Nonlocality of Star-Shaped Correlation Tensors Based on the Architecture of a General Multi-Star-Network

Abstract

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.