标题： 极端量子态和过程，以及有限维代数中一般谱多面体的极点 显示英文标题

标题： Extreme quantum states and processes, and extreme points of general spectrahedra in finite dimensional algebras

摘要： 凸集在量子态和过程中起着核心作用，在量子理论和量子信息中。量子理论中许多重要的凸集例子是半定域，即在仿射约束下的正算子集合。这些例子包括具有某些感兴趣的可观测量期望值的量子态集合，具有给定边缘的多部分量子态集合，量子测量、通道和多时间量子过程的集合，以及高阶量子映射和量子因果结构的集合。本贡献提供了对一般半定域极端点的表征，以及对应算子秩的界限。一般结果被应用于几个特殊情况，并用于重新获得经典结果，如Choi对极端量子通道的表征，Parhasarathy对具有给定边缘的极端量子态的表征，以及针对qubit单位通道的Birkhoff定理的量子版本。最后，我们提出了一个具有通用仿射约束的正算子值测度（POVMs）的概念，并表征了极值POVMs。 显示英文摘要

摘要： Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators subject to affine constraints. These examples include sets of quantum states with given expectation values of some observables of interest, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels, and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterisation of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and are used to retrieve classic results such as Choi's characterisation of the extreme quantum channels, Parhasarathy's characterisation of the extreme quantum states with given marginals, and the quantum version of Birkhoff's theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalisation, and we characterise the extremal POVMs.