A spectral gap for POVMs

TL;DR

This paper introduces the spectral gap for POVMs, linking it to quantum noise, Markov chains, and geometric transforms, and explores its properties and robustness in quantum measurement contexts.

Contribution

It defines the spectral gap for POVMs and connects it to geometric and probabilistic concepts, providing a new invariant for quantum measurement analysis.

Findings

Spectral gap relates to quantum noise and measurement unsharpness.

The spectral gap can be described geometrically in metric measure spaces.

The spectral gap is stable under Wasserstein perturbations.

Abstract

For a class of positive operator valued measures, we introduce the spectral gap, an invariant which shows up in a number of contexts: the quantum noise operator responsible for the unsharpness of quantum measurements, the Markov chain describing the state reduction for repeated quantum measurements, and the Berezin transform on compact Kahler manifolds. The spectral gap admits a transparent description in terms of geometry of certain metric measure spaces, is related to the diffusion distance, and exhibits a robust behaviour under perturbations in the Wasserstein metric.