Positive Operator Valued Measures and Feller Markov kernels

Abstract

A Positive Operator Valued Measure (POVM) is a map F:B(X)→Ls+(H) from the Borel σ-algebra of a topological space X to the space of positive self-adjoint operators on a Hilbert space 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 ν 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 [21], they are important for a range of quantum information processing protocols, where classical post-processing plays a role [27]. 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.