A dynamical approach to quantum optimal transport

Abstract

We begin with a brief overview of measure theory and the theory of optimal transport. We then proceed to study a special class of quantum states represented by quantum Markov semi-groups (QMS) on a finite dimensional C*-algebra. We show that these semi-groups are ergodic and have a unique stationary state. We then proceed to define a notion of quantum detailed balance and show that these semi-groups satisfy this detailed balance condition with respect to the unique stationary state. This condition characterises the form of the generator of the QMS. Starting from the form of this generator we proceed to show how one can construct the operators of multiplication, gradient and divergence acting on a direct sum of Hilbert spaces. These notions are then used to obtain a quantum mechanical analog of the continuity equation for probability densities. We define a Riemannian manifold of density matrices and proceed to show that for a given metric, the time evolution of our quantum states can be written as gradient flow for the relative entropy functional. This is a direct quantum analog to the time evolution of probability densities on Rn, which can be written as gradient flow for the Wasserstein metric.