AI 中文总结
该研究将经典优化理论推广至非交换自伴矩阵元组，提出非受控优化理论，推导量子信道存在的充要条件，还探讨了其计算验证难度及推广应用。
AI 中文摘要
我们引入非受控优化理论，将Hardy、Littlewood和Pólya提出的经典优化理论推广至不一定交换的自伴矩阵元组。我们定义并刻画了非受控优化序，该序扩展了经典优化序，其定义基于凸非受控函数，我们利用Davidson与第一作者近期提出的非受控凸性理论和非受控Choquet理论。作为应用，我们得到了保迹完全正映射（即量子信道）存在的新充要条件，该映射可在两个有限矩阵集之间插值。我们给出例子表明，即使在局部，该映射也不一定能选为混合幺正的。我们还探讨了验证非受控优化序的计算难度。我们的结果可推广至非迹情形，得到更一般的结果，刻画了保任意忠实态的单位完全正映射甚至单位完全正映射存在的情况。
英文摘要
We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.
Comments33 pages
