A new bound on quantum version of Wielandt inequality for positive (not necessarily completely positive) maps has been established. Also bounds for entanglement breaking and PPT channels are put forward which are better bound than the previous bounds known. We prove that a primitive positive map $\mathcal {E}$ acting on $\mathcal {M}_{d}$ that satisfies the Schwarz inequality becomes strictly positive after at most $2(d-1)^{2}$ iterations. This is to say that after $2(d-1)^{2}$ iterations, such a map sends every positive semidefinite matrix to a positive definite one. This finding does not depend on the number of Kraus operators as the map may not admit any Kraus decomposition. The motivation of this work is to provide an answer to a question raised by Sanz-García-Wolf and Cirac in their work on quantum Wielandt bound.

中文翻译：

---

量子维兰特不等式的新界限

已经建立了正（不一定完全正）映射的维兰特不等式量子版本的新界限。还提出了解缠结和 PPT 通道的边界，它们比以前已知的边界更好。我们证明了一个原始的正映射 $\mathcal {E}$ 作用于 $\mathcal {M}_{d}$ 满足 Schwarz 不等式在至多之后变为严格正 $2(d-1)^{2}$ 迭代。这就是说之后 $2(d-1)^{2}$ 迭代，这样的映射将每个正半定矩阵发送到一个正定矩阵。这一发现不依赖于克劳斯算子的数量，因为地图可能不允许任何克劳斯分解。这项工作的动机是为 Sanz-García-Wolf 和 Cirac 在他们关于量子维兰特束缚的工作中提出的问题提供答案。