论文标题
线性图的马尔可夫分裂性的必要标准
Necessary Criteria for Markovian Divisibility of Linear Maps
论文作者
论文摘要
表征与马尔可夫时间演变相对应的那些量子通道是量子信息理论中的一个开放问题，甚至存在不同的量子马尔可维亚性概念。一种这样的观念是Arxiv中引入的量子通道的无限马尔可夫分裂性：Math-PH/0611057。尽管无穷小马尔可夫分式量子通道具有完整的表征，但除了确定性非负性的必要性外，没有必要或足够的标准以较高的尺寸知道。 我们描述了如何将无穷小马尔可夫分隔的概念扩展到一般线性图以及封闭和凸的发电机集。我们为证明涉及线性图的奇异值的（无限）马克维亚分裂性的必要标准提供了一种普遍的方法。通过这种方法，我们证明了在任何有限尺寸$ d $中起作用的量子通道的无限分组的必要标准：在$θ（d）$的$θ（d）$的功率方面，在$θ（d）$θ（d）（d）$最小的singular valuess的产品方面，上限是$θ（d）$。我们的标准使我们能够在任何给定的维度上分析构建一组含有无限的马尔可夫分割的渠道。 我们还讨论了这种情况的经典对应物，即与过渡速率矩阵给出的发电机的随机矩阵。在这里，我们表明，证明量子通道的形式的无限马尔可夫分裂性没有必要的标准。但是，我们描述了所有过渡速率矩阵的子集，可以将我们的推理应用于马尔可夫分裂性的必要条件。
Characterizing those quantum channels that correspond to Markovian time evolutions is an open problem in quantum information theory, even different notions of quantum Markovianity exist. One such notion is that of infinitesimal Markovian divisibility for quantum channels introduced in arXiv:math-ph/0611057. Whereas there is a complete characterization for infinitesimal Markovian divisible qubit channels, no necessary or sufficient criteria are known for higher dimensions, except for necessity of non-negativity of the determinant. We describe how to extend the notion of infinitesimal Markovian divsibility to general linear maps and closed and convex sets of generators. We give a general approach towards proving necessary criteria for (infinitesimal) Markovian divisibility that involve singular values of the linear map. With this approach, we prove two necessary criteria for infinitesimal divisibility of quantum channels that work 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. Our criteria allow us to analytically construct, in any given dimension, a set of channels that contains provably non infinitesimal Markovian divisible ones. We also discuss the classical counterpart of this scenario, i.e., stochastic matrices with the generators given by transition rate matrices. Here, we show that no necessary criteria for infinitesimal Markovian divisibility of the form proved for quantum channels can hold in general. However, we describe subsets of all transition rate matrices for which our reasoning can be applied to obtain necessary conditions for Markovian divisibility.
