{"title":"人鱼多面体量子计算与基础","authors":"Cihan Okay, Ho Yiu Chung, Selman Ipek","doi":"10.26421/qic23.9-10-2","DOIUrl":null,"url":null,"abstract":"Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\\MP_\\beta$ obtained from the Mermin scenario, parametrized by a function $\\beta$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\\MP_0$ and $\\MP_1$ depending on the parity of $\\beta$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\\MP_1$ can be seen as a nonlocal toy version of $\\Lambda$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $\\Lambda$-polytope using $\\MP_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.","PeriodicalId":54524,"journal":{"name":"Quantum Information & Computation","volume":"31 1","pages":"0"},"PeriodicalIF":0.7000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Mermin polytopes quantum computation and foundations\",\"authors\":\"Cihan Okay, Ho Yiu Chung, Selman Ipek\",\"doi\":\"10.26421/qic23.9-10-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\\\\MP_\\\\beta$ obtained from the Mermin scenario, parametrized by a function $\\\\beta$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\\\\MP_0$ and $\\\\MP_1$ depending on the parity of $\\\\beta$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\\\\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\\\\MP_1$ can be seen as a nonlocal toy version of $\\\\Lambda$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $\\\\Lambda$-polytope using $\\\\MP_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.\",\"PeriodicalId\":54524,\"journal\":{\"name\":\"Quantum Information & Computation\",\"volume\":\"31 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quantum Information & Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26421/qic23.9-10-2\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Information & Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26421/qic23.9-10-2","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Mermin polytopes quantum computation and foundations
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\MP_\beta$ obtained from the Mermin scenario, parametrized by a function $\beta$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\MP_0$ and $\MP_1$ depending on the parity of $\beta$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\MP_1$ can be seen as a nonlocal toy version of $\Lambda$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $\Lambda$-polytope using $\MP_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.
期刊介绍:
Quantum Information & Computation provides a forum for distribution of information in all areas of quantum information processing. Original articles, survey articles, reviews, tutorials, perspectives, and correspondences are all welcome. Computer science, physics and mathematics are covered. Both theory and experiments are included. Illustrative subjects include quantum algorithms, quantum information theory, quantum complexity theory, quantum cryptology, quantum communication and measurements, proposals and experiments on the implementation of quantum computation, communications, and entanglement in all areas of science including ion traps, cavity QED, photons, nuclear magnetic resonance, and solid-state proposals.