Uniform continuity of POVMs

TL;DR

This paper investigates the uniform continuity of POVMs, linking it to Feller Markov kernels in the commutative case, and applies these concepts to analyze physical examples like phase space localization and unsharp observables.

Contribution

It extends the theoretical framework of uniform continuity of POVMs and explores its implications for important physical measurement models.

Findings

Uniform continuity characterized by Feller Markov kernels in the commutative case.

Analysis of phase space localization, unsharp phase, and number observables regarding their uniform continuity.

Conditions for the norm-1 property and existence of Feller Markov kernels in studied examples.

Abstract

Recently a characterization of uniformly continuous POVMs and a necessary condition for a uniformly continuous POVM $F$ to have the norm-1 property have been provided. Moreover it was proved that in the commutative case, uniform continuity corresponds to the existence of a Feller Markov kernel. We apply such results to the analysis of some relevant physical examples; i.e., the phase space localization observables, the unsharp phase observable and the unsharp number observable of which we study the uniform continuity, the norm-1 property and the existence of a Feller Markov kernel.