In this work, we discuss two types of trilocality of probability tensors (PTs) P=〚P(a1a2a3)〛 over an outcome set Ω3 and correlation tensors (CTs) P=〚P(a1a2a3|x1x2x3)〛 over an outcome-input set Δ3 based on a triangle network and described by continuous (integral) and discrete (sum) trilocal hidden variable models (C-triLHVMs and D-triLHVMs). We say that a PT (or CT) P is C-trilocal (resp. D-trilocal) if it can be described by a C-triLHVM (resp. D-triLHVM). It is proved that a PT (resp. CT) is D-trilocal if and only if it can be realized in a triangle network by three shared separable states and a local POVM (resp. a set of local POVMs) performed at each node; a CT is C-trilocal (resp. D-trilocal) if and only if it can be written as a convex combination of the product deterministic CTs with a C-trilocal (resp. D-trilocal) PT as a coefficient tensor. Some properties of the sets consisting of C-trilocal and D-trilocal PTs (resp. C-trilocal and D-trilocal CTs) are proved, including their path-connectedness and partial star-convexity.
In this work, we discuss two types of trilocality of probability tensors (PTs) P=〚P(a1a2a3)〛 over an outcome set Ω3 and correlation tensors (CTs) P=〚P(a1a2a3|x1x2x3)〛 over an outcome-input set Δ3 based on a triangle network and described by continuous (integral) and discrete (sum) trilocal hidden variable models (C-triLHVMs and D-triLHVMs). We say that a PT (or CT) P is C-trilocal (resp. D-trilocal) if it can be described by a C-triLHVM (resp. D-triLHVM). It is proved that a PT (resp. CT) is D-trilocal if and only if it can be realized in a triangle network by three shared separable states and a local POVM (resp. a set of local POVMs) performed at each node; a CT is C-trilocal (resp. D-trilocal) if and only if it can be written as a convex combination of the product deterministic CTs with a C-trilocal (resp. D-trilocal) PT as a coefficient tensor. Some properties of the sets consisting of C-trilocal and D-trilocal PTs (resp. C-trilocal and D-trilocal CTs) are proved, including their path-connectedness and partial star-convexity.
Abstract Bell’s inequalities are linear and apply for cases of two entangled bodies. In this work, we consider the case of entanglement among three bodies as previously discussed in [Renou, et al Phys. Rev. Lett., 123, 140 401 (2019)] and based on triangle network. By discussing the question whether a sparse probability tensor (SPT) can be represented by a discrete trilocal hidden variable model (D-triLHVM), we show that every SPT having a D-triLHVM satisfies a set of concrete equalities and a nonlinear inequality, which can be used to detect whether a D-triLHVM can describe the network completely. As an application, we re-explore the D-nontrilocality of the correlations studied by Renou et al and that of the triangle network with shared entangled pure states. We also leave open questions about the closednees of the set of all D-trilocal probability tensors and the description with a continuous trilocal hidden variable model.
In this work, we study the nonlocality of star-shaped correlation tensors (SSCTs) based on a general multi-star-network MSN(m,n1,…,nm). Such a network consists of 1+m+n1+⋯+nm nodes and one center-node A that connects to m star-nodes B1,B2,…,Bm while each star-node Bj has nj+1 star-nodes A,C1j,C2j,…,Cnjj. By introducing star-locality and star-nonlocality into the network, some related properties are obtained. Based on the architecture of such a network, SSCTs including star-shaped probability tensors (SSPTs) are proposed and two types of localities in SSCTs and SSPTs are mathematically formulated, called D-star-locality and C-star-locality. By establishing a series of characterizations, the equivalence of these two localities is verified. Some necessary conditions for a star-shaped CT to be D-star-local are also obtained. It is proven that the set of all star-local SSCTs is a compact and path-connected subset in the Hilbert space of tensors over the index set ΔS and has least two types of star-convex subsets. Lastly, a star-Bell inequality is proved to be valid for all star-local SSCTs. Based on our inequality, two examples of star-nonlocal MSN(m,n1,…,nm) are presented.
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.
Correlations in quantum networks with independent sources exhibit a completely novel form of nonclassicality in the sense that the nonlocality of such correlations can be demonstrated in fixed local input scenarios. Before the pioneering work of Renou et al. [Phys. Rev. Lett. 123, 140401 (2019)], the nonlocal feature of such network correlations was directly attributable to standard Bell nonlocality. In the above-mentioned reference, the authors provided some of the first examples of triangle network correlations, whose nonlocality cannot be deduced from Bell-CHSH nonlocality. To date, a complete characterization of such scenarios has yet to be provided. The present work characterizes correlations arising due to fixed local measurements in a triangle network under a source independence assumption. Precisely speaking, a set of criteria is framed in the form of Bell-type inequalities, each of which is necessarily satisfied by trilocal correlations. Possible quantum violation of at least one criterion from the set is analyzed, which in turn points out the utility of the set of criteria to detect nonlocality (nontrilocality) in quantum triangle networks. Interestingly, measurement on a local product state basis turns out to be sufficient to generate nontrilocal correlations in some quantum networks. Noise tolerance of the detection criteria is discussed followed by a generalization of the framework for demonstrating correlations in any n-sided polygon where n is finite.
Defining nonlocality in a no-input closed quantum network scenario is a new area of interest nowadays. Gisin, in[Entropy 21, 325 (2019)], proposed a possible condition for non-tri-locality of the trivial no-input closed network scenario, triangle network, by introducing a new kind of joint measurement bases and a probability bound. In[npj Quantum Information (2020) 6:70] they found a shred of numerical evidence in support of Gisin's probability bound. Now based on that probability bound, we find the nature of the correlation in a triangle network scenario. We here observe how far the probability lies from that Gisin's bound with every possible combination of entangled and local pure states distributed from three independent quantum sources. Here we use the generalized Elegant Joint Measurements bases for each party and find that there is a dependency of non-locality on the entanglement of these joint measurement bases. We also check the probability bound for the polygon structure.
Abstract The correlations in quantum networks have attracted strong interest due to the fact that linear Bell inequalities derived from one source are useless for characterizing multipartite correlations of general quantum networks. In this paper, a type of multi-star-shaped quantum networks are introduced and discussed. Such a network consists of three-grade nodes: the first grade is named party (node) A , the second one consists of m nodes marked B 1 , B 2 , …, B m , which are stars of A and the third one consists of m 2 nodes C k j ( j , k = 1 , 2 , … , m ) , where C k j ( k = 1 , 2 , … , m ) are stars of B j . We call such a network a three-grade m -star quantum network and denoted by SQN (3, m ), being as a natural extension of bilocal networks and star-shaped networks. We introduce and discussed the locality and strong locality of a SQN (3, m ) and derive the related nonlinear Bell inequalities, called (3, m )-locality inequalities and (3, m )-strong locality inequalities. To compare with the bipartite locality of quantum states, we define the separability of SQN (3, m ) that imply the locality and then locality of SQN (3, m ). When all of the shared states of the network are pure ones, we prove that SQN (3, m ) is nonlocal if and only if it is entangled.