论文标题
严格的积极性和$ d $ -jorization
Strict Positivity and $D$-Majorization
论文作者
论文摘要
由量子热力学的启发，我们首先研究了严格阳性的概念，即线性图，这些图将正定状态映射到再次确定的东西。我们表明，严格的积极性是由对任何全等级状态的行动决定的，而非刻板图的图像生活在低维的亚词法中。这意味着此类地图与身份通道的距离是由一个界限的。 当将大量订购概括为正面矢量$ d $的真实载体上，就正方矩阵上的大量化而言，就积极的确定矩阵$ d $而言，严格的积极性概念派上用场。对于二维情况，我们通过有限的许多跟踪规范不等式给出了此排序的表征，此外，还研究了其某些顺序属性。特别是它承认独特的最小和最大元素。后者也是唯一的，并且只有当$ d $的最小特征值具有多重性时，后者也是独特的。
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.
