Abstract
When an n-partite physical system is measured by n observers, the joint probabilities of outcomes conditioned on the observables chosen by the n parties form a nonnegative tensor, called an n-partite correlation tensor (CT). In this paper, we aim to establish some characterizations of nonsignaling and Bell locality of an n-partite CT, respectively. By placing CTs within the linear space of correlation-type tensors (CTTs), we prove that every n-partite nonsignaling CTT can be decomposed as a linear combination of all local deterministic CTs using single-value decomposition of matrices and mathematical induction. As a consequence, we prove that an n-partite CT is nonsignaling (resp. Bell local) if and only if it can be written as a quasi-convex (resp. convex) combination of the outer products of deterministic CTs, implying that an n-partite CT is nonsignaling if and only if it has a local hidden variable model governed by a quasi-probability distribution. As an application of these results, we prove that a CT is nonsignaling if and only if it can be written as a quasi-convex of two Bell local ones, revealing a close relationship between nonsignaling CTs and Bell local ones.
1 Introduction
Quantum nonlocality was first discovered by Einstein, Podolsky, and Rosen (EPR) in 1935 [1], including quantum entanglement, quantum steering, and Bell nonlocality. They formulated an apparent paradox of quantum theory (EPR paradox) and gave a “thought” experiment that argues the wave function description in quantum mechanics is incomplete.
Bell nonlocality originated from the Bell’s 1964 paper [2]. 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. [3] for the “behaviors” P(ab|xy) (correlations), a terminology introduced by Tsirelson (1993) [4], but not for quantum states.
The result obtained by Bell [2] was named Bell’s theorem, which states that quantum predictions are incompatible with a local hidden variable description and are a cornerstone of quantum theory and at the center of many quantum information processing protocols. Over the years, different perspectives on non-locality have been put forward, including different ways to detect non-locality and quantify it.
Usually, Bell nonlocality for quantum states is detected by violation of some of Bell’s inequalities, such as Clause-Horne-Shimony-Holt (CHSH) inequality for two qubits. A proof of nonlocality without inequalities for two particles had been given earlier by Heywood and Redhead [5], which was much simplified by Brown and Svetlichny [6]. Greenberger, Horne, and Zeilinger (GHZ) [7] gave a proof of nonlocality but without using inequalities, in which a minimum of three particles was required in their proof. Mermin [8] provided a simple unified form for the major no-hidden-variables theorems by two examples. Hardy in [9, 10] proposed the two-particle 2-dimensional 2-setting Hardy paradox and gave the maximum probability of Bell’s nonlocality. Hardy et al. [11] discovered the two-particle 2-dimensional k-setting Hardy paradox. Aravind [12] established a Bell’s theorem without inequalities and only two distant observers. Dong et al. obtained in [13] some methods for detecting Bell nonlocality based on the Hardy Paradox. Chen et al. [14] proved that Bell nonlocal states can be constructed from some steerable states. They also established in [15] a mapping criteria between nonlocality and steerability. Jiang et al. [16] proposed a generalized Hardy’s paradox, and Yang et al. [17] presented a stronger Hardy-type paradox based on the Bell inequality and its experimental test. Cao and Guo [18] introduced mathematically the Bell locality and the unsteerability of a bipartite state for a given measurement setting and established their characterizations.
Viewed as joint outcome probabilities (correlations) for a specific experimental configuration as a vector of a Euclidean space , Pironio [19] proved that a Bell inequality defining a facet of the polytope of Bell local correlations can be lifted to one that also defines a facet of the more complex polytope, and established a formula for finding the affine dimension of .
