Positive Operator Valued Measures: A General Setting for Frames

TL;DR

This paper explores the deep connections between positive operator-valued measures (POVMs) and frames in Hilbert spaces, introducing the concept of framed POVMs to unify various frame types and leverage POVM theory.

Contribution

It introduces the concept of framed POVMs, unifying classical frames, fusion frames, and generalized frames under a common framework, enabling new insights from POVM theory.

Findings

Frames can be represented as framed POVMs.

Classical and generalized frames are special cases of framed POVMs.

POVM theory offers new perspectives for frame analysis.

Abstract

This paper presents an overview of close parallels that exist between the theory of positive operator-valued measures (POVMs) associated with a separable Hilbert space and the theory of frames on that space, including its most important generalizations. The concept of a framed POVM is introduced, and classical frames, fusion frames, generalized frames, and other variants of frames are all shown to to arise as framed POVMs. This observation allows drawing on a rich existing theory of POVMs to provide new perspectives in the study of frames.