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Lobachevskii Journal of Mathematics

On Construction of Positive Operator-valued Measures from Frames in a Hilbert Space

Grigori Gennadievich Amosov

Published Jun 1, 2026• 0 citations
Publisher

Abstract

Abstract The task of finding POVMs including informationally complete POVMs from frames in a Hilbert space is investigated. The case in which frames are obtained from projective unitary representations of groups is considered.
DOI
10.1134/s199508022661859x

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We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.
The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.