# Concentration of quantum states from quantum functional and transportation cost inequalities

## TL;DR

This paper introduces quantum analogs of transportation cost inequalities and explores their relationships with quantum functional inequalities, leading to concentration results and applications in quantum parameter estimation.

## Contribution

It develops quantum versions of TC1 and TC2 inequalities using new quantum Wasserstein distances, linking them to existing quantum functional inequalities.

## Key findings

- Quantum TC1 and TC2 inequalities are related as in the classical case.
- These inequalities imply concentration results for quantum states.
- Application to quantum parameter estimation provides error bounds.

## Abstract

Quantum functional inequalities (e.g. the logarithmic Sobolev- and Poincar\'e inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincar\'e inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation.