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

Eric A. Carlen, Jan Maas

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

Let C denote the Clifford algebra over Rn, which is the von Neumann algebra generated by n self-adjoint operators Qj, j = 1,...,n satisfying the canonical anticommutation relations, Qi Qj + Qj Qi = 2δij I, and let τ denote the normalized trace on C. This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let P denote the set of all positive operators ρ ∈ C such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space (C,τ). 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 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.

Original languageEnglish (US)
Pages (from-to)887-926
Number of pages40
JournalCommunications In Mathematical Physics
Volume331
Issue number3
DOIs
StatePublished - Nov 2014

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Noncommutative Probability
Wasserstein Metric
Gradient Flow
Fokker-Planck Equation
Entropy
Denote
Analogue
Metric
Hypercontractivity
Algebra
Sharp Inequality
Relative Entropy
Clifford Algebra
Probability Space
Positive Operator
Von Neumann Algebra
Riemannian Metric
Self-adjoint Operator
Probability Density
Quantum Mechanics

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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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",
abstract = "Let C denote the Clifford algebra over Rn, which is the von Neumann algebra generated by n self-adjoint operators Qj, j = 1,...,n satisfying the canonical anticommutation relations, Qi Qj + Qj Qi = 2δij I, and let τ denote the normalized trace on C. This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let P denote the set of all positive operators ρ ∈ C such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space (C,τ). 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 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.",
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