{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:42Z","timestamp":1760147382979,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11871318","12271325","1503070117"],"award-info":[{"award-number":["11871318","12271325","1503070117"]}]},{"name":"Special Plan for Young Top-Notch Talent of Shaanxi Province","award":["11871318","12271325","1503070117"],"award-info":[{"award-number":["11871318","12271325","1503070117"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we discuss two types of trilocality of probability tensors (PTs) P=\u301aP(a1a2a3)\u301b over an outcome set \u03a93 and correlation tensors (CTs) P=\u301aP(a1a2a3|x1x2x3)\u301b over an outcome-input set \u03943 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.<\/jats:p>","DOI":"10.3390\/e25020273","type":"journal-article","created":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T02:21:12Z","timestamp":1675304472000},"page":"273","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Two Types of Trilocality of Probability and Correlation Tensors"],"prefix":"10.3390","volume":"25","author":[{"given":"Shu","family":"Xiao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710119, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2141-825X","authenticated-orcid":false,"given":"Huaixin","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710119, China"}]},{"given":"Zhihua","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710119, China"}]},{"given":"Kanyuan","family":"Han","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shaanxi Normal University, Xi\u2019an 710119, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1038\/nature07127","article-title":"The quantum internet","volume":"453","author":"Kimble","year":"2008","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1103\/RevModPhys.83.33","article-title":"Quantum repeaters based on atomic ensembles and linear optics","volume":"83","author":"Sangouard","year":"2011","journal-title":"Rev. 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