### Abstract

Let C denote the Clifford algebra over R^{n}, 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 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 language | English (US) |
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Pages (from-to) | 887-926 |

Number of pages | 40 |

Journal | Communications In Mathematical Physics |

Volume | 331 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2014 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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**An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy.** / Carlen, Eric A.; Maas, Jan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy

AU - Carlen, Eric A.

AU - Maas, Jan

PY - 2014/11

Y1 - 2014/11

N2 - 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.

AB - 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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U2 - https://doi.org/10.1007/s00220-014-2124-8

DO - https://doi.org/10.1007/s00220-014-2124-8

M3 - Article

VL - 331

SP - 887

EP - 926

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -