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 ${\mathfrak{C}}$ denote the Clifford algebra over ${\mathbb{R}^n}$, which is the von Neumann algebra generated by n self-adjoint operators Q _j , j = 1,…,n satisfying the canonical anticommutation relations, Q _i Q _j  + Q _j Q _i =  2δ _ij I, and let τ denote the normalized trace on ${\mathfrak{C}}$. This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let ${\mathfrak{P}}$ denote the set of all positive operators ${\rho\in\mathfrak{C}}$ such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space ${(\mathfrak{C},\tau)}$. 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 ${\mathfrak{P}}$ 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.