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.

中文翻译：

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通过非对易多项式优化对纠缠维度和量子图参数的限制

在本文中，我们使用 trac 非对易多项式优化技术研究与二部量子相关性相关的优化问题。首先，我们考虑找到这种相关性的最小纠缠维度的问题。我们构建了一个半定规划下限的层次结构，并展示了对一个新参数的收敛性：最小平均纠缠维数，它测量在免费访问共享随机性时重现量子相关性所需的纠缠量。然后我们研究了由量子图参数引起的同步量子相关性的优化问题。我们引入半定规划层次结构，并通过将它们置于 trac 多项式优化的框架中来统一量子色度数和量子稳定性数的现有界限。