Deviation bounds and concentration inequalities for quantum noises

TL;DR

This paper develops deviation bounds and concentration inequalities for quantum noise processes using non-commutative Dirichlet forms, with applications to quantum optics and quantum Markov semigroups.

Contribution

It introduces new non-commutative functional inequalities and derives optimal deviation bounds for quantum stochastic processes.

Findings

Derived finite time deviation bounds for quantum processes.

Established concentration inequalities for tensor products of quantum Markov semigroups.

Applied bounds to Gibbs samplers above a threshold temperature.

Abstract

We provide a stochastic interpretation of non-commutative Dirichlet forms in the context of quantum filtering. For stochastic processes motivated by quantum optics experiments, we derive an optimal finite time deviation bound expressed in terms of the non-commutative Dirichlet form. Introducing and developing new non-commutative functional inequalities, we deduce concentration inequalities for these processes. Examples satisfying our bounds include tensor products of quantum Markov semigroups as well as Gibbs samplers above a threshold temperature.