Learning quantum Gibbs states in the Wasserstein distance

Speaker: Daniel Stilck Franca (École Normale Supérieure de Lyon)

Abstract: In this talk, I will discuss recent advances in the learning of quantum Gibbs states. I will begin by motivating the problem, highlighting its interpretation as a noncommutative extension of the classical task of estimating graphical models. Following this, I will review the recently developed noncommutative versions of the Wasserstein distance of order 1, emphasizing their relevance as operationally motivated metrics for this problem.

Next, I will demonstrate how the combination of functional inequalities, specifically transportation inequalities of order 1, with maximum entropy estimation techniques, enables an exponential reduction in the sample complexity required to estimate these states, compared to other commonly studied metrics such as the trace distance.

Finally, I will outline methods to derive these functional inequalities for broad classes of quantum Gibbs states, either through semigroup techniques or under the assumption of exponential decay of correlations for the underlying states.
This presentation is based on joint work with C. Rouzé, J. Watson, and E. Onorati.