Characterizations of bilocality and n-locality of correlation tensors Shu Xiao Huaixin Cao Zhihua Guo Kanyuan Han_2

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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 P=[P(abc|xyz)], 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) P=[P(abc)], includ-

ing equivalent characterizations and properties of bilocal PTs. Lastly,

we obtain corresponding results about n-locality of n+ 1-partite CTs

P=[P(ab|xy)]and PTs P=[P(ab)], respectively.

Keywords. bilocality; n-locality; correlation tensor; probability ten-

sor; 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 mea-

sured, 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]. Usually, Bell nonlocality can be checked by

violation of some types Bell inequalities [19–29].

arXiv:2210.04207v1 [quant-ph] 9 Oct 2022

Quantum systems that have never interacted can become nonlocally corre-

lated through a process called entanglement swapping. To characterize non-

locality 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 lim1sends particles to Alice and

Bob, and a separate source λ2sends particles to Charles and Bob. All parties

can perform measurements on their system, labeled x, y, and zfor 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) = ZZΛ1×Λ2

ρ1(λ1)ρ2(λ2)PA(a|x, λ1)PB(b|y, λ1λ2)PC(c|z, λ2)dλ1dλ2

(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

λ1and λ2may be “continuous” ones. Branciard et al. proved that all bilocal

correlations satisfy a quadratic inequality I≤1 + E2[30, Eq. (10)]. To com-

pare 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 √I+√J≤1,which was proved to be valid

for every bilocal P=[P(abc|xyz)]. Gisin et al. [32] proved that all entangled

pure quantum states violate the bilocality inequality. Importantly, bilocality in-

equality is related to the 2-locality approach for detecting quantum correlations

in networks, especially, in star networks [33–43]. 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 ρAiBiwith each Aiof nedge nodes (referred to as the

Alices, denoted by A1, A2, . . . , An).

Usually, both bilocality and n-locality are described by integration local hid-

den variable models (LHVMs), called C-biLHVMs and C-nLHVMs, rather than



1 1

A B



2 ,3



2 2

A B



3 3

A B



4 4

A B



Figure 2: A star-network with a central node Band nstar-nodes A1, A2, . . . , An

where n= 4.

by summations LHVMs, called D-biLHVMs and D-nLHVMs. 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-

nLHVMs and D-nLHVMs. In Sect. 2, we ﬁrst ﬁx the concept of bilocality of

a tripartite correlation tensor (CT) P=[P(abc|xyz)], 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) P=[P(a, b, c)]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 P=[P(ab|xy)]and PTs P=[P(ab)], respectively.

2 Bilocality of tripartite correlation tensors

In what follows, we use [n] to denote the set {1,2, . . . , n}. When a tripartite

system is measured by separated three parties A, B and Cwith measurements

labeled by x∈[mA], y ∈[mB] and z∈[mC], respectively, the joint probability

distribution P(abc|xyz) of obtaining outcomes a∈[oA], b ∈[oB] and c∈[oC]

forms a tensor P=[P(abc|xyz)]over ∆3= [oA]×[oB]×[oC]×[mA]×[mB]×

[mC], we call it a correlation tensor (CT) [46], just like a matrix. Abstractly, a

tripartite CT over ∆3is a function P: ∆3→Rsuch that

P(abc|xyz)≥0(∀x, y, z, a, b, c) and X

a,b,c

P(abc|xyz) = 1(∀x, y, z).

Any function P: ∆3→Ris called a correlation-type tensor (CTT) [46] over

∆3. We use T(∆3) and CT (∆3) to denote the sets of all CTTs and CTs over

∆3, respectively.

For any two elements P=[P(abc|xyz)]and Q=[Q(abc|xyz)]of T(∆3),

deﬁne

sP+tQ=[sP (abc|xyz) + tQ(abc|xyz)],

hP|Qi=X

a,b,c,x,y,z

P(abc|xyz)Q(abc|xyz),

then T(∆3) becomes a ﬁnite dimensional Hilbert space over R. Clearly, the

norm-convergence of a sequence in T(∆3) is just the pointwise-convergence and

then CT (∆3) becomes a compact convex set in T(∆3).

We ﬁx the concept of the bilocality of a CT over ∆3according to [30, 31].

Deﬁnition 2.1. A CT P=[P(abc|xyz)]over ∆3is said to be bilocal if it

has a “continuous” bilocal hidden variable model (C-biLHVM):

P(abc|xyz) = ZZΛ1×Λ2

q1(λ1)q2(λ2)PA(a|x, λ1)PB(b|y, λ1λ2)PC(c|z, λ2)dµ1(λ1)dµ2(λ2)

(2.1)

for a measure space (Λ1×Λ2,Ω1×Ω2, µ1×µ2) and for all a, b, c, x, y, z, where

(a) q1(λ1) and PA(a|x, λ1)(x∈[mA], a ∈[oA]) are Ω1-measurable on Λ1,

q2(λ2) and PC(c|z, λ2)(z∈[mC], c ∈[oC]) are Ω2-measurable on Λ2, and

PB(b|y, λ1λ2)(y∈[mB], b ∈[oB]) are Ω1×Ω2-measurable on Λ1×Λ2;

(b) qi(λi), PA(a|x, λ1), PB(b|y, λ1λ2) and PC(c|z, λ2) are probability distri-

butions (PDs) of λi, a, b, c, respectively.

