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

摘要

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 -> B(H) has an integral representation of the form disp-formula id="Equa"F(E)=mml:munderover Sigma k=1mmml:munderover integral EGk(omega)circle times Gk(omega)mml:mspace width="0.2em"mml:mspaced mu(omega) for some weakly measurable maps Gkmml:mspace width="0.25em"mml:mspace>(1 <= k <= m) from a measurable space Omega to a Hilbert space 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.