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 : Ω → B ( H ) has an integral representation of the form F ( E ) = ∑ k = 1 m ∫ E G k ( ω ) ⊗ G k ( ω ) d μ ( ω ) for some weakly measurable maps G k ( 1 ≤ k ≤ m ) from a measurable space Ω to a Hilbert space ℋ and some positive measure μ on Ω . 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.