i}|x_{i}}){\otimes}N_{b|y})\rho_{n}]$
$\displaystyle=$
$\displaystyle\sum_{\lambda_{1},\ldots,\lambda_{n}}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|x_{i},\lambda_{i})\times
P_{B}(b|y,\lambda_{1},\ldots,\lambda_{n})$

for all ${\bf{x}},{\bf{a}},y,b$, where

$P_{i}(a_{i}|x_{i},\lambda_{i})=\left\langle
e^{(i)}_{\lambda_{i}}|M_{a_{i}|x_{i}}|e^{(i)}_{\lambda_{i}}\right\rangle,$
$P_{B}(b|y,\lambda_{1},\ldots,\lambda_{n})=\left\langle
f^{(1)}_{\lambda_{1}}\ldots
f^{(n)}_{\lambda_{n}}|N_{b|y}|f^{(1)}_{\lambda_{1}}\ldots
f^{(n)}_{\lambda_{n}}\right\rangle.$

This shows that ${\bf{P}}$ has a D-$n$LVHM.

$(v)\Rightarrow(i):$ Use Remark 4.2. The proof is completed.

As an application of Theorem 4.1, we obtain the following corollary.

Corollary 4.1. A CT ${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over
$\Delta_{n+1}$ has a C-$n$LHVM (4.4) if and only if it has a D-$n$LHVM (4.5)
if and only if it can be written as

$P({\bf{a}}b|{\bf{x}}y)=\sum_{k_{1}=1}^{N_{1}}\cdots\sum_{k_{n}=1}^{N_{n}}\sum_{j=1}^{N_{B}}q(k_{1},\ldots,k_{n},j)\prod_{i=1}^{n}\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}\times\delta_{b,K_{j}(y)}$
(4.13)

for all ${\bf{x}},{\bf{a}},y,b,$ where

$q(k_{1},\ldots,k_{n},j)=\pi_{1}(k_{1})\ldots\pi_{n}(k_{n})p(j|k_{1},\ldots,k_{n}),$
(4.14)

$\pi_{i}(k_{i})$ and $p(j|k_{1},\ldots,k_{n})$ are PDs of $k_{i}$ and $j$,
respectively.

From this corollary, we see that when a CT ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over $\Delta_{n+1}$ is $n$-local, it has a
D-$n$LHVM (4.5). Tracing out any Alice, say $A_{1}$, yields

$\displaystyle P(a_{2}\ldots a_{n}b|x_{2}\ldots x_{n}y)$ $\displaystyle=$
$\displaystyle\sum_{\lambda_{2}=1}^{d_{2}}\ldots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=2}^{n}q_{i}(\lambda_{i})$
(4.15) $\displaystyle\times\prod_{i=2}^{n}P_{i}(a_{i}|x_{i},\lambda_{i})\times
P^{\prime}_{B}(b|y,\lambda_{2}\ldots\lambda_{n})$

for all ${a_{2},\ldots,a_{n}},{x_{2},\ldots,x_{n}},y,b,$ where

$P^{\prime}_{B}(b|y,\lambda_{2}\cdots\lambda_{n})=\sum_{\lambda_{1}=1}^{d_{1}}q_{i}(\lambda_{i})P_{B}(b|y,\lambda),$

which is a PD of $b$ for all $y,\lambda_{2}\cdots\lambda_{n}$. This means that
the marginal distribution ${\bf{P}}_{A_{2}\cdots A_{n}B}$ of an $n$-local CT
${\bf{P}}$ is an $n-1$-local CT.

It was proved in [46, Theorem 5.1] that an $n+1$-partite CT ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over $\Delta_{n+1}$ is Bell local if and only
if Eq. (4.13) holds for some PD $q(k_{1},\ldots,k_{n},j)$ of
$k_{1},\ldots,k_{n},j$, which is not necessarily of the form (4.14). This
characterization implies that the set $\mathcal{CT}^{\textrm{Bell-
local}}(\Delta_{n+1})$ of all $n+1$-partite Bell local CTs over $\Delta_{n+1}$
forms a convex compact set. Thus, Corollary 2.1 implies that every $n$-local
CT ${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over $\Delta_{n+1}$ must
be $n+1$-partite Bell local CT. It follows from Remark 4.1 that the set
$\mathcal{CT}^{\textrm{Bell-local}}(\Delta_{n+1})$ is just the convex hull of
the set $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ of all $n$-local CTs
over $\Delta_{n+1}$. That is,

$\mathcal{CT}^{\textrm{Bell-
local}}(\Delta_{n+1})={\rm{conv}}(\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})).$
(4.16)

The characterization (4.8) of an $n$-local CT shows that all $n$-local CTs can
be represented by D-$n$LHVMs in which local hidden variables
$k_{1},k_{2},\ldots,k_{n}$ have the same dimensions
$N_{1},N_{2},\ldots,N_{n}$. This advantage will serve to the proof of the
closedness of $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$.

Corollary 4.2. $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is a compact
set in the Hilbert space $\mathcal{T}(\Delta_{n+1})$.

Proof. Let ${\bf{P}}^{(m)}=\Lbrack
P^{(m)}({\bf{a}}b|{\bf{x}}y)\Rbrack\in\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$
for all $m=1,2,\ldots,$ with
$\textbf{P}^{(m)}\rightarrow\textbf{P}\in\mathcal{T}(\Delta_{n+1})$ as
$m\rightarrow\infty$, i.e., $P^{(m)}({\bf{a}}b|{\bf{x}}y)\rightarrow
P({\bf{a}}b|{\bf{x}}y)$ as $m\rightarrow\infty$ for all possible variables
$({\bf{a}},b,{\bf{x}},y)$. Clearly, $\textbf{P}\in\mathcal{CT}(\Delta_{n+1})$.
According to Theorem 4.1, each $\textbf{P}^{(m)}$ can be written as

$P^{(m)}({\bf{a}}b|{\bf{x}}y)=\sum_{k_{1}=1}^{N_{1}}\cdots\sum_{k_{n}=1}^{N_{n}}\prod_{i=1}^{n}\pi^{(m)}_{i}(k_{i})\times\prod_{i=1}^{n}\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}\times
P^{(m)}_{B}(b|y,k_{1}\ldots k_{n})$ (4.17)

for all $m=1,2,\ldots$ and possible variables $({\bf{a}},b,{\bf{x}},y)$, where
$\\{\pi^{(m)}_{i}(k_{i})\\}_{k_{i}\in[N_{i}]}(\forall i\in[n])$ and
$\\{P^{(m)}_{B}(b|y,k_{1}\ldots k_{n})\\}_{b\in[o_{B}]}(\forall
y\in[m_{B}],k_{i}\in[N_{i}])$ are PDs. By taking subsequences if necessary, we
may assume that

$\lim_{m\rightarrow\infty}\pi^{(m)}_{i}(k_{i})=\pi_{i}(k_{i})(\forall
i\in[n]),$ $\lim_{m\rightarrow\infty}P^{(m)}_{B}(b|y,k_{1}\ldots
k_{n})=P_{B}(b|y,k_{1}\ldots k_{n})(\forall y,b,k_{1},\ldots,k_{n}).$

Obviously, $\\{\pi_{i}(k_{i})\\}_{i\in[N_{i}]}$ and $\\{P_{B}(b|y,k_{1}\ldots
k_{n})\\}_{b\in[o_{B}]}(\forall y,k_{1},\ldots,k_{n})$ are PDs. Letting
$m\rightarrow\infty$ in Eq. (4.17) yields that

$P({\bf{a}}b|{\bf{x}}y)=\sum_{k_{1}=1}^{N_{1}}\cdots\sum_{k_{n}=1}^{N_{n}}\prod_{i=1}^{n}\pi_{i}(k_{i})\times\prod_{i=1}^{n}\delta_{a_{i},J^{(i)}_{k_{i}}(x_{i})}\times
P_{B}(b|y,k_{1}\ldots k_{n})$

for all possible ${\bf{x}},{\bf{a}},y,b$. Using Theorem 4.1 again implies that
${\bf{P}}\in\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$. This shows that
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is closed and so compact since
it is also a bounded set of the finite dimensional Hilbert space
$\mathcal{T}(\Delta_{n+1})$. The proof is completed.

