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

Shu Xiao, Zhihua Guo (), Huaixin Cao (), Kanyuan Han and Ying Yang
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Shu Xiao: School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
Zhihua Guo: School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
Huaixin Cao: School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
Kanyuan Han: School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China
Ying Yang: School of Mathematics and Information Technology, Yuncheng University, Yuncheng 044000, China

Mathematics, 2023, vol. 11, issue 7, 1-33

Abstract: In this work, we study the nonlocality of star-shaped correlation tensors (SSCTs) based on a general multi-star-network M S N ( m , n 1 , … , n m ) . Such a network consists of 1 + m + n 1 + ⋯ + n m nodes and one center-node A that connects to m star-nodes B 1 , B 2 , … , B m while each star-node B j has n j + 1 star-nodes A , C 1 j , C 2 j , … , C n j j . 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 Δ S 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 M S N ( m , n 1 , … , n m ) are presented.

Keywords: multi-star-network; star-shaped correlation tensor; star-locality; star-Bell inequality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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