--- _id: '2133' abstract: - lang: eng text: "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." author: - first_name: Eric full_name: Carlen, Eric last_name: Carlen - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 citation: ama: Carlen E, Maas J. An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy. Communications in Mathematical Physics. 2014;331(3):887-926. doi:10.1007/s00220-014-2124-8 apa: Carlen, E., & Maas, J. (2014). An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2124-8 chicago: Carlen, Eric, and Jan Maas. “An Analog of the 2-Wasserstein Metric in Non-Commutative Probability under Which the Fermionic Fokker-Planck Equation Is Gradient Flow for the Entropy.” Communications in Mathematical Physics. Springer, 2014. https://doi.org/10.1007/s00220-014-2124-8. ieee: E. Carlen and J. Maas, “An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy,” Communications in Mathematical Physics, vol. 331, no. 3. Springer, pp. 887–926, 2014. ista: Carlen E, Maas J. 2014. An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy. Communications in Mathematical Physics. 331(3), 887–926. mla: Carlen, Eric, and Jan Maas. “An Analog of the 2-Wasserstein Metric in Non-Commutative Probability under Which the Fermionic Fokker-Planck Equation Is Gradient Flow for the Entropy.” Communications in Mathematical Physics, vol. 331, no. 3, Springer, 2014, pp. 887–926, doi:10.1007/s00220-014-2124-8. short: E. Carlen, J. Maas, Communications in Mathematical Physics 331 (2014) 887–926. date_created: 2018-12-11T11:55:54Z date_published: 2014-11-01T00:00:00Z date_updated: 2021-01-12T06:55:30Z day: '01' doi: 10.1007/s00220-014-2124-8 extern: '1' intvolume: ' 331' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: 'http://arxiv.org/abs/1203.5377 ' month: '11' oa: 1 oa_version: Submitted Version page: 887 - 926 publication: Communications in Mathematical Physics publication_status: published publisher: Springer publist_id: '4901' quality_controlled: '1' status: public 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 type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 331 year: '2014' ...