By placing quantum possibilities within a wider context, Barrett et al. [20] investigated the polytope of no-signaling correlations, which contains the quantum correlations as a proper subset, determined the vertices of in the some special cases, and discussed how interconversions between different sorts of correlations may be achieved. They also considered some multipartite examples. Barrett et al. [21] introduced a version of the chained Bell inequality for an arbitrary number of measurement outcomes and use it to give a simple proof that the maximally entangled state of two d-dimensional quantum systems has no local component. Masanes et al. [22] considered nonlocality of n-partite correlations and identified a series of properties common to all theories that do not allow for superluminal signaling and predict the violation of Bell inequalities. They observed that intrinsic randomness, uncertainty due to the incompatibility of two observables, monogamy of correlations, impossibility of perfect cloning, privacy of correlations, and bounds in the shareability of some states are solely a consequence of the no-signaling principle and nonlocality. Loubenets [25] proved that the probabilistic description of an arbitrary multipartite correlation scenario admits a local quasi hidden variable (LqHV) simulation if and only if all joint probability distributions of this scenario satisfy the general nonsignaling condition formulated in [23, 24] using the notions of an LqHV model and a deterministic LqHV given by integrals rather than sums. Loubenets [26] also proved that the probabilistic description of any quantum multipartite correlation scenario with an arbitrary number of settings and outcomes at each site does admit an LqHV model. In an LqHV model given in [23, 24, 26], locality inherent to an LHV model is preserved but the basic concept of Kolmogorov’s probability model [27], a probability space, is replaced by a measure space with a normalized bounded real-valued measure not necessarily positive. Mndez, J. Urías [28] formulated the set of half-spaces describing the polytope of no-signaling probability states that are admitted by the most general class of Bell scenarios, presented a computational tool to solve the no-signaling description for the elements, which are the pure no-signaling boxes and the facets of Bell polytopes. Chaves and Budroni [29] introduced the concept of entropic nonsignaling correlations, and characterized and showed the relevance of these entropic correlations in a variety of different scenarios, ranging from typical Bell experiments to more refined descriptions such as bilocality and information causality. They applied the framework to derive the first entropic inequality testing genuine tripartite nonlocality in quantum systems of arbitrary dimension and also proved the first known monogamy relation for entropic Bell inequalities. Cope and Colbeck [30] found a series of Bell inequalities from no-signaling distributions by exploiting knowledge of the set of extremal no-signaling distributions. Eli et al. [31] characterized Bell nonlocality of bipartite correlations using tensor networks [32] and sparse recovery and proved that nonsignaling bipartite correlations can be described by local hidden variable models (LHVMs) governed by a quasi-probability distribution.
In the present paper, we continue to discuss nonsignaling and Bell nonlocality of n-partite correlations in order to generalize the Eli’s result to a multipartite case. Such correlations define the entries of a nonnegative tensor P of order 2n, which we call an n-partite correlation tensor (CT). In Section 2, we review some concepts and notations about tensors used later. In Section 3, n-partite nonsignaling correlation tensors are recalled and some observations are obtained. Also, correlation-type tensors are introduced as an extension of correlation tensors. In Section 4, a tensor-network decomposition of an n-partite nonsignaling CT is deduced using the singular-value-decomposition theorem of matrices and a decomposition lemma of row-stochastic matrices (RSMs) into a convex combination of {0, 1}-RSMs. In Section 5, we discuss Bell locality of an n-partite CT P and establish a relationship between Bell local CTs and nonsignaling ones.
2 Tensors and Their Operations
In what follows, we use the notation [m] = {1, 2, …, m} for every positive integer m.
Let and be orthonormal bases for and , respectively. Then forms an orthonormal basis for . Thus, every state ρAB of the system AB can be represented asThis implies that every state ρAB is determined by a set of complex coefficients ρijkℓ labeled by four indices i, j, k and ℓ, which defines a complex tensor Tρ = [[ρijkℓ]] of order 4.
Generally, a complex tensor is a multi-dimensional array of complex numbers and the order (rank) of a tensor is the number of indices [34]. Equivalently, we refer to a complex (or real) tensor of order k as a function T from an index set DT = [d1] × [d2] ×⋯ × [dk] into (or ), denoted by , where , the value of the function T at (i1, i2, …, ik), called the (i1, i2, …, ik)-entry of T. We also call such a T a (d1, d2, …, dk)-dimensional tensor of order k, or a rank-k tensor over DT. Thus, a rank-0 tensor is a scalar x, a d-dimensional tensor of order 1 is a d-dimensional vector (v1, v2, …, vd), and an (m, n)-dimensional tensor of order 2 is just an m × n matrix [Aij].
Two tensors A and B are said to be equal, denoted by A = B, if they are equal as functions, having the same domain of definition D and taking the same values at each index (i1, i2, …, ik) in D. A and B are said to contractive if they share at least one index. The contraction of A and B is the tensor A⋄B whose entries are the sum over all the possible values of the repeated indices of A and B. For instance, when A = [[Aij]] and B = [[Bik]] are tensors over [m] × [n] and [m] × [p], respectively, they are contractive with the contraction C = [[Cjk]] whereThat is, A⋄B = C, which is just the matrix product of matrices AT and B. In this case, B and A are also contractive with the contraction D = [[Dkj]] wherewhich is just the matrix product of matrices BT and A. Generally, A⋄B ≠ B⋄A.
