Universal Randomization of Quantum Observables

Abstract

Commutative POVMs are smearing of spectral measures with the smearing realized by a Markov kernel. In the case of POVMs defined on the Borel σ-algebra of the reals, the existence of a universal Markov kernel has been established. Here we show that every weak Markov kernel is functionally subordinated to the universal Markov kernel. Then, we analyze the implications of that result to the characterization of a Markov kernel as a Choquet’s integral on the space of deterministic Markov kernels.