# A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy

@article{Wirth2018ANT, title={A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy}, author={Melchior Wirth}, journal={arXiv: Operator Algebras}, year={2018} }

We study quantum Dirichlet forms and the associated symmetric quantum Markov semigroups on noncommutative $L^2$ spaces. It is known from the work of Cipriani and Sauvageot that these semigroups induce a first order differential calculus, and we use this differential calculus to define a noncommutative transport metric on the set of density matrices. This construction generalizes both the $L^2$-Wasserstein distance on a large class of metric spaces as well as the discrete transport distance…

## 24 Citations

### Complete Gradient Estimates of Quantum Markov Semigroups

- MathematicsCommunications in Mathematical Physics
- 2021

A complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state is introduced, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance.

### The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups

- Mathematics
- 2022

A BSTRACT . We show that the generator of a GNS-symmetric quantum Markov semigroup can be written as the square of a derivation. This generalizes a result of Cipriani and Sauvageot for tracially…

### Complete logarithmic Sobolev inequalities via Ricci curvature bounded below

- MathematicsAdvances in Mathematics
- 2022

### Ju l 2 02 0 COMPLETE GRADIENT ESTIMATES OF QUANTUM MARKOV SEMIGROUPS

- Mathematics
- 2020

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity…

### Curvature-dimension conditions for symmetric quantum Markov semigroups

- MathematicsAnnales Henri Poincaré
- 2022

Two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras are introduced and a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommuter setting are proved.

### On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups

- Mathematics
- 2022

Cipriani and Sauvageot have shown that for any L 2 -generator L (2) of a tracially symmetric quantum Markov semigroup on a C*-algebra A there exists a densely deﬁned derivation δ from A to a Hilbert…

### Quantum optimal transport for approximately finite-dimensional $C^{*}$-algebras

- Mathematics
- 2019

We introduce optimal transport of states on AF-$C^{*}$-algebras, using a dynamic formulation of the metric in the spirit of Benamou-Brenier. Essential for having energy functionals is an extension of…

### Ricci curvature of quantum channels on non-commutative transportation metric spaces

- Mathematics
- 2021

Following Ollivier’s work [57], we introduce the coarse Ricci curvature of a quantum channel as the contraction of non-commutative metrics on the state space. These metrics are defined as a…

### RICCI CURVATURE OF QUANTUM CHANNELS ON NON-COMMUTATIVE TRANSPORTATION METRIC SPACES

- Mathematics
- 2021

Following Ollivier’s work [57], we introduce the coarse Ricci curvature of a quantum channel as the contraction of non-commutative metrics on the state space. These metrics are defined as a…

### Complete order and relative entropy decay rates

- Mathematics
- 2022

We prove that the complete modified log-Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its completely bounded mixing time. This implies that the complete modified…

## References

SHOWING 1-10 OF 134 REFERENCES

### Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

- Mathematics
- 2016

### Metric measure spaces with Riemannian Ricci curvature bounded from below

- Mathematics
- 2014

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler…

### L p -WASSERSTEIN DISTANCES ON STATE AND QUASI-STATE SPACES OF C ∗ -ALGEBRAS

- Mathematics
- 2015

We construct an analogue of the classical L p -Wasserstein distance for the state space of a C � -algebra. Given an abstract Lipschitz gauge on a C � -algebra A in the sense of Rieffel, one can…

### Ricci Curvature of Finite Markov Chains via Convexity of the Entropy

- Mathematics
- 2012

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott,…

### Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

- Mathematics
- 2014

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A general…

### Dirichlet Forms on Noncommutative Spaces

- Mathematics
- 2008

We show how Dirichlet forms provide an approach to potential theory of noncommutative spaces based on the notion of energy. The correspondence with KMS-symmetric Markovian semigroups is explained in…

### On some topics of analysis on noncommutative spaces

- Mathematics
- 2016

We consider a conservative Markov semigroup on a semi-finite W ∗-algebra. It is known that under some reasonable assumptions it is enough to determine a kind of differential structure on such a…

### A new class of transport distances between measures

- Mathematics
- 2009

We introduce a new class of distances between nonnegative Radon measures in $${\mathbb{R}^d}$$ . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances…

### Gradient flows of the entropy for jump processes

- Mathematics, Computer Science
- 2012

A new transportation distance between probability measures that is built from a Levy jump kernel is introduced via a non-local variant of the Benamou-Brenier formula and it is shown that the entropy is convex along geodesics.