有限维量子测量的重心分解
Barycentric decomposition of quantum measurements in finite dimensions
摘要：
在给定的局部紧Hausdorff空间中，分析了有限维量子系统上表示量子测量的正算子值测度（POVM）集的凸结构.凸集的极值点是集中在结果空间的k个\le d^2点的有限集合上的算子值测度，d< \infty是希尔伯特空间的维数。我们证明了第二可数结果空间的任何POVM承认一个Choquet表示作为重心的极值点的集合相对于一个合适的概率测度。在一般情况下，引用Krein-Milman定理将POVM表示为集中在结果空间的k个\le d^2点上的某个POVM集合的重心。
英文摘要：
We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of k \le d^2 points of the outcome space, d< \infty being the dimension of the Hilbert space. We prove that for second countable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein-Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on k \le d^2 points of the outcome space.
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