An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy

作者：Carlen Eric A^*; Maas Jan

来源：Communications in Mathematical Physics, 2014, 331(3): 887-926.

DOI：10.1007/s00220-014-2124-8

Let denote the Clifford algebra over , which is the von Neumann algebra generated by n self-adjoint operators Q (j) , j = 1,aEuro broken vertical bar,n satisfying the canonical anticommutation relations, Q (i) Q (j) + Q (j) Q (i) = 2 delta (ij) I, and let tau denote the normalized trace on . This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let denote the set of all positive operators such that tau(rho) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space . 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 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.

• 出版日期2014-11