From Definition 4.1, we observe that when
${\bf{P}}\in\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$, the marginal
distributions satisfy ${\bf{P}}_{A_{1}\ldots
A_{n}}={\bf{P}}_{A_{1}}{\otimes}\ldots{\otimes}{\bf{P}}_{A_{n}},$ i.e.,

${{P}}_{A_{1}\ldots A_{n}}(a_{1}\ldots a_{n}|x_{1}\ldots
x_{n})={{P}}_{A_{1}}(a_{1}|x_{1})\ldots{{P}}_{A_{n}}(a_{n}|x_{n}),\ \forall
a_{i},x_{i};$

particularly,

${\bf{P}}_{A_{1}A_{2}}={\bf{P}}_{A_{1}}{\otimes}{\bf{P}}_{A_{2}}.$ (4.18)

It is easy to find a Bell local ${\bf{P}}$ such that the property (4.18) is
not satisfied and then is not $n$-local. This shows that not all Bell local
CTs are $n$-local. By noticing that a convex combination of two product CTs is
not necessarily a product CT, we conclude from the property (4.18) that the
set $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is not convex. However, it
has many subsets that are star-convex.

Next, let us discuss the weak star-convexity of the set
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ by finding star-convex subsets
of it. To do so, for a fixed $1\leq k\leq n$, put

${\widehat{A_{k}}}=A_{1}\ldots A_{k-1}A_{k+1}\ldots A_{n},$
${\bf{a}}_{k}=a_{1}\ldots a_{k-1}a_{k+1}\ldots a_{n},{\bf{x}}_{k}=x_{1}\ldots
x_{k-1}x_{k+1}\ldots x_{n},$

and take an $n-1$-partite product CT ${\bf{E}}_{k}=\Lbrack
E_{k}({\bf{a}}_{k}|{\bf{x}}_{k})\Rbrack$ over $\prod_{i\neq
k}[o_{i}]\times\prod_{i\neq k}[m_{i}]$ with

$E_{k}({\bf{a}}_{k}|{\bf{x}}_{k})=\prod_{i\neq k}Q_{i}(a_{i}|x_{i}),$

and define a set

$\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})=\left\\{{\bf{P}}\in\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1}):{\bf{P}}_{\widehat{A_{k}}}={\bf{E}}_{k}\right\\},$
(4.19)

where ${\bf{P}}_{\widehat{A_{k}}}$ denotes the marginal distribution of
${\bf{P}}$ on the subsystem $\widehat{A_{k}}$, i.e.,

${{P}}_{\widehat{A_{k}}}({\bf{a}}_{k}|{\bf{x}}_{k})=\sum_{a_{k},b}P({\bf{a}}b|{\bf{x}}b),$

which is independent of the choices of $x_{k}$ and $y$ whenever ${\bf{P}}$ is
$n$-local and then nonsignaling. Define ${\bf{S}}_{k}=\Lbrack
S_{k}({\bf{a}}b|{\bf{x}}y)\Rbrack$ by
$S_{k}({\bf{a}}b|{\bf{x}}y)=\prod_{i=1}^{n}Q_{i}(a_{i}|x_{i})\times\frac{1}{o_{B}},$
where $Q_{k}(a_{k}|x_{k})\equiv\frac{1}{o_{k}},$ then
${\bf{S}}_{k}\in\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})$
(Remark 4.1).

Corollary 3.3. For any $k=1,2,\ldots,n,$ the set
$\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})$ is star-convex
with a sun ${\bf{S}}_{k}$, i.e.,

$(1-t){\bf{S}}_{k}+t\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})\subset\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1}),\
\forall t\in[0,1].$

See Figure 6.

Figure 6: Star-convexity of the set
$\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})$.

Proof. Let ${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ be any element of
$\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})$. Then Theorem
4.1 implies that ${\bf{P}}$ has a D-$n$LHVM:

$P({\bf{a}}b|{\bf{x}}y)=\sum_{\lambda_{1}=1}^{d_{1}}\cdots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|x_{i},\lambda_{i})\times
P_{B}(b|y,\lambda)$ (4.20)

for all ${\bf{x}},{\bf{a}},y,b,$ where

$\\{q_{i}(\lambda_{i})\\}_{\lambda_{i}=1}^{n_{i}},\\{P_{i}(a_{i}|x_{i},\lambda_{i})\\}_{a_{i}=1}^{o_{i}}{\rm{\
and\ }}\\{P_{B}(b|y,\lambda)\\}_{b=1}^{o_{B}}$

are PDs. For every $t\in[0,1]$, put

$f_{k}(\lambda_{k},s)=\left\\{\begin{array}[]{cc}q_{k}(\lambda_{k})(1-t),&s=0;\\\
q_{k}(\lambda_{k})t,&s=1,\end{array}\right.$ (4.21)
$P_{B}(b|y,\lambda_{1}\ldots\lambda_{k-1}(\lambda_{k},s)\lambda_{k+1}\ldots\lambda_{n})=\left\\{\begin{array}[]{cc}\frac{1}{o_{B}},&s=0;\\\
P_{B}(b|y,\lambda_{1}\ldots\lambda_{n}),&s=1,\end{array}\right.$ (4.22)
$P_{k}(a_{k}|x_{k},(\lambda_{k},s))=\left\\{\begin{array}[]{cc}\frac{1}{o_{k}},&s=0;\\\
P_{k}(a_{k}|x_{k},\lambda_{k}),&s=1,\end{array}\right.$ (4.23)

which are PDs of $(\lambda_{k},s),b$ and $a_{k}$, respectively. Put

$\displaystyle Q^{t}_{k}({\bf{a}}b|{\bf{x}}y)$ $\displaystyle=$
$\displaystyle\sum_{\lambda_{i}(i\neq k)}\prod_{i\neq
k}q_{i}(\lambda_{i})\times\sum_{\lambda_{k},s}f_{k}(\lambda_{k},s)\times\prod_{i\neq
k}P_{i}(a_{i}|x_{i},\lambda_{i})$ $\displaystyle\times
P_{k}(a_{k}|x_{k},(\lambda_{k},s))P_{B}(b|y,\lambda_{1}\ldots\lambda_{k-1}(\lambda_{k},s)\lambda_{k+1}\ldots\lambda_{n}),$

then Theorem 4.1 implies that ${\bf{Q}}^{t}_{k}=\Lbrack
Q^{t}_{k}({\bf{a}}b|{\bf{x}}y)\Rbrack\in\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$.
On the other hand, for all ${\bf{x}},{\bf{a}},y,b,$ we compute that

