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Title: Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures
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 $F: \Omega \to B(H)$ 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 $\Omega$ to a Hilbert space $\mathcal{H}$ and some positive measure $\mu$ on $\Omega$. 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.
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Acta Applicandae Mathematicae
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National Science Foundation
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