In the present paper we show a link between bistochastic quantum channels and classical maps. The primary goal of this work is to analyse the multiplicative structure of the Birkhoff polytope of order 3 (the simplest nontrivial case). A suitable complex parametrization of the Birkhoff polytope is proposed, which reveals several its symmetries and characteristics, in particular: (i) the structure of Markov semigroups inside the Birkhoff polytope, (ii) the relation between the set of Markov time evolutions, the set of positive definite matrices and the set of divisible matrices. A condition for Markov time evolution of semigroups in the set of symmetric bistochastic matrices is derived, which leads to an universal conserved quantity for all Markov evolutions. Finally, the complex parametrization is extended to the Birkhoff polytope of order 4.

中文翻译：

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3 阶 Birkhoff 多面体中的动态半群作为分析量子通道的工具

在本文中，我们展示了双随机量子通道和经典映射之间的联系。这项工作的主要目标是分析 3 阶 Birkhoff 多面体的乘法结构（最简单的非平凡情况）。提出了一个合适的 Birkhoff 多面体的复杂参数化，它揭示了它的几个对称性和特征，特别是：（i）Birkhoff 多面体内部的马尔可夫半群的结构，（ii）马尔可夫时间演化的集合之间的关系，集合正定矩阵和整除矩阵的集合。推导了对称双随机矩阵集合中半群的马尔可夫时间演化的条件，这导致了所有马尔可夫演化的普遍守恒量。最后，将复参数化扩展到 4 阶 Birkhoff 多面体。