## An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy Journal Article

 Author(s): Carlen, Eric A; Maas, Jan Article Title: An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy Affiliation Abstract: Let $\Cl$ denote the Clifford algebra over $\R^n$, which is the von Neumann algebra generated by n self-adjoint operators Qj, j=1,...,n satisfying the canonical anticommutation relations, QiQj+QjQi=2δijI, and let τ 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 τ(ρ)=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. Journal Title: Communications in Mathematical Physics Volume: 331 Issue 3 ISSN: 1432-0916 Publisher: Springer Date Published: 2014-07-29 Start Page: 887 End Page: 926 Sponsor: J. Maas’s work was partially supported by Rubicon subsidy 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO). E. A. Carlen’s work was partially supported by US National Science Foundation Grant DMS 0901632. URL: DOI: 10.1007/s00220-014-2124-8 Open access: yes (repository)
IST Austria Authors
1. 26 Maas