We describe how to extend the notion of infinitesimal Markovian divisibility from quantum channels to general linear maps and compact and convex sets of generators. We give a general approach toward proving necessary criteria for (infinitesimal) Markovian divisibility. With it, we prove two necessary criteria for infinitesimal divisibility of quantum channels in any finite dimension d: an upper bound on the determinant in terms of a Θ(d)-power of the smallest singular value and in terms of a product of Θ(d) smallest singular values. These allow us to analytically construct, in any given dimension, a set of channels that contains provably non-infinitesimal Markovian divisible ones. Moreover, we show that, in general, no such non-trivial criteria can be derived for the classical counterpart of this scenario.

中文翻译：

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线性映射的马氏可除性的必要条件

我们描述了如何将无限小马尔可夫可除性的概念从量子通道扩展到一般的线性映射以及生成器的紧集和凸集。我们为证明（无限小）马尔可夫可除性的必要标准提供了一种通用方法。有了它，我们证明了在任何有限维d中量子通道的无穷小可除性的两个必要标准：行列式的上限，以最小奇异值的Θ（d）幂表示，以Θ（d）的乘积表示d）最小的奇异值。这些使我们能够在任何给定的维度上分析性地构建一组包含可证明是非无限小的马尔可夫可分割通道的通道。而且，我们表明，一般而言，对于这种情况的经典对应物，无法导出这样的非平凡标准。