PSD秩的上下界
Some upper and lower bounds on PSD-rank
摘要：
半正定秩（PSD秩）是一个相对较新的量，它在组合优化和通信复杂性中有着广泛的应用。我们首先研究了PSD秩的几个基本性质，然后发展了新的技术来显示PSD秩的下界。所有这些界限都是基于将矩阵$M$的半正定因式分解视为量子通信协议。这些下界取决于矩阵的元素，而不仅仅是它的支持（零/非零模式），克服了一些以前的技术的限制。我们比较这些新的下界与已知的界限，并给出新的更好的例子。作为应用，我们确定了一些常见矩阵（近似）的PSD秩。
英文摘要：
Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix $M$ as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.
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