Furthermore, the outer product (also called the tensor product) A ⊗B of two tensors A and B is the tensor whose entries are the products of entries of A and B. Say, when A = [[Aijk]] and B = [[Bxyzuv]] are tensors over DA and DB, respectively, the outer product of A and B is the tensorwhich is a rank-8 tensor over DA × DB. And that of A and B readswhich is a rank-8 tensor over DB × DA. Generally, A ⊗B ≠ B ⊗A.
3 Correlation and Correlation-Type Tensors
3.1 Correlation Tensors
Let us consider n parties A1, A2, …, An, each Ai possessing a physical system Si, which can be measured with different observables. Denote by xk the observable chosen (the label of observables or measurements) by party k, and by ak the corresponding measurement outcome. Let xk and ak take mk and ok values, respectively, and denote byThe joint probability for the outcomes a1, a2, ⋯, an, conditioned on the observables x1, x2, ⋯, xn chosen by the n parties. Then it holds thatThis gives a function , called a correlation function of the n-partite physical system S1S2⋯ Sn.
A tensor of order 2noveris said to be an n-partite correlation tensor (CT) over Δ2n if its entries are of the formsfor some correlation function P of an n-partite physical system S1S2⋯Sn. Especially, when P(a1a2⋯an|x1x2⋯xn) ∈ {0, 1} for all xk and ak, equivalently, there exists a function such thatFor all xk and ak, P is said to be an n-partite deterministic correlation tensor (DCT) induced by J and is written as P = PJ.
According to the special relativity, an n-partite CT P of order 2n given by (3.3) is said to be nonsignaling, or no-signaling [22, 31] if for each nonempty proper subset Δ = {k1, k2,⋯,kd}(k1 < k2 <⋯< kd) of [n] with the complement Δ′ = [n] \Δ, the sumdepends only on xj(j ∈ Δ) and aj(j ∈ Δ), being independent of xj(j ∈ Δ′). We call this condition the nonsignaling condition (NSC). Physically, the NSC says that the marginal distribution for each subset of parties {A1, A2,⋯, An} only depends on its corresponding inputs, i.e., for each nonempty proper subset Δ = {k1, k2,⋯,kd} of [n] with k1 < k2 <⋯< kd, it holds thatfor all xj(j ∈ Δ′), where
For example, a 2-partite CT P = [[P(ab|xy)]] over Δ4 = [mA] × [oA] × [mB] × [oB] is nonsignaling ifThat is, the marginal probability distribution of Alice (Bob) does not depend on the input used by Bob (Alice).
A 3-partite CT P = [[P(abc|xyz)]] over Δ6 = [m1] × [o1] × [m2] × [o2] × [m3] × [o3] is nonsignaling if and only if the following six equations are satisfied:
Indeed, the conditions (3.12–3.14) imply the conditions (3.9–3.11). For example, if (3.12 and 3.13) are satisfied, then we have ∀x, x′, y, y′, z, c,This implies (3.11).
Generally, we have the following characterization of nonsignaling [22].
Proposition 3.1.Ann-partite CTPover Δ2ngiven by(3.3)is nonsignaling if and only if for eachk ∈ [n], the marginal distribution obtained when tracing outakis independent ofxk:For alland allaj ∈ [mj](j ≠ k).
The following proposition characterizes nonsignaling property of a deterministic CT (DCT) PJ induced by a map , in such a way that
Proposition 3.2.A DCTPJis nonsignaling if and only if there exist mapsJk : [mk] → [ok](∀k ∈ [n]) such thatfor all . In that case,for all xk ∈ [mk], ak ∈ [ok](k = 1, 2,⋯, n).
Proof. The sufficiency is clear. Next, we show the necessity. To do this, we assume that PJ is nonsignaling. We can write J asfor all and for some maps where k = 1, 2,⋯, n. Thenfor all xk, ak and soSince PJ is nonsignaling, we have for all . This implies that f1(x1, x2,⋯,xn) is independent of the choice of x2,⋯, xn and depends only on x1. Similarly, one can see that fk(x1, x2,⋯,xn) is independent of the choice of xj(j ≠ k) and depends only on xk for each k = 2, 3,⋯,n. This enables us to define a map Jk : [mk] → [ok] for each k ∈ [n] byNow, Eq. (3.18) implies Eq. (3.16). Obviously, Eq. (3.17) yields Eq. (3.16). The proof is completed.
