标题： 关于量子Wielandt不等式的注记 显示英文标题

标题： A note on the quantum Wielandt inequality

摘要： 在本笔记中，我们证明了任何本原归一的施瓦兹映射的本原性指数至多为$2(D-1)^2$，其中$D$是底层矩阵代数的维数。 这一不等式最初由拉哈曼对于既是归一又是迹保持的施瓦兹映射证明。 正如我们所展示的，归一性的假设基本上是无害的，但在一般情况下，并非所有的本原归一施瓦兹映射都是迹保持的。 因此，本文的确切目的是展示如何将拉哈曼的方法应用于那些不保持迹的本原归一施瓦兹映射。 作为该定理的一个推论，我们证明了任何本原2-正映射的本原性指数至多为$2(D-1)^2$，因此这个界也适用于任意本原完全正映射。 我们简要讨论了这与佩雷斯-加西亚、韦斯特罗特、沃尔夫和西拉克的猜想之间的关系。 显示英文摘要

摘要： In this note, we prove that the index of primitivity of any primitive unital Schwarz map is at most $2(D-1)^2$, where $D$ is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. As we show, the assumption of unitality is basically innocuous, but in general not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the method of Rahaman to unital primitive Schwarz maps that don't preserve trace. As a corollary of this theorem, we show that the index of primitivity of any primitive 2-positive map is at most $2(D-1)^2$, so in particular this bound holds for arbitrary primitive completely positive maps. We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.