An Analog of the 2-Wasserstein Metric in Non-commutative Probability under which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy

Abstract: Let $\Cl$ denote the Clifford algebra over $\R^n$, which is the von Neumann algebra generated by $n$ self-adjoint operators $Q_j$, $j=1,...,n$ satisfying the canonical anticommutation relations, $Q_iQ_j+Q_jQ_i = 2\delta_{ij}I$, and let $\tau$ denote the normalized trace on $\Cl$. This algebra arises in quantum mechanics as the algebra of observables generated by $n$ Fermionic degrees of freedom. Let $\Dens$ denote the set of all positive operators $\rho\in\Cl$ such that $\tau(\rho) =1$; these are the non-commutative analogs of probability densities in the non-commutative probability space $(\Cl,\tau)$. The Fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on $\Dens$ that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the Fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.