Here is an illustration of Proposition 3.2 for the case where n = 2. If we identify the fk used in Eq. (3.18) with the m1 × m2 matrix [fk(x1, x2)], whose (x1, x2)-entries are fk(x1, x2) ∈ [ok], i.e., fk ≡ [fk(x1, x2)], then the condition (3.16) is equivalent to J(x1, x2) = (f1(x1, x2), f2(x1, x2)) whereandThat is, f1 is row-constant and f2 is column-constant.
For example, when m1 = m2 = o1 = o2 = 2 andthat is,the DCT PJ induced by J with J(x1, x2) = (f1(x1, x2), f2(x1, x2)) is not nonsignaling. Whenthat is,the DCT PJ induced by J with J(x1, x2) = (f1(x1, x2), f2(x1, x2)) is nonsignaling.
3.2 Correlation-Type Tensors
Generalizing the concept of correlation tensors, let us introduce the concepts of correlation-type tensors.
Let T = [[T(a1a2⋯an|x1x2⋯xn)]] be a real tensor of order 2n, where ai ∈ [oi], xi ∈ [mi] for all i ∈ [n]. We call such a tensor T an n-partite correlation-type tensor (n-partite CTT). It is said to be nonsignaling (or, an NSCTT) if for each k ∈ [n], the sumis independent of xk, i.e., for all , it holds thatfor all xj ∈ [mj], aj ∈ [oj](j ≠ k).
Similar to the characterization of an NSCT (Proposition 3.1), one can show that T is an NSCTT if and only if for each nonempty proper subset Δ of [n] with the complement Δ′ = [n] \Δ, the sumdepends only on xj(j ∈ Δ) and aj(j ∈ Δ), being independent of xj(j ∈ Δ′).
Obviously, NSCTs are special NSCTTs.
4 Decomposition of n-partite NSCTTs
A tensor-network decomposition of a bipartite nonsignaling correlation was given in [31]. The following technical Lemma 4.2 was proved and used there. We rewrite it and give an alternative proof. To do so, let us recall a result proved by Li et al. [33], which implies that the set of all extreme points of the set CSM(m,n) of all nonnegative column-stochastic matrices (CSMs) of order m × n are exactly mn {0, 1}-CSMs of order m × n. Since an m × n nonnegative matrix A = [aij] is column-stochastic, i.e., ∑iaij = 1 for all j, if and only if its transpose AT = [aji] is row-stochastic, we get immediately the following result.
Lemma 4.1.The set of all extreme points of the setRSM(m,n) of all nonnegative row-stochastic matrices (RSMs) of orderm × nare exactlynm {0, 1}-RSMs of orderm × n:whereare the set of all maps from [m] into [n].
The following lemma was given in [31]. Here, we give a detailed proof based on Lemma 4.1.
Lemma 4.2. [31] An m × o matrix M = [Mij] has constant row sums (independent ofi) if and only if it can be decomposed aswith, where \{Jk : k ∈ [om]} denotes the set of all maps from [m] into [o]. If all Mij ≥ 0, then we can choose allck ≥ 0.
Proof. The sufficiency is clear. To prove the necessity, we assume that M = [Mij] has the desired property. Put a = min{Mij : i ∈ [m], j ∈ [o]} and use to denote the m × o matrix whose entries are all 1. Then becomes a nonnegative matrix with constant row sums C − oa ≥ 0.
Case 1.C − oa = 0. In this case, . Since is a row stochastic matrix, Lemma 3.1 implies that it can be written as a convex combination of (0, 1)-RSMs:where and all bk ≥ 0. Thus,with . Clearly, all aobk ≥ 0 if all Mij ≥ 0.
Case 2.C − oa > 0. In this case, is a row stochastic matrix and so it can be written as a convex combination of (0, 1)-RSMs (Lemma 3.1):where and all dk ≥ 0. It follows from Eq. 4.3 thatwhere ck = oabk + (C − oa)dk being of sum C. Clearly, when all Mij ≥ 0, we have a ≥ 0 and so all ck ≥ 0. The proof is completed.
Theorem 4.1. For any , an NSCTT T = [[T(a1⋯an|x1⋯xn)]] of order 2n can be decomposed asfor all xi ∈ [mi], ai ∈ [oi](i = 1, 2,⋯n), where , and denotes the set of all maps from [mi] into .