$\displaystyle Q^{t}_{k}({\bf{a}}b|{\bf{x}}y)$ $\displaystyle=$
$\displaystyle\sum_{\lambda_{i}(i\neq k)}\prod_{i\neq
k}q_{i}(\lambda_{i})\times\sum_{\lambda_{k}}f_{k}(\lambda_{k},0)\times\prod_{i\neq
k}P_{i}(a_{i}|x_{i},\lambda_{i})$ $\displaystyle\times
P_{k}(a_{k}|x_{k},(\lambda_{k},0))P_{B}(b|,y\lambda_{1}\ldots\lambda_{k-1}(\lambda_{k},0)\lambda_{k+1}\ldots\lambda_{n})$
$\displaystyle+\sum_{\lambda_{i}(i\neq k)}\prod_{i\neq
k}q_{i}(\lambda_{i})\sum_{\lambda_{k}}f_{k}(\lambda_{k},1)\times\prod_{i\neq
k}P_{i}(a_{i}|x_{i},\lambda_{i})$ $\displaystyle\times
P_{k}(a_{k}|x_{k},(\lambda_{k},1))P_{B}(b|,y\lambda_{1}\ldots\lambda_{k-1}(\lambda_{k},1)\lambda_{k+1}\ldots\lambda_{n})$
$\displaystyle=$
$\displaystyle(1-t)P_{\widehat{A_{k}}}({\bf{a}}_{k}|{\bf{x}}_{k})\times\frac{1}{o_{k}}\times\frac{1}{o_{B}}+tP({\bf{a}}b|{\bf{x}}y)$
$\displaystyle=$
$\displaystyle(1-t)S_{k}({\bf{a}}b|{\bf{x}}y)+tP({\bf{a}}b|{\bf{x}}y).$

This shows that $(1-t){\bf{S}}_{k}+t{\bf{P}}={\bf{Q}}^{t}_{k}$, which is an
$n$-local CT over $\Delta_{n+1}$. Clearly,

$({\bf{Q}}^{t}_{k})_{\widehat{A_{k}}}=(1-t)({\bf{S}}_{k})_{\widehat{A_{k}}}+t{\bf{P}}_{\widehat{A_{k}}}={\bf{E}}_{k}$

and so ${\bf{Q}}^{t}_{k}$, i.e., $(1-t){\bf{S}}_{k}+t{\bf{P}}$ is an element
of $\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1}).$ This shows
that

$(1-t){\bf{S}}_{k}+t\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})\subset\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1}),\
\ \forall t\in[0,1],$

and so $\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1})$ is star-
convex with a sun ${\bf{S}}_{k}$. The proof is completed.

Corollary 4.4. The set $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is
path-connected. See Figure 7.

Figure 7: Path-connectedness of the set
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$.

Proof. Put $I({\bf{a}}b|{\bf{x}}y)\equiv\frac{1}{o_{1}o_{2}\ldots
o_{n}o_{B}}$, then ${\bf{I}}:=\Lbrack I({\bf{a}}b|{\bf{x}}y)\Rbrack$ is an
element of $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$. Let
${\bf{P}}=\Lbrack P({\bf{a}}b|{\bf{x}}y)\Rbrack$ and ${\bf{Q}}=\Lbrack
Q({\bf{a}}b|{\bf{x}}y)\Rbrack$ be any two elements of
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$. Theorem 4.1 implies that
${\bf{P}}$ and ${\bf{Q}}$ admit D-$n$LHVMs:

$P({\bf{a}}b|{\bf{x}}y)=\sum_{\lambda_{1}=1}^{d_{1}}\cdots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=1}^{n}p_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|x_{i},\lambda_{i})\times
P_{B}(b|y,\lambda),$ (4.24)
$Q({\bf{a}}b|{\bf{x}}y)=\sum_{\xi_{1}=1}^{r_{1}}\cdots\sum_{\xi_{n}=1}^{r_{n}}\prod_{i=1}^{n}q_{i}(\xi_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|x_{i},\xi_{i})\times
P_{B}(b|y,\xi)$ (4.25)

where $\xi=\xi_{1}\xi_{2}\ldots\xi_{n}$. For every $t\in[0,1/2]$, set

$P^{t}_{i}(a_{i}|x_{i},\lambda_{i})=(1-2t)P_{i}(a_{i}|x_{i},\lambda_{i})+2t\frac{1}{o_{i}}(i=1,2,\ldots,n);$
$P^{t}_{B}(b|y,\lambda)=(1-2t)P_{B}(b|y,\lambda)+2t\frac{1}{o_{B}},$

which are clearly PDs of $a_{i}$ and $b$, respectively. Put

$\displaystyle
P^{t}({\bf{a}}b|{\bf{x}}y)=\sum_{\lambda_{1}=1}^{d_{1}}\cdots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=1}^{n}p_{i}(\lambda_{i})\times\prod_{i=1}^{n}P^{t}_{i}(a_{i}|x_{i},\lambda_{i})\times
P^{t}_{B}(b|y,\lambda),$

then ${f}(t):=\Lbrack P^{t}({\bf{a}}b|{\bf{x}}y)\Rbrack$ is an $n$-local CT
over $\Delta_{n+1}$ for all $t\in[0,1/2]$ with ${f}(0)={\bf{P}}$ and
${f}(1/2)={\bf{I}}$. Obviously, the map $t\mapsto{f}(t)$ from $[0,1/2]$ into
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is continuous.

Similarly, for every $t\in[1/2,1]$, set

$Q^{t}_{i}(a_{i}|x_{i},\xi_{i})=(2t-1)Q_{i}(a_{i}|x_{i},\xi_{i})+2(1-t)\frac{1}{o_{i}}(i=1,2,\ldots,n),$
$Q^{t}_{B}(b|y,\xi)=(2t-1)Q_{B}(b|y,\xi)+2(1-t)\frac{1}{o_{B}},$

which are clearly PDs of $a_{i}$ and $b$, respectively. Put

$\displaystyle
Q^{t}({\bf{a}}b|{\bf{x}}y)=\sum_{\xi_{1}=1}^{r_{1}}\cdots\sum_{\xi_{n}=1}^{r_{n}}\prod_{i=1}^{n}q_{i}(\xi_{i})\times\prod_{i=1}^{n}Q^{t}_{i}(a_{i}|x_{i},\xi_{i})\times
P^{t}_{B}(b|y,\xi),$

then ${g}(t):=\Lbrack Q^{t}({\bf{a}}b|{\bf{x}}y)\Rbrack$ is an $n$-local CT
over $\Delta_{n+1}$ for all $t\in[1/2,1]$ with ${g}(1/2)={\bf{I}}$ and
${g}(1)={\bf{Q}}$. Obviously, the map $t\mapsto{g}(t)$ from $[1/2,1]$ into
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is continuous. Thus, the
function $p:[0,1]\rightarrow\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$
defined by

$p(t)=\left\\{\begin{array}[]{cc}{f}(t),&t\in[0,1/2];\\\
{g}(t),&t\in(1/2,1],\end{array}\right.$

is continuous everywhere and then induces a path $p$ in
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ with $p(0)={\bf{P}}$ and
$p(1)={\bf{Q}}$. This shows that
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is path-connected. The proof
is completed.