A CT P=[P(abc|xyz)]over ∆3is said to be non-bilocal if it is not bilocal.

We use CT bilocal(∆3) to denote the set of all bilocal CTs over ∆3.

Remark 2.1. By Deﬁnition 2.1, when a CT P=[P(abc|xyz)]over ∆3

is a product of three conditional probability distributions PA(a|x), PB(b|y) and

can rewrite it as

P(abc|xyz) =

λ1λ2=1

q1(λ1)q2(λ2)PA(a|x, λ1)PB(b|y, λ1λ2)PC(c|z, λ2)

where qk(λk) = 1(k= 1,2) and

Thus, P=[P(abc|xyz)]can be written as (2.1) for the counting measures µk

on the set Λk={1}and then is bilocal.

Remark 2.2. From deﬁnition, we observe that when a CT P=[P(abc|xyz)]

over ∆3is bilocal, marginal distributions satisfy:

PAC (ac|xz) = PA(a|x)PC(c|z),∀x, z, a, c. (2.2)

By using this property of a bilocal CT, we can ﬁnd that not all Bell local CTs

over ∆3are bilocal.

Example 2.1. Let mX=oX= 2(X=A, B, C) and take

PB(1|1) = 1/2, PB(2|1) = 1/2, PB(1|2) = 1/2, PB(2|2) = 1/2,

A(1|1) = 1, P 0

A(2|1) = 0, P 0

A(1|2) = 1, P 0

A(2|2) = 0,

P00

A(1|1) = 0, P 00

A(2|1) = 1, P 00

A(1|2) = 0, P 00

A(2|2) = 1,

C(1|1) = 1, P 0

C(2|1) = 0, P 0

C(1|2) = 0, P 0

C(2|2) = 1,

P00

C(1|1) = 0, P 00

C(2|1) = 1, P 00

C(1|2) = 1, P 00

C(2|2) = 0,

and deﬁne

P=1

2P0

A⊗PB⊗P0

C+1

2P00

A⊗PB⊗P00

that is,

P(abc|xyz) = 1

2P0

A(a|x)PB(b|y)P0

C(c|z) + 1

2P00

A(a|x)PB(b|y)P00

C(c|z).

Clearly, P=[P(abc|xyz)]is a Bell local CT. Note that

4|[P0

A(a|x)−P00

A(a|x)][P0

C(c|z)−P00

C(c|z)]| ≡ 1

we see that

PAC (ac|xz)6=PA(a|x)PC(c|z),∀x, z, a, c.

Thus, P/∈ CT bilocal(∆3). Moreover, from Remark 2.1, we see that P0

A⊗PB⊗P0

and P00

A⊗PB⊗P00

Care in CT bilocal(∆3). This shows that the set CT bilocal(∆3)

is not convex in the Hilbert space T(∆3).

Generally, the ﬁve PDs in a C-biLHVM (2.1) for Pare not necessarily unique

and depend on P. The following proposition ensures that any two bilocal CTs

over ∆3can be represented by C-biLHVMs with the same measure space and

the same PDs of the same hidden variables.

Proposition 2.1. Let P=[P(abc|xyz)]and P0=[P0(abc|xyz)]be any

two bilocal CTs over ∆3. Then there is a product measure space (S1×S2, T1×

T2, γ1×γ2)and PDs fk(sk)of sk(k= 1,2) such that

P(abc|xyz) = ZZS1×S2

f1(s1)f2(s2)PA(a|x, s1)

×PB(b|y, s1s2)PC(c|z, s2)dγ1(s1)dγ2(s2),(2.3)

P0(abc|xyz) = ZZS1×S2

f1(s1)f2(s2)P0

A(a|x, s1)

×P0

B(b|y, s1s2)P0

C(c|z, s2)dγ1(s1)dγ2(s2) (2.4)

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

Proof. By deﬁnition, Pand P0can be represented as

P(abc|xyz) = ZZΛ1×Λ2

q1(λ1)q2(λ2)PA(a|x, λ1)

×PB(b|y, λ1λ2)PC(c|z, λ2)dµ1(λ1)dµ2(λ2) (2.5)

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Characterizationsofbilocalityandn-localityofcorrelationtensorsShuXiao,HuaixinCao,ZhihuaGuo,KanyuanHanSchoolofMathematicsandStatistics,ShaanxiNormalUniversityXi'an710119,ChinaAbstractIntheliterature,bilocalityandn-localityofcorrelationtensors(CTs)aredescribedbyintegrationlocalhiddenvariablemodels(cal...

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Characterizations of bilocality and n-locality of correlation tensors Shu Xiao Huaixin Cao Zhihua Guo Kanyuan Han_2

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