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 measure \(F: \varOmega \to B(H)\) has an integral representation of the form$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). 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.

中文翻译：

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正算子值测度的框架和有限秩积分表示

离散帧和连续帧可以视为具有正运算符值的量度（POVM），具有使用秩一运算符的整数表示。但是，并非每个POVM都有一个完整的表示形式。本文的一个目标是研究具有有限秩积分表示的POVM。更准确地说，我们提出了一个必要的充分条件，在该条件下，正算子值测度\（F：\ varOmega \ to B（H）\）具有形式为$$ F（E）= \ sum_ {k的整数表示= 1} ^ {m} \ int _ {E} G_ {k}（\ omega）\ otimes G_ {k}（\ omega）\，d \ mu（\ omega）$$对于某些弱可测地图\（G_ {k} \（1 \ leq k \ leq m）\）从可测空间\（\ varOmega \）到希尔伯特空间ℋ和一些正值\（\ mu \）在\（\ varOmega \）上。对于投影值测度，也可以获得类似的特征。作为表征的特殊结果，我们否定了Ehler和Okoudjou有关概率POVM的概率框架表示的问题，并证明了当且仅当相应的度量为时，积分可表示的概率POVM才能扩张为积分可表示的投影值度量。纯粹是原子的。