Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures
Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures
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DOI：
10.1007/s10440-019-00252-6
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发表时间：
2019-04
影响因子：
1.6
通讯作者：
J. Gabardo;D. Han
中科院分区：
文献类型：
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作者：
J. Gabardo;D. Han
离散和连续框架可以被认为是正算子值测度（POVM），它们具有使用秩1算子的积分表示。然而，并不是每个POVM都有积分表示。本文的一个目标是研究具有有限秩积分表示的POVM。更确切地说，我们给出了一个充要条件，在此条件下，从可测空间到Hilbert空间的某些弱可测映射的正算子值测度具有如下形式的积分表示：对于投影值测度也得到了类似的刻画。作为我们刻画的特殊结果，我们否定地解决了Ehler和Okoudjou关于概率POVM的概率框架表示的问题，并证明了一个积分可表示的概率POVM可以扩张为一个积分可表示的投影值测度当且仅当相应的测度是纯原子的。
Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measurehas an integral representation of the formfor some weakly measurable mapsfrom a measurable spaceto a Hilbert space ℋ and some positive measureon. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.