##  5 $n$-Locality of $n+1$-partite PTs

In this section, we introduce and discuss $n+1$-partite PT ${\bf{P}}=\Lbrack
P(a_{1},\ldots,a_{n},b):=\Lbrack P({\bf{a}},b)\Rbrack\equiv\Lbrack
P({\bf{a}}b)\Rbrack$ over $O_{n+1}=\prod_{i=1}^{n}[o_{i}]\times[o_{B}]$ where

${\bf{a}}=(a_{1},\ldots,a_{n})\equiv a_{1}\ldots
a_{n}\in[o_{1}]\times\ldots\times[o_{n}].$

It is defined as a tensor with index set $O_{n+1}$, equivalently, a function
$P:O_{n+1}\rightarrow{\mathbb{R}}$, satisfying

$P({\bf{a}},b)\geq 0(\forall{\bf{a}},b),\ \sum_{{\bf{a}},b}P({\bf{a}},b)=1.$
(5.1)

We use $\mathcal{PT}(O_{n+1})$ to denote the set of all PTs over $O_{n+1}$
(functions $P$ with condition (5.1)).

To consider operations of PTs, we use $\mathcal{T}(O_{n+1})$ to denote the set
of all tensors (i.e., real functions) over $O_{n+1}$. For any two elements
${\bf{P}}=\Lbrack P({\bf{a}},b)\Rbrack$ and ${\bf{Q}}=\Lbrack
Q({\bf{a}},b)\Rbrack$ of $\mathcal{T}(O_{n+1})$, define

$s{\bf{P}}+t{\bf{Q}}=\Lbrack sP({\bf{a}},b)+tQ({\bf{a}},b)\Rbrack,\
\langle{\bf{P}}|{\bf{Q}}\rangle=\sum_{{\bf{a}},b}P({\bf{a}},b)Q({\bf{a}},b).$

Then $\mathcal{T}(O_{n+1})$ becomes a real Hilbert space. Clearly, the norm-
topology of $\mathcal{T}(O_{n+1})$ is just the pointwise convergent topology
on $O_{n+1}$. Thus, $\mathcal{PT}(O_{n+1})$ forms a compact convex subset of
$\mathcal{T}(O_{n+1})$.

Definition 5.1. An $n+1$-partite PT ${\bf{P}}=\Lbrack P({\bf{a}},b)\Rbrack$
over $\Delta_{n+1}$ is said to $n$-bilocal if it admits a continuous $n$-LHVM
(C-$n$LHVM):

$P({\bf{a}},b)=\int_{\Lambda}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|\lambda_{i})\times
P_{B}(b|\lambda){\rm{d}}\mu(\lambda)$ (5.2)

for some product measure space

$(\Lambda,\Omega,\mu)=\left(\Lambda_{1}\times\ldots\times\Lambda_{n},\Omega_{1}\times\ldots\times\Omega_{n},\mu_{1}\times\ldots\times\mu_{n}\right),$

where
${\lambda}=(\lambda_{1}\lambda_{2},\ldots,\lambda_{n})\equiv\lambda_{1}\lambda_{2}\ldots\lambda_{n}\in\Lambda_{1}\times\ldots\times\Lambda_{n},$
and

(a) $q_{i}(\lambda_{i})$ and $P_{i}(a_{i}|\lambda_{i})(a_{i}\in[o_{i}])$ are
nonnegative $\Omega_{i}$-measurable functions on $\Lambda_{i}$, and
$P_{B}(b|\lambda)(b\in[o_{B}])$ are nonnegative $\Omega$-measurable functions
on $\Lambda$;

(b) $q_{i}(\lambda_{i}),P_{i}(a_{i}|\lambda_{i})$ and $P_{B}(b|\lambda)$ are
PDs of $\lambda_{i},a_{i}$ and $b$, respectively, for all $i\in[n]$ and all
$\lambda\in\Lambda$.

Remark 5.1. If an $n+1$-partite PT ${\bf{P}}=\Lbrack P({\bf{a}},b)\Rbrack$
over $O_{n+1}$ is product, i.e., $P({\bf{a}}b|{\bf{x}}y)$ is a product of
$n+1$ CTs $P_{1}(a_{1}),\ldots,P_{n}(a_{n})$ and $P_{B}(b)$, then it can be
written as the form of (4.4) by taking the counting measures $\mu_{i}$ on
$\Lambda_{i}=\\{1\\}$ and

$q_{i}(\lambda_{i})=1,P_{1}(a_{i}|\lambda_{i})=P_{i}(a_{i})(i=1,2,\ldots,n),P_{B}(b|\lambda)=P_{B}(b).$

Thus, every product PT over $O_{n+1}$ is $n$-local, but not the converse.

Remark 5.2. If an $n+1$-partite PT ${\bf{P}}=\Lbrack P({\bf{a}},b)\Rbrack$
over $O_{n+1}$ admits a discrete $n$-LHVM (D-$n$LHVM):

$P({\bf{a}},b)=\sum_{\lambda_{1}=1}^{d_{1}}\cdots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|\lambda_{i})\times
P_{B}(b|\lambda)$ (5.3)

for all ${\bf{a}},b,$ where $q_{i}(\lambda_{i}),P_{i}(a_{i}|\lambda_{i})$ and
$P_{B}(b|\lambda)$ are PDs of $\lambda_{i},a_{i}$ and $b$, respectively, for
all $i\in[n]$ and all $\lambda\in\Lambda$, then Eq. (5.2) holds for the
counting measures $\mu_{i}$ on $\Lambda_{i}=[d_{i}](i=1,2,\ldots,n)$. This
shows that if ${\bf{P}}$ has a D-$n$LHVM (5.3), then it has a C-$n$LHVM (5.2).
Indeed, the converse is also valid, see the following theorem which can be
viewed as a special case of Theorem 4.1 with $m_{i}=1$ and $N_{i}=o_{i}$ for
all $i=1,2,\ldots,n$.

Theorem 5.1. For a PT ${\bf{P}}=\Lbrack P({\bf{a}},b)\Rbrack$ over $O_{n+1}$,
the following statements $(i)$-$(iv)$ are equivalent.

(i) ${\bf{P}}$ is $n$-local, i.e., it can be written as (5.2).

(ii) ${\bf{P}}$ admits a D-$n$LHVM:

$P({\bf{a}},b)=\sum_{k_{1}=1}^{N_{1}}\cdots\sum_{k_{n}=1}^{N_{n}}\prod_{i=1}^{n}\pi_{i}(k_{i})\times\prod_{i=1}^{n}\delta_{a_{i},{k_{i}}}\times
P_{B}(b|k_{1},\ldots,k_{n})$ (5.4)

where $\\{\pi_{i}(k_{i})\\}_{k_{i}\in[N_{i}]}(\forall i\in[n])$ and
$\\{P_{B}(b|k_{1},\ldots,k_{n})\\}_{b\in[o_{B}]}(\forall k_{i}\in[N_{i}])$ are
PDs.

