---
res:
bibo_abstract:
- "Let ℭ denote the Clifford algebra over ℝ\U0001D45B, 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 ℭ. This algebra arises in quantum mechanics as the algebra
of observables generated by n fermionic degrees of freedom. Let \U0001D513 denote
the set of all positive operators \U0001D70C∈ℭ such that τ(ρ) = 1; these are the
non-commutative analogs of probability densities in the non-commutative probability
space (ℭ,\U0001D70F). 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 \U0001D513 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.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Eric
foaf_name: Carlen, Eric
foaf_surname: Carlen
- foaf_Person:
foaf_givenName: Jan
foaf_name: Maas, Jan
foaf_surname: Maas
foaf_workInfoHomepage: http://www.librecat.org/personId=4C5696CE-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-0845-1338
bibo_doi: 10.1007/s00220-014-2124-8
bibo_issue: '3'
bibo_volume: 331
dct_date: 2014^xs_gYear
dct_language: eng
dct_publisher: Springer@
dct_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@
...