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

  • Jean-Pierre GabardoEmail author
  • Deguang Han
Article
  • 19 Downloads

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.

Keywords

Positive operator valued measures Frames Integral representations Frames Dilations 

Mathematics Subject Classification

42C15 46C05 47B10 

Notes

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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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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