Proof. When n = 1, to get the decomposition of an NSCTT T, let us consider T = [[T(a1|x1)]] as a matrix with (x1, a1)-entry T(a1|x1). Using Lemma 4.2 yields thatwhere and denotes the set of all maps from [m1] into [o1].
Suppose that for some n ≥ 1, a decomposition (4.4) exists for any n-partite NSCTT T.
Let T = [[T(a1⋯an+1|x1⋯xn+1)]] be an (n + 1)-partite NSCTT. To consider T as a matrix, we choose bijections α1 : [m1] × [o1] → [m1o1],and then define an (m1o1) × (m2o2⋯mn+1on+1) matrix with (i1, i2)-entry ifLet be a singular value decomposition (SVD) of where U = [Uiλ] and V = [Vλj] are some real orthogonal matrices of orders m1o1 and m2o2⋯mn+1on+1, respectively, and S is one of the following matrices:where Σ is a nonnegative diagonal matrix. Without loss of generality, we assume that S is the first case. Thus,where . Put if i = α1(x1, a1), 1 ≤ λ ≤ m1o1; and if j = α2(x2,⋯,xn+1, a2,⋯,an+1), 1 ≤ λ ≤ m2o2⋯mn+1on+1. Then the SVD of yields thatThe nonsignaling condition on T implies thatfor all and for each k = 2, 3,⋯, n + 1,for all . By writingwhich is a vector in and lettingwhich is a vector in , we get from (4.7) thatfor all x2,⋯,xn+1, a2,⋯,an+1 and . Since the column vectors of the unitary matrix V form an orthonormal basisfor , we conclude that , i.e.,Since dλ > 0 for each λ = 1, 2,⋯, r, we obtainUsing Lemma 4.2 yields that for each λ = 1, 2,⋯, r,where , is the set of all maps from [m1] into [o1]. Similarly, by writingand for fixed 2 ≤ k ≤ n + 1, xj ∈ [mj](j ≠ k), aj ∈ [oj](j ≠ k), lettingwe get from (4.8) that for all x1, a1 and . Since the row vectors ’s of the unitary matrix U form an orthonormal basis for , we conclude that , i.e.,for all . Since dλ > 0 for each λ = 1, 2,⋯, r, we obtainfor all . This shows thatdefines an n-partite NSCTT. It follows from the assumption of induction thatfor all xi ∈ [mi], ai ∈ [oi](i = 2,⋯, n + 1). Now, we obtain from Eqs 4.6, 4.10, 4.12 thatfor all xi ∈ [mi], ai ∈ [oi](i = 1, 2,⋯, n + 1), whereThis shows that a decomposition (4.4) exists for an (n + 1)-partite NSCTT T. The proof is completed.
Theorem 4.2.For a CTTT = [[T(a1a2⋯an|x1x2⋯xn)]], the following statements are equivalent.
(1) Tis nonsignaling.
(2) Thas a decomposition (4.4).
(3) T has the following generalized LHV model:for all xi ∈ [mi], ai ∈ [oi], where , are PDs for all xi, λ.
(4) Thas the form ofwhere.
(5) The following tensor-network decomposition holds:where is a real tensor of order n and is the tensor of order 3 with (ki, xi, ai)-entries for all i = 1, 2,⋯ n.
Proof. Theorem 4.1 says that (1) and (2) are equivalent. (2) yields (3) clearly. Lemma 4.2 implies that (3) yields (2). The proof is completed.
Loubenets [25] proved that a CT admits a local quasi hidden variable (LqHV) simulation if and only if all joint probability distributions of this scenario satisfy the general nonsignaling condition formulated in [23, 24, 26] using the notions of an LqHV model and a deterministic LqHV given by integrals rather than sums. As a consequence of Theorem 4.2, we obtain the following corollary, which means that a CT is nonsignaling if and only if it has a generalized LHV model given by a “discrete” sum instead of a “continuous” integral.
Corollary 4.1. For an n-partite CT P = [[P(a1a2⋯an|x1x2⋯xn)]], the following statements are equivalent.
(1) Pis nonsignaling.
(2) P has a decomposition (4.4) in which .
(3) P has a generalized LHV model (4.13) in which .
(4) P has the form of (4.14) in which .
(5) P has a tensor-network decomposition (4.15) with .
5 Characterization of a Bell Local CT
An n-partite CToveris said to be Bell local if there exists a PD such thatfor all xi ∈ [mi], ai ∈ [oi](i = 1, 2,⋯,n), where is a PD for each i ∈ [n], each xi ∈ [mi], and each λ ∈ [d]. P is said to be Bell nonlocal if it not Bell local.
