Deviation bounds and concentration inequalities for quantum noises

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.

Concentration Inequalities for Output Statistics of Quantum Markov Processes

“…4, we show how perturbation theory and spectral techniques can be used to derive concentration inequalities for the case of quantum counting processes too. This result integrates the bounds obtained in [18], providing a simple bound also for the case of counting processes and non-self-adjoint generators. Moreover, it bypasses the problem of establishing functional inequalities and estimating the constants appearing in the inequalities.…”

“…5.2. Despite the fact that the asymptotic behaviour of the process (f (X n )) n∈N is rather well understood, less is known about its finite time properties, with the notable exception of the recent concentration results for continuous-time Markov dynamics [18]. The main goal of the present work is to derive alternative concentration bounds, i.e.…”

“…In this section, we consider continuous-time concentration bounds for counting measurements in the output of a quantum Markov process. In this context, alternative concentration bounds have been recently obtained in [18]. In that paper, the authors first prove a bound for general quantum measurement processes which is sharp, but depends on quantities which may be hard to calculate in practical situations (Theorem 6); then, using functional inequalities, they are also able to derive further concentration bounds involving easily computable quantities, but only in the diffusive case for symmetric generators (Theorems 8 and 9 and Corollary 2).…”

“…Hypothesis (H ) means that (Φ t ) t≥0 is irreducible. There are many ways of defining quantum counting processes: we will follow the formulation of Davis and Srinivas [32,67] and we refer to [18] and references therein for their definition through quantum stochastic calculus and quantum filtering or how to characterize them using stochastic Schrödinger equations. Before doing that, it is convenient to introduce some notation: we define the completely positive map J i (x) = L * i xL i and the semigroup of completely positive maps e tL0 (x) = e tG * xe tG , where…”

“…In contrast to the recent progress in understanding the asymptotics of the outcome process, much less is known on the finite-time properties. Notable recent results in this direction are the deviation bounds and concentration inequalities for quantum counting processes and homodyne measurements obtained in [18] (hence regarding continuous-time models). The main aim of the present work is to provide new classes of concentration inequalities for additive functionals of the finite-time measurement process, which complement the results of [18] and offer more explicit bounds.…”