Positive Operator Valued Measures and Feller Markov Kernels

TL;DR

This paper characterizes when a POVM can be represented as a smearing of a spectral measure using Feller Markov kernels, linking mathematical structure to physical measurement noise and precision.

Contribution

It proves that commutative POVMs are exactly those obtained by smearing spectral measures with Feller Markov kernels, and characterizes the continuity conditions for strong Feller kernels.

Findings

Commutative POVMs are characterized by smearing spectral measures.

Strong Feller Markov kernels correspond to uniformly continuous POVMs.

Norm bounded POVMs admit strong Feller Markov kernels.

Abstract

A Positive Operator Valued Measure (POVM) is a map $F:\mathcal{B}(X)\to\mathcal{L}_s^+(\mathcal{H})$ from the Borel $\sigma$-algebra of a topological space $X$ to the space of positive self-adjoint operators on a Hilbert space $\mathcal{H}$. We assume $X$ to be Hausdorff, locally compact and second countable and prove that a POVM $F$ is commutative if and only if it is the smearing of a spectral measure $E$ by means of a Feller Markov kernel. Moreover, we prove that the smearing can be realized by means of a strong Feller Markov kernel if and only if $F$ is uniformly continuous. Finally, we prove that a POVM which is norm bounded by a finite measure $\nu$ admits a strong Feller Markov kernel. That provides a characterization of the smearing which connects a commutative POVM $F$ to a spectral measure $E$ and is relevant both from the mathematical and the physical viewpoint since smearings of spectral measures form a large and very relevant subclass of POVMs: they are paradigmatic for the modeling of certain standard forms of noise in quantum measurements, they provide optimal approximators as marginals in joint measurements of incompatible observables \cite{Busch}, they are important for a range of quantum information processing protocols, where classical post-processing plays a role \cite{Heinosaari}. The mathematical and physical relevance of the results is discussed and particular emphasis is given to the connections between the Markov kernel and the imprecision of the measurement process.