Operators and POVMs generated by Parseval frames

TL;DR

This paper explores the relationship between operators and POVMs generated by Parseval frames, extending classical spectral measure concepts from orthonormal bases to frames in Hilbert spaces.

Contribution

It introduces a construction of operators and POVMs from Parseval frames and analyzes their relationship, generalizing the spectral measure framework.

Findings

Established a link between operators and POVMs from Parseval frames.

Extended classical spectral measure correspondence to frame-generated operators.

Provided mathematical foundations for frame-based quantum measurement models.

Abstract

Let $F$ be a Parseval frame in a Hilbert space and let $E$ be a set of real numbers. From these data, we construct an operator $H_{E,e}$ and a positive operator-valued measure (POVM) $F_{E,e}$. This paper investigates in detail the relationship between the operator $H_{E, e}$ and the POVM $F_{E,e}$. Our results extend the classical correspondence between a self-adjoint operator generated by an orthonormal basis and its associated projection-valued (spectral) measure.