Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

Abstract

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.

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Acknowledgement

The authors would like to thank the referees to carefully reading the manuscript and giving helpful comments that help to improve the quality of the paper.

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Correspondence to Jean-Pierre Gabardo.

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Jean-Pierre Gabardo is supported by an NSERC Discovery grant.

Deguang Han is partially supported by NSF grant DMS-1712602.

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Gabardo, J., Han, D. Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures. Acta Appl Math 166, 11–27 (2020). https://doi.org/10.1007/s10440-019-00252-6

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Mathematics Subject Classification

  • 42C15
  • 46C05
  • 47B10

Keywords

  • Positive operator valued measures
  • Frames
  • Integral representations
  • Frames
  • Dilations