Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

• 成果类型: Article; Proceedings Paper
• 署名作者: Gribling, Sander; de Laat, David; Laurent, Monique
• 署名单位: Centrum Wiskunde & Informatica (CWI); Tilburg University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1287-z
• 发表日期: 2018
• 页码: 5-42
• 关键词: connes embedding conjecture; chromatic number; relaxations

摘要

In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.