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，我们展示了如何扩展Rahaman引入的算子代数方法，以证明任意原始Schwarz映射的本原性指数至多为$2(D-1)^2$，其中$D$是基础矩阵代数的维度。 Rahaman首次证明了这一不等式，适用于既是保单位又是迹保持的Schwarz映射。 我们在此表明，对于原始Schwarz映射而言，保单位性的假设是自动成立的（归一化后），但一般情况下，并非所有原始保单位Schwarz映射都是迹保持的。 因此，本文的目的在于展示如何将Rahaman的证明应用于任意原始Schwarz映射。 作为该定理的一个推论，我们证明了任意原始2-正性映射的本原性指数至多为$2(D-1)^2$，因此，特别地，这个界限也适用于任意原始完全正性映射。 我们简要讨论了这与Perez-Garcia、Verstraete、Wolf和Cirac关于矩阵乘积态母哈密顿量性质猜想之间的关系。 显示英文摘要

摘要： In this note, we show how to extend operator algebraic methods introduced by Rahaman to prove that the index of primitivity of any primitive 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. We show here that the assumption of unitality is automatic (up to normalization) for primitive Schwarz maps, 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 proof of Rahaman to arbitrary primitive Schwarz maps. 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 concerning properties of parent Hamiltonians of matrix product states.