Title: Concentration of quantum states from quantum functional and transportation cost inequalities,
Abstract: Quantum functional inequalities (e.g. the logarithmic Sobolev- and Poincaré
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 Poincaré 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.