(iii) ${\bf{P}}$ is “separable quantum”, i.e., it can be generated by a local
POVM

$M=\\{M_{a_{1}}{\otimes}\ldots{\otimes}M_{a_{n}}{\otimes}N_{b}\\}_{(a_{1},\ldots,a_{n},b)\in[o_{1}]\times\ldots\times[o_{n}]\times[o_{B}]}$

on a Hilbert space
$\left({\otimes}_{i=1}^{n}{\mathcal{H}}_{A_{i}}\right)\otimes{\mathcal{H}}_{B}$
(${\mathcal{H}}_{B}={\otimes}_{i=1}^{n}{\mathcal{H}}_{B_{i}}$) together with
$n$ separable states $\rho_{A_{i}B_{i}}$ of systems
${\mathcal{H}}_{A_{i}}\otimes{\mathcal{H}}_{B_{i}}$, in such a way that

$P({\bf{a}},b)={\rm{tr}}[(({\otimes}_{i=1}^{n}M_{a_{i}}){\otimes}N_{b})\rho_{n}],\
\ \forall{\bf{a}},b,$ (5.5)

where

$\rho_{n}=\mathcal{T}(\rho_{A_{1}B_{1}}{\otimes}\ldots{\otimes}\rho_{A_{n}B_{n}})\mathcal{T}^{\dagger},$
(5.6)

and $\mathcal{T}$ denotes the canonical unitary isomorphism from
${\otimes}_{i=1}^{n}({\mathcal{H}}_{A_{i}}\otimes{\mathcal{H}}_{B_{i}})$ onto
$\left({\otimes}_{i=1}^{n}{\mathcal{H}}_{A_{i}}\right)\otimes\left({\otimes}_{i=1}^{n}{\mathcal{H}}_{B_{i}}\right)$.

(iv) P admits a D-$n$LHVM:

$P({\bf{a}},b)=\sum_{\lambda_{1}=1}^{d_{1}}\cdots\sum_{\lambda_{n}=1}^{d_{n}}\prod_{i=1}^{n}q_{i}(\lambda_{i})\times\prod_{i=1}^{n}P_{i}(a_{i}|\lambda_{i})\times
P_{B}(b|\lambda),\ \ \forall{\bf{a}},b,$ (5.7)

where
$\\{q_{i}(\lambda_{i})\\}_{\lambda_{i}=1}^{n_{i}},\\{P_{i}(a_{i}|\lambda_{i})\\}_{a_{i}=1}^{o_{i}}{\rm{\
and\ }}\\{P_{B}(b|\lambda)\\}_{b=1}^{o_{B}}$ are PDs.

As special cases of corresponding conclusions in Section 4, we obtain the
following corollaries.

Corollary 5.1. A PT over $O_{n+1}$ has a C-$n$LHVM (5.2) if and only if it has
a D-$n$LHVM (5.3).

An $n$-local PT ${\bf{P}}=\Lbrack P({\bf{a}},b)\Rbrack$ over $O_{n+1}$ is said
to be Bell local if the CT $\Lbrack P({\bf{a}},b|{\bf{x}},y)\Rbrack:=\Lbrack
P({\bf{a}},b)\Rbrack$ over $\Delta_{n+1}$ with $m_{i}=m_{B}=1(i=1,2,\ldots,n)$
is Bell local, i.e., it can be written as

$P({\bf{a}},b)=\sum_{\lambda\in
O_{n+1}}q(\lambda)\prod_{i=1}^{n}P_{i}(a_{i}|\lambda)\times P_{B}(b|\lambda),$
(5.8)

where $q(\lambda),P_{i}(a_{i}|\lambda),P_{B}(b|\lambda)$ are PDs of
$\lambda,a_{i}$ and $b$, respectively. Indeed, every PT ${\bf{P}}=\Lbrack
P({\bf{a}},b)\Rbrack$ over $O_{n+1}$ is Bell local since it can be written as
(5.8) where
$P_{i}(a_{i}|\lambda)=\delta_{a_{i}|\lambda_{i}},P_{B}(b|\lambda)=\delta_{b,\lambda_{n+1}}$
and

$\lambda=(\lambda_{1},\ldots,\lambda_{n},\lambda_{n+1}),q(\lambda)=P(\lambda_{1},\ldots,\lambda_{n},\lambda_{n+1}).$

Using (4.16) yields that

$\mathcal{PT}(O_{n+1})=\mathcal{PT}^{\textrm{Bell-
local}}(O_{n+1})={\rm{conv}}(\mathcal{PT}^{n\textrm{-local}}(O_{n+1})).$ (5.9)

Corollary 5.2. The set $\mathcal{PT}^{n\textrm{-local}}(O_{n+1})$ is compact
in the Hilbert space $\mathcal{T}(O_{n+1})$.

From Definition 5.1, we observe that when
${\bf{P}}\in\mathcal{PT}^{n\textrm{-local}}(O_{n+1})$, the marginal
distributions satisfy ${\bf{P}}_{A_{1}\ldots
A_{n}}={\bf{P}}_{A_{1}}{\otimes}\ldots{\otimes}{\bf{P}}_{A_{n}},$ i.e.,

${{P}}_{A_{1}\ldots
A_{n}}(a_{1},\ldots,a_{n})={{P}}_{A_{1}}(a_{1})\ldots{{P}}_{A_{n}}(a_{n}),\
\forall a_{i}\in[o_{i}];$

especially,

${\bf{P}}_{A_{1}A_{2}}={\bf{P}}_{A_{1}}{\otimes}{\bf{P}}_{A_{2}}.$ (5.10)

It is easy to construct a PT ${\bf{P}}$ that has no the property (5.10) and
then is not $n$-local. This shows that not all Bell local PTs are $n$-local.
By noticing that a convex combination of two product PTs is not necessarily a
product PT, we conclude from the property (5.10) that the set
$\mathcal{PT}^{n\textrm{-local}}(O_{n+1})$ is not convex. However, it has many
subsets that are star-convex.

Next, let us discuss the weak star-convexity of the set
$\mathcal{PT}^{n\textrm{-local}}(O_{n+1})$ by finding star-convex subsets of
it. To do so, for a fixed $1\leq k\leq n$, put

${\widehat{A_{k}}}=A_{1}\ldots A_{k-1}A_{k+1}\ldots
A_{n},{\bf{a}}_{k}=a_{1}\ldots a_{k-1}a_{k+1}\ldots a_{n},$

and take an $n-1$-partite product PT ${\bf{E}}_{k}=\Lbrack
E_{k}({\bf{a}}_{k})\Rbrack$ over $\prod_{i\neq k}[o_{i}]$ with
$E_{k}({\bf{a}}_{k})=\prod_{i\neq k}Q_{i}(a_{i}),$ and define a set

$\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(O_{n+1})=\left\\{{\bf{P}}\in\mathcal{PT}^{n\textrm{-local}}(O_{n+1}):{\bf{P}}_{\widehat{A_{k}}}={\bf{E}}_{k}\right\\},$
(5.11)

where ${\bf{P}}_{\widehat{A_{k}}}$ denotes the marginal distribution of
${\bf{P}}$ on the subsystem $\widehat{A_{k}}$, i.e.,
${{P}}_{\widehat{A_{k}}}({\bf{a}}_{k})=\sum_{a_{k},b}P({\bf{a}},b).$ Define
$Q_{k}(a_{k})\equiv\frac{1}{o_{k}}$ and ${\bf{S}}_{k}=\Lbrack
S_{k}({\bf{a}},b)\Rbrack$ by
$S_{k}({\bf{a}},b)=\prod_{i=1}^{n}Q_{i}(a_{i})\times\frac{1}{o_{B}},$ then
${\bf{S}}_{k}\in\mathcal{PT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(O_{n+1})$
(Remark 5.1).

Corollary 5.3. For any $k=1,2,\ldots,n,$ the set
$\mathcal{PT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(O_{n+1})$ is star-convex with
a sun ${\bf{S}}_{k}$, i.e.,

$(1-t){\bf{S}}_{k}+t\mathcal{PT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(O_{n+1})\subset\mathcal{PT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(O_{n+1}),\
\forall t\in[0,1].$

Corollary 5.4. The set $\mathcal{PT}^{n\textrm{-local}}(O_{n+1})$ is path-
connected.

##  6 Conclusions

In this work, we have discussed the bilocality of tripartite correlation
tensors (CTs) ${\bf{P}}=\Lbrack P(abc|xyz)\Rbrack$ over
$\Delta_{3}=[o_{A}]\times[o_{B}]\times[o_{C}]\times[m_{A}]\times[m_{B}]\times[m_{C}]$
and probability tensors (PTs) ${\bf{P}}=\Lbrack P(a,b,c)\Rbrack$ over
$O_{3}=[o_{A}]\times[o_{B}]\times[o_{C}]$, respectively, as well as the
$n$-locality of $n+1$-partite CTs ${\bf{P}}=\Lbrack
P({\bf{a}}b|{\bf{x}}y)\Rbrack$ over
$\Delta_{n+1}=\prod_{i=1}^{n}[o_{i}]\times[o_{B}]\times\prod_{i=1}^{n}[m_{i}]\times[m_{B}]$
and PTs ${\bf{P}}=\Lbrack P({\bf{a}}b)\Rbrack$ over
$O_{n+1}=\prod_{i=1}^{n}[o_{i}]\times[o_{B}]$, where $[n]$ stands for the set
consisting of $1,2,\ldots,n$. Based on a convex-decomposition lemma on a
measurable function row-stochastic matrix, we have established a series of
characterizations and properties of bilocal and $n$-local CTs and PTs, and
obtained the following conclusions.

(1) Integration and summation descriptions (which we named C-biLHVM and
D-biLHVM) for bilocality of a tripartite CT and PT are equivalent, denoted by
C-biLHVM=D-biLHVM.

(2) A tripartite CT (resp. PT) ${\bf{P}}$ is bilocal if and only if it is
“separable quantum”, i.e., it can be generated by a pair of separable shared
states together with a set of local POVMs (resp. a local POVM).

(3) The set $\mathcal{CT}^{\textrm{Bell-local}}(\Delta_{3})$ of tripartite
Bell local CTs with the same size $\Delta_{3}$ is just the convex hull of the
set $\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ of all bilocal CTs over
$\Delta_{3}$, while the convex hull of the set
$\mathcal{PT}^{\textrm{bilocal}}(\Delta_{3})$ of all bilocal PTs over $O_{3}$
is just the set of all PTs over $O_{3}$.

(4) The set $\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3})$ forms a compact
path-connected set w.r.t. the pointwise convergent topology on the index set
$\Delta_{3}$ and has many star-convex subsets:

$\mathcal{CT}_{A-{\bf{E}}}^{\textrm{bilocal}}(\Delta_{3}):=\left\\{{\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3}):\
{\bf{P}}_{A}={\bf{E}}\right\\},\
\forall{\bf{E}}\in\mathcal{CT}([o_{A}]\times[m_{A}]),$
$\mathcal{CT}_{C-{\bf{F}}}^{\textrm{bilocal}}(\Delta_{3}):=\left\\{{\bf{P}}\in\mathcal{CT}^{\textrm{bilocal}}(\Delta_{3}):\
{\bf{P}}_{C}={\bf{F}}\right\\},\
\forall{\bf{F}}\in\mathcal{CT}([o_{C}]\times[m_{C}]).$

(5) Corresponding conclusions have been obtained for $n$-locality of
$n+1$-partite CTs and PTs, including: C-$n$LHVM=D-$n$LHVM; ${\bf{P}}$ is in
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ if and only if it is
“separable quantum”; $\mathcal{CT}^{\textrm{Bell-local}}(\Delta_{n+1})$ is the
convex hull of $\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$;
$\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1})$ is a compact path-connected
set w.r.t. the pointwise convergent topology on $\Delta_{n+1}$ and has many
star-convex subsets:

$\mathcal{CT}_{{\bf{E}}_{k}}^{n\textrm{-local}}(\Delta_{n+1}):=\left\\{{\bf{P}}\in\mathcal{CT}^{n\textrm{-local}}(\Delta_{n+1}):{\bf{P}}_{\widehat{A_{k}}}={\bf{E}}_{k}\right\\}$

for all ${\bf{E}}_{k}\in\mathcal{CT}(\prod_{i\neq k}[o_{i}]\times\prod_{i\neq
k}[m_{i}])$, where ${\bf{P}}_{\widehat{A_{k}}}$ denotes the marginal
distribution of ${\bf{P}}$ on the subsystem $\prod_{i\neq k}A_{i}$.

It is remarkable to point out that the bilocality is not a special case that
$n=2$ of the $n$-locality discussed here, but it is a special case where $n=2$
of the $n$-locality ($n\geq 2$) introduced in [33], described by the following
C-LHVM:

$\displaystyle P(a_{1}\ldots a_{n+1}|x_{1}\ldots x_{n+1})$ $\displaystyle=$
$\displaystyle\int{\rm{d}}\lambda_{1}\cdots\int{\rm{d}}\lambda_{n}\prod_{i=1}^{n}\rho_{i}(\lambda_{i})\times
P_{1}(a_{1}|x_{1},\lambda_{1})$
$\displaystyle\times\prod_{i=2}^{n}P_{i}(a_{i}|x_{i},\lambda_{i-1}\lambda_{i})\times
P_{n}(a_{n+1}|x_{n+1},\lambda_{n}).$

Indeed, the $n$-locality here is based on the star network configuration [36],
while that of the work [33] is a direct generalization of the bilocal scenario
considered in [30].

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Nos. 11871318, 12271325), the Fundamental Research Funds for the Central
Universities (GK202103003, GK202107014) and and the Special Plan for Young
Top-notch Talent of Shaanxi Province (1503070117).

