ABSTRACT

Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector d to majorization on square matrices with respect to a positive definite matrix D. For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of D has multiplicity one.

摘要

受量子热力学的启发，我们首先研究严格正性的概念，即线性映射，它将正定状态再次映射到正定状态。我们证明了严格的积极性是由对任何秩状态的作用决定的，并且非严格的正映射的图像生活在一个较低维的子代数中。这意味着此类映射到身份通道的距离以1为下限。当将实向量相对于正向量d的归一化顺序推广到平方矩阵相对于正定矩阵D的化归类化时，严格正定性的概念派上用场。对于二维情况，我们通过有限的多个迹范不等式给出了这种排序的特征，此外，还研究了其一些排序特性。特别地，它接受唯一的最小和最大元素。当且仅当D的最小特征值具有多重性1时，后者也是唯一的。