From Proposition 3.2, we see that an n-partite DCT PJ is nonsignaling if and only if it is Bell local. Generally, using Lemma 4.1 for mi × oi RSMs Pi≔[Pi(ai|xi, λ)] with (xi, ai)-entries Pi(ai|xi, λ) implies that local probabilities in (5.3) can written asfor all λ ∈ [d], xi ∈ [mi], ai ∈ [oi](i = 1, 2,⋯, n). This yields the following conclusion, which gives a characterization of Bell locality of an n-partite CT.
Theorem 5.1. An n-partite CT P = [[P(a1a2⋯an|x1x2⋯xn)]] over Δ2n is Bell local if and only if it has the form ofFor all xi ∈ [mi], ai ∈ [oi](∀i ∈ [n]), equivalently, the following tensor-network decomposition holds:where is a nonnegative real tensor of order n such thatand is the tensor of order 3 with (ki, xi, ai)-entries for all i = 1, 2,⋯, n.
Combining Theorem 5.1 with Corollary 4.1 shows that every Bell local CT must be nonsignaling, while a nonsignaling CT is not necessarily Bell local (e.g., the PR box). Furthermore, for a nonsignaling Bell nonlocal CT P = [[P(a1⋯an|x1⋯xn)]] over Δ2n, we see from Corollary 4.1 that P has a decompositionfor all xi, ai where . Putthen with for all k1,⋯, kn. Lettingand using (5.7) yield thatfor all possible xi, ai, where q+ − q− = 1 andUsing Theorem 5.1, we see that bothare Bell local CTs, satisfying P = q+P+ − q−P−. When P is Bell local, the last decomposition is also valid for P+ = P− = P and q+ = 1, q− = 0.
As a conclusion, we obtain the following theorem, which reveals a relationship between nonsignaling CTs and Bell local ones.
Corollary 5.1.A CTP over Δ2n is nonsignaling if and only if it can be written aswhere bothP+andP− are Bell local CTs, q+, q−≥ 0 with q+ − q− = 1.
This implies the affine hull ah(BL(Δ2n)) of the convex compact set contains the polytope NS(Δ2n) of NSCTs over Δ2n. That is, NS(Δ2n) ⊂ ah(BL(Δ2n)).
6 Summary and Conclusion
Bell nonlocality is a cornerstone of quantum theory and at the center of many quantum information processing protocols. As the number of subsystems increases, deciding whether a given state w.r.t. a measurement setting is local or nonlocal becomes computationally intractable. To overcome this difficulty, Elins et al. have proposed a method for analyzing Bell nonlocality of a nonsignaling correlation using tensor network and sparse recovery. Motivated by this work, we have discussed nonsignaling and Bell locality of n-partite correlations in teams of tensor decompositions of the corresponding correlation tensors.
Consider n parties A1,⋯, An, each Ak possessing a physical system Sk, which can be measured with mk different observables xk = 1, 2,⋯, mk and the corresponding outcomes ak = 1, 2,⋯, ok. Conditioned on the observables chosen by the n parties, the joint probability distribution (JPD) P(a1⋯an|x1⋯xn) for the outcomes is obtained. Thus, such a JPD is just a function P from into [0, 1], which we called a correlation function (CF). One way to represent a JPD is vector-representation (VR) [19], i.e., a way to represent joint probabilities P(a1⋯an|x1⋯xn) as a high dimensional vector in where , called a correlation vector (CV). With this representation, the set of all Bell local CVs forms a polytope in with the dimension [19, Theorem 1]. For a bipartite correlation P(ab|xy), a useful notation was introduced and used by Tsirelson and Cope [4, 30], which represents it as a matrix Π = [Pxy] with Pxy = [P(ab|xy)] as the (x, y)-block with (a, b)-entries P(ab|xy). We call Π = [Pxy] a correlation matrix (CM). In the present paper, we have represented a JPD P(a1⋯an|x1⋯xn) as a nonnegative tensor P of order 2n with (x1, a1,⋯xn, an)-entries, which we named an n-partite correlation tensor (CT).