## Appendix

The proof of Lemma 2.1. Let $N=n^{m}$ and $R_{k}=[\delta_{j,J_{k}(i)}]$, where
$J_{k}({k\in[N]})$ are all of the mappings from $[m]$ into $[n]$. Here, we use
$\max^{*}\\{a_{1},a_{2},\ldots,a_{n}\\}$ to denote the first maximal number of
$a_{1},a_{2},\ldots,a_{n}$ and define
$B^{(0)}(\lambda)=[b_{ij}^{(0)}(\lambda)]=B(\lambda).$ For each $i\in[m]$, put

$b_{i,j^{(0)}_{i}(\lambda)}^{(0)}(\lambda)={\max}^{*}\left\\{b_{i1}^{(0)}(\lambda),b_{i2}^{(0)}(\lambda),\ldots,b_{in}^{(0)}(\lambda)\right\\},$

which is the first maximum value of entries in the $i$-th row of the matrix
$B^{(0)}$, which is clearly $\Omega$-measurable w.r.t. $\lambda$ on $\Lambda$.
Define

$h_{ij}(\lambda)=\left\\{\begin{array}[]{cc}1,&j=j^{(0)}_{i}(\lambda);\\\
0,&{j\neq j^{(0)}_{i}(\lambda)}\end{array}\right.\
R_{s_{1}(\lambda)}=[h_{ij}(\lambda)],$
$\alpha_{1}(\lambda)=\min\left\\{b_{i,j^{(0)}_{i}(\lambda)}^{(0)}(\lambda)~{}|~{}i\in[m]\right\\},$
$B^{(1)}(\lambda)=[b^{(1)}_{ij}(\lambda)]=B^{(0)}(\lambda)-\alpha_{1}(\lambda)R_{s_{1}(\lambda)}.$

Then $\alpha_{1}(\lambda)$ is an entry of $B^{(0)}(\lambda)=B(\lambda)$ and
$B^{(1)}(\lambda)$ is a nonnegative matrix, and satisfies

$B^{(0)}(\lambda)=\alpha_{1}(\lambda)R_{s_{1}(\lambda)}+B^{(1)}(\lambda),\ \
{\rm{zero}}(B^{(0)}(\lambda))<{\rm{zero}}(B^{(1)}(\lambda)),$ $None$

where ${\rm{zero}}(A)$ denotes the number of zero-entries of a matrix $A$.
Since $b_{ij}(\lambda)$’s are $\Omega$-measurable w.r.t. $\lambda$ on
$\Lambda$, we see that $\alpha_{1}(\lambda)$ and entries of $B^{(1)}(\lambda)$
are $\Omega$-measurable w.r.t. $\lambda$ on $\Lambda$. Clearly,
$R_{s_{1}(\lambda)}$ is a $\\{0,1\\}$-RS matrix, depending on $\lambda$. To
our aim, we have to replace it with a convex combination of $R_{k}$’s with
$\Omega$-measurable coefficients. To do this, we put

$\Lambda^{(1)}_{k}=\\{\lambda\in\Lambda:R_{s_{1}(\lambda)}=R_{k}\\}(k=1,2,\ldots,N),$
$None$

then $\lambda\in\Lambda^{(1)}_{k}$ if and only if
$b_{ij^{(0)}_{i}(\lambda)}^{(0)}(\lambda)=b_{i,J_{k}(i)}^{(0)}(\lambda)$ for
all $i\in[m]$. Hence, $\Lambda^{(1)}_{k}$ is an $\Omega$-measurable subset of
$\Lambda$ and so its characteristic function $\chi_{\Lambda^{(1)}_{k}}$ is
$\Omega$-measurable on $\Lambda$ for each $k\in[N]$. Thus,
$c^{(1)}_{k}(\lambda):=\chi_{\Lambda^{(1)}_{k}}(\lambda)\alpha_{1}(\lambda)$
is nonnegative and $\Omega$-measurable w.r.t. $\lambda$ on $\Lambda$. Since
${\Lambda^{(1)}_{k}}\cap{\Lambda^{(1)}_{j}}=\emptyset(k\neq j)$ and
$\cup_{k\in[N]}{\Lambda^{(1)}_{k}}=\Lambda$, we see that

$\sum_{k\in[N]}\chi_{\Lambda^{(1)}_{k}}(\lambda)\alpha_{1}(\lambda)=\alpha_{1}(\lambda),\
\sum_{k\in[N]}\chi_{\Lambda^{(1)}_{k}}(\lambda)R_{k}=R_{s_{1}(\lambda)},\ \
\forall\lambda\in\Lambda.$ $None$

Using Eqs. (A.1) and (A.3) yields that

$B^{(0)}(\lambda)=\sum_{k\in[N]}c^{(1)}_{k}(\lambda)R_{k}+B^{(1)}(\lambda),\ \
\forall\lambda\in\Lambda.$ $None$

Similarly, we can decompose $B^{(1)}(\lambda)$ in (A.4) as

$B^{(1)}(\lambda)=\sum_{k\in[N]}c^{(2)}_{k}(\lambda)R_{k}+B^{(2)}(\lambda),\
\forall\lambda\in\Lambda,$ $None$

where $c^{(2)}_{k}(\lambda)$ and $B^{(2)}(\lambda)$ are nonnegative and
$\Omega$-measurable w.r.t. $\lambda$ on $\Lambda$ with the property that
${\rm{zero}}(B^{(1)}(\lambda))<{\rm{zero}}(B^{(2)}(\lambda))$ for all
$\lambda\in\Lambda$. Thus,

$B^{(0)}(\lambda)=\sum_{k\in[N]}c^{(1)}_{k}(\lambda)R_{k}+\sum_{k\in[N]}c^{(2)}_{k}(\lambda)R_{k}+B^{(2)}(\lambda),\
\forall\lambda\in\Lambda.$ $None$

Continuously, we can find nonnegative $\Omega$-measurable functions
$c^{(1)}_{k},c^{(2)}_{k},\ldots,c^{(r)}_{k}$ and nonnegative
$\Omega$-measurable matrices $B^{(1)},\ldots,B^{(r-1)},B^{(r)}$ such that

$B^{(0)}(\lambda)=\sum_{k\in[N]}\sum_{t=1}^{r-1}c^{(t)}_{k}(\lambda)R_{k}+B^{(r)}(\lambda),\
\ \forall\lambda\in\Lambda,$

with the property that

${\rm{zero}}(B^{(0)}(\lambda))<{\rm{zero}}(B^{(1)}(\lambda))<{\rm{zero}}(B^{(2)}(\lambda))<\cdots<{\rm{zero}}(B^{(r-1)}(\lambda))\leq
mn.$

Thus, after doing at most $mn$ steps, we arrive at the case where $B^{(r)}=0$
and then obtain that

$B(\lambda)=B^{(0)}(\lambda)=\sum_{k=1}^{N}\alpha_{k}(\lambda)R_{k},\ \
\forall\lambda\in\Lambda,$

where the coefficient functions
$\alpha_{k}=\sum_{t=1}^{r-1}c^{(t)}_{k}(k\in[N])$, which are nonnegative and
$\Omega$-measurable functions on $\Lambda$. The proof is completed.

