Some Measurement-Based Characterizations of Separability of Bipartite States

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 $B^1,B^2,\ldots,B^m$, which are stars of $A$ and the third one consists of $m^2$ nodes $C^j_k (j,k=1,2,\ldots,m)$, where $C^j_k (k=1,2,\ldots,m)$ are stars of $B^j$. We call such a network a $3$-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.

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Two Types of Trilocality of Probability and Correlation Tensors

Xiao

Cao

Guo

et al. 2023

Entropy

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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.

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Some Measurement-Based Characterizations of Separability of Bipartite States

Cited by 3 publications

References 56 publications

Some characterizations of fully separability for multipartite quantum states

Some characterizations of fully separability for multipartite quantum states

Nonlocality of a type of multi-star-shaped quantum networks

Two Types of Trilocality of Probability and Correlation Tensors

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