Generally, nonnegativity and normalization condition makes it that an n + 1-partite CT could not be written as a convex combination of k-partite CTs some k ≤ n. Thus, it is almost impossible to extend Eli’s decomposition [31] of a bipartite nonsignaling CT (NSCT) to multi-partite case by means of mathematical induction. To overcome this difficulty, we have placed all n-partite CTs within the linear space of correlation-type tensors (CTTs) of the form P = [[P(a1a2⋯an|x1x2⋯xn)]] with real entries (not necessarily nonnegative and normalized) and induced the nonsignaling property of them. This enables us to prove that every nonsignaling n-partite CTT can be decomposed as a linear combination of local deterministic CTs (LDCTs) using single-value decomposition of matrices and mathematical induction. This decomposition theorem is particularly valid for any nonsignaling n-partite CT. As a consequence, we have proved that a CT P is nonsignaling if and only if it can be written as a quasi-convex combination of the outer products of deterministic CTs D1(k1),⋯Dn(kn) of order 2 and that P is Bell local if and only if the decomposition is valid for a probability tensor . Also, such a decomposition suggests close relationships between nonsignaling CTs P and quasi-probability tensor q, as well as Bell local CTs P and probability tensor q.
As an application of these results, we have observed that a CT P is nonsignaling if and only if it can be written aswhere P+ and P− are Bell local CTs, ɛ ≥ 0. This gives a close relationship between nonsignaling CTs and Bell local CTs. Moreover, the last decomposition shows that the set of all n-partite nonsignaling CTs is contained in the affine hull [19] of the compact convex set of all n-partite Bell local CTs. Clearly, and are not the same since the former is a compact convex set and the latter is unbounded.
Statements
Data availability statement
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors.
Author contributions
The authors contributed equally to this work.
Funding
This work was supported by the National Natural Science Foundation of China (Nos. 11871318, 11771009, 12001480), the Fundamental Research Funds for the Central Universities (GK202103003, GK202107014) and the Special Plan for Young Top-notch Talent of Shaanxi Province (1503070117).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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References
1.
EinsteinAPodolskyBRosenN. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?Phys Rev (1935) 47:777–80. 10.1103/physrev.47.777
2.
BellJS. On the Einstein Podolsky Rosen Paradox. Physics (1964) 1:195–200. 10.1103/physicsphysiquefizika.1.195
3.
BrunnerNCavalcantiDPironioSScaraniVWehnerS. Bell Nonlocality. Rev Mod Phys (2014) 86:419–78. 10.1103/revmodphys.86.419
4.
TsirelsonBS. Some Results and Problems on Quantum Bell-type Inequalities. Hadronic J Suppl (1993) 8:329–45.
5.
HeywoodPRedheadMLG. Nonlocality and the Kochen-Specker Paradox. Found Phys (1983) 13:481–99. 10.1007/bf00729511
6.
BrownHRSvetlichnyG. Nonlocality and Gleason's Lemma. Part I. Deterministic Theories. Found Phys (1990) 20:1379–87. 10.1007/bf01883492
7.
GreenbergerDMHorneMAShimonyAZeilingerA. Bell's Theorem without Inequalities. Am J Phys (1990) 58:1131–43. 10.1119/1.16243
8.
MerminND. Simple Unified Form for the Major No-Hidden-Variables Theorems. Phys Rev Lett (1990) 65:3373–6. 10.1103/physrevlett.65.3373
9.
HardyL. Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories. Phys Rev Lett (1992) 68:2981–4. 10.1103/physrevlett.68.2981
10.
HardyL. Nonlocality for Two Particles without Inequalities for Almost All Entangled States. Phys Rev Lett (1993) 71:1665–8. 10.1103/physrevlett.71.1665
11.
BoschiDBrancaSDe MartiniFHardyL. Ladder Proof of Nonlocality without Inequalities: Theoretical and Experimental Results. Phys Rev Lett (1997) 79:2755–8. 10.1103/physrevlett.79.2755
12.
AravindPK. Bell’s Theorem without Inequalities and Only Two Distant Observers. Found Phys Lett (2002) 15:397–405. 10.1023/a:1021272729475
13.
DongZYangYCaoH. Detecting Bell Nonlocality Based on the Hardy Paradox. Int J Theor Phys (2020) 59:1644–56. 10.1007/s10773-020-04432-1
14.
ChenJ-LRenCChenCYeX-JPatiAK. Bell's Nonlocality Can Be Detected by the Violation of Einstein-Podolsky-Rosen Steering Inequality. Sci Rep (2016) 6:39063. 10.1038/srep39063
15.
ChenCBRenCLYeXJChenJL. Mapping Criteria between Nonlocality and Steerability in Qudit-Qubit Systems and between Steerability and Entanglement in Qubit-Qudit Systems. Phys Rev A (2018) 98:052114. 10.1103/physreva.98.052114
16.