## References

  * [1] J.S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1 195-200 (1964).
  * [2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86 419-478 (2014).
  * [3] B.S. Tsirelson, Some results and problems on quantum Bell-type inequalities, Hadronic J. Suppl. 8 329-345 (1993).
  * [4] J. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23 880 (1969).
  * [5] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40 4277 (1989).
  * [6] D. Kaszlikowski, P. Gnaci$\acute{\rm{n}}$ski, M. $\dot{\rm{Z}}$ukowski, W. Miklaszewski, A. Zeilinger, Violations of local realism by two entangled $N$-dimensional systems are stronger than for two qubits, Phys. Rev. Lett. 85 4418 (2000).
  * [7] R.F. Werner, M.M. Wolf, All-multipartite Bell-correlation inequalities for two dichotomic observables per site, Phys. Rev. A 64 032112 (2001).
  * [8] D. Collins, N. Gisin, N. Linden, S. Massar, S. Popescu, Bell inequalities for arbitrarily high-dimensional systems, Phys. Rev. Lett. 88 040404 (2002).
  * [9] M. $\dot{\rm{Z}}$ukowski, $\breve{\rm{C}}$. Brukner, W. Laskowski, M. Wie$\acute{\rm{s}}$niak, Do all pure entangled states violate Bell’s inequalities for correlation functions? Phys. Rev. Lett. 88 210401 (2002).
  * [10] J, Barrett, Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality, Phys. Rev. A 65 042302 (2002).
  * [11] J.L. Chen, C.F. Wu, C. Kwek, C.H. Oh, Gisin’s theorem for three qubits, Phys. Rev. Lett. 93 140407 (2004).
  * [12] W. Laskowski, T. Paterek, M. $\dot{\rm{Z}}$ukowski, and $\breve{\rm{C}}$. Brukner, Tight multipartite Bell’s inequalities involving many measurement settings, Phys. Rev. Lett. 93 200401 (2004).
  * [13] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, D. Roberts, Nonlocal correlations as an information-theoretic resource, Phys. Rev. A 71 022101 (2005).
  * [14] H.M. Wiseman, S.J. Jones, A.C. Doherty, Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox, Phys. Rev. Lett. 98 140402 (2007).
  * [15] S.K. Choudhary, S. Ghosh, G. Kar, R. Rahaman, Complete proof of Gisin’s theorem for three qubits, Phys. Rev. A, 81 042107 (2010).
  * [16] H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Nonlocality and communication complexity, Rev. Mod. Phys. 82 665-698 (2010).
  * [17] H.X. Cao, Z.H. Guo, Characterizing Bell nonlocality and EPR steering, Sci. China-Phys. Mech. Astron. 62 030311 (2019).
  * [18] A. Fine, Hidden Variables, Joint probability, and the Bell inequalities, Phys. Rev. Lett. 48 291 (1982).
  * [19] N. Gisin, Bell’s inequality holds for all non-product states, Phys. Lett. A 154 201 (1991).
  * [20] L. Khalfin, B. Tsirelson, Quantum/classical correspondence in the light of Bell’s inequalities, Found. Phys. 22 879-948 (1992).
  * [21] N. Gisin and A. Peres, Maximal violation of Bell’s inequality for arbitrarily large spin, Phys. Lett. A 162 15 (1992).
  * [22] M. Ardehali, Bell inequalities with a magnitude of violation that grows exponentially with the number of particles, Phys. Rev. A 46 5375 (1992).
  * [23] A.V. Belinskii, D.N. Klyshko, Interference of light and Bell’s theorem, Phys. Usp. 36 653-693 (1993).
  * [24] R. Horodecki, M. Horodecki, P. Horodecki, Teleportation, Bell’s inequalities and inseparability, Phys. Lett. A 222 21-25 (1996).
  * [25] V.A. Andreev, Generalized Bell inequality and a method for its verification, Theor. Math. Phys. 152, 1286-1298 (2007).
  * [26] T. V$\acute{\rm{e}}$rtesi, More efficient Bell inequalities for Werner states, Phys. Rev. A 78 032112 (2008).
  * [27] S.X. Yu, Q. Chen, C.J. Zhang, C.H. Lai, C.H. Oh, All entangled pure states violate a single Bell’s inequality, Phys. Rev. Lett. 109 120402 (2012).
  * [28] M. Li, S.M. Fei, X. Li-Jost, Bell inequality, separability and entanglement distillation, Chinese Sci. Bull. 56 945-954 (2011).
  * [29] M.J. Hoban, J.J. Wallman, D.E. Browne, Generalized Bell-inequality experiments and computation, Phys. Rev. A 84 062107 (2011).
  * [30] C. Branciard, N. Gisin, and S. Pironio, Characterizing the nonlocal correlations created via entanglement swapping, Phys. Rev. Lett. 104 170401 (2010).
  * [31] Cyri Branciard, Denis Rosset, Nicolas Gisin, and Stefano Pironio, Bilocal versus nonbilocal correlations in entanglement-swapping experiments, Phys. Rev. A 85 032119 (2012)
  * [32] N. Gisin, Q.X. Mei, A. Tavakoli, M.O. Renou, N. Brunner, All entangled pure quantum states violate the bilocality inequality, Phys. Rev. A 96 020304(R) (2017).
  * [33] A. Tavakoli, P. Skrzypczyk, D. Cavalcanti, and A. Acin, Nonlocal correlations in the star-network configuration, Phys. Rev. A 90 062109 (2014).
  * [34] K. Mukherjee, B. Paul, and D. Sarkar, Correlations in $n$-local scenario, Quantum Inf. Process. 14 2025 (2015).
  * [35] N. Gisin, The elegant joint quantum measurement and some conjectures about $N$-locality in the triangle and other configurations, arXiv:1708.05556 (2017).
  * [36] A. Tavakoli, M. O. Renou, N. Gisin and N. Brunner, Correlations in star networks: from Bell inequalities to network inequalities, New J. Phys. 19  073003 (2017).
  * [37] M.X. Luo, Computationally efficient nonlinear Bell inequalities for quantum networks, Phys. Rev. Lett. 120 140402 (2018).
  * [38] F. Andreoli, G. Carvacho, L. Santodonato, R. Chaves and F. Sciarrino, Maximal qubit violation of $n$-locality inequalities in a star-shaped quantum network, New J. Phys. 19 113020 (2017).
  * [39] T. Kri$\acute{\rm{v}}$achy, Y. Cai, D. Cavalcanti, A. Tavakoli, N. Gisin, and N. Brunner, A neural network oracle for quantum nonlocality problems in networks, npj Quantum Inf. 6 (2020).
  * [40] N. Gisin, Entanglement 25 years after quantum teleportation: Testing joint measurements in quantum networks, Entropy 21 325 (2019).
  * [41] K. Mukherjee, B. Paul, and A. Roy, Characterizing quantum correlations in a fixed-input $n$-local network scenario, Phys. Rev. A 101 032328 (2020).
  * [42] Y. Yang, H.X. Cao, Digraph states and their neural network representations, Chin. Phys. B 31 060303 (2022).
  * [43] M.O. Renou, E. B$\ddot{\rm{a}}$umer, S. Boreiri, N. Brunner, N. Gisin, and S. Beigi, Genuine quantum nonlocality in the triangle network, Phys. Rev. Lett. 123 140401 (2019).
  * [44] W. Rudin, Real and Complex Analysis, McGraw-Hill Education, 1986\.
  * [45] T. Fritz, Beyond Bell’s theorem: correlation scenarios, New J. Phys. 14 103001 (2012).
  * [46] L.H. Bai, S. Xiao, Z.H. Guo, and H.X. Cao, Decompositions of $n$-partite nonsignaling correlation-type tensors with applications, Front. Phys. 10 864452 (2022).