严格正性与$D$-优化排序
Strict Positivity and $D$-Majorization
AI总结：
该研究受量子热力学启发，探究严格正性的判定与性质，并将其用于推广$D$-优化排序，给出二维情形下该排序的迹范数刻画及序性质。
摘要：
受量子热力学启发，我们首先研究严格正性的概念，即能将正定态映射为正定对象的线性映射。我们证明，严格正性可通过对任意满秩态的作用来判定，且非严格正映射的像存在于低维子代数中。这意味着此类映射与恒等信道的距离下界为1。\n在将实向量上关于正向量$d$的优化排序推广为方阵上关于正定矩阵$D$的优化排序时，严格正性的概念十分有用。针对二维情形，我们通过有限个迹范数不等式给出了该排序的刻画，并进一步研究了它的若干序性质。特别地，它存在唯一的极小元和一个极大元；当且仅当$D$的最小特征值的重数为1时，该极大元也是唯一的。
英文摘要：
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.
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