JiangSHXuZPSuHYPatiAKChenJL. Generalized Hardy's Paradox. Phys Rev Lett (2018) 120:050403. 10.1103/PhysRevLett.120.050403
17.
YangMMengHXZhouJXuZPXiaoYSunKet alStronger Hardy-type Paradox Based on the Bell Inequality and its Experimental Test. Phys Rev A (2019) 99:032103. 10.1103/physreva.99.032103
18.
CaoHXGuoZH. Characterizing Bell Nonlocality and EPR Steering. Sci China-phys Mech Astron (2019) 62:030311. 10.1007/s11433-018-9279-4
19.
PironioS. Lifting Bell Inequalities. J Math Phys (2005) 46:062112. 10.1063/1.1928727
20.
BarrettJLindenNMassarSPironioSPopescuSRobertsD. Nonlocal Correlations as an Information-Theoretic Resource. Phys Rev A (2005) 71:022101. 10.1103/physreva.71.022101
21.
BarrettJKentAPironioS. Maximally Nonlocal and Monogamous Quantum Correlations. Phys Rev Lett (2006) 97:170409. 10.1103/physrevlett.97.170409
22.
MasanesLAcinAGisinN. General Properties of Nonsignaling Theories. Phys Rev A (2006) 73:012112. 10.1103/physreva.73.012112
23.
LoubenetsER. On the Probabilistic Description of a Multipartite Correlation Scenario with Arbitrary Numbers of Settings and Outcomes Per Site. J Phys A: Math Theor (2008) 41(23pp):445303. 10.1088/1751-8113/41/44/445303
24.
LoubenetsER. Multipartite Bell-type Inequalities for Arbitrary Numbers of Settings and Outcomes Per Site. J Phys A: Math Theor (2008) 41:445304. 10.1088/1751-8113/41/44/445304
25.
LoubenetsER. Nonsignaling as a Consistency Condition for a Local Quasi Hidden Variable (LqHV) Simulation of a General Multipartite Correlation Scenario (2011). arXiv:1109.0225.
26.
LoubenetsER. Local Quasi Hidden Variable Modelling and Violations of Bell-type Inequalities by a Multipartite Quantum State. J Math Phys (2012) 53:022201. 10.1063/1.3681905
27.
KolmogorovAN. Foundations of the Theory of Probability (English Translation). New York, Chelsea (1950).
28.
MéndezJMUríasJ. On the No-Signaling Approach to Quantum Nonlocality. J Math Phys (2015) 56:032101. 10.1063/1.4914336
29.
ChavesRBudroniC. Entropic Nonsignaling Correlations. Phys Rev Lett (2016) 116:240501. 10.1103/physrevlett.116.240501
30.
CopeColbeckTR. Bell Inequalities from No-Signaling Distributions. Phys Rev A (2019) 100:022114. 10.1103/physreva.100.022114
31.
ElinsISBritoSGAChavesR. Bell Nonlocality Using Tensor Networks and Sparse Recovery. Phys Rev Res (2020) 2:023198.
32.
OrsR. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Ann Phys (2014) 349:117–58.
33.
LiC-KTamB-STsingN-K. Linear Maps Preserving Permutation and Stochastic Matrices. Linear Algebra its Appl (2002) 341:5–22. 10.1016/s0024-3795(00)00242-1
34.
CichockiAZdunekRPhanAHAmariS. Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-Way Data Analysis and Blind Source Separation. Chichester, UK: Wiley (2009).
Summary
Keywords
nonsignaling, Bell locality, correlation tensor, correlation-type tensor, Bell nonlocality
Citation
Bai L, Xiao S, Guo Z and Cao H (2022) Decompositions of n-Partite Nonsignaling Correlation-Type Tensors With Applications. Front. Phys. 10:864452. doi: 10.3389/fphy.2022.864452
Received
28 January 2022
Accepted
18 February 2022
Published
19 April 2022
Volume
10 - 2022
Edited by
Masahito Hayashi, Southern University of Science and Technology, China
Reviewed by
Baichu Yu, Southern University of Science and Technology, China
JIng-Ling Chen, Nankai University, China
Updates
Copyright
© 2022 Bai, Xiao, Guo and Cao.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Zhihua Guo, guozhihua@snnu.edu.cn; Huaixin Cao, caohx@snnu.edu.cn
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
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