Let K be an irreducible and reversible Markov kernel on a finite set X . We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in R n by Jordan, Kinderlehrer and Otto (1998). The metric W is similar to, but different from, the L 2 -Wasserstein metric, and is defined via a discrete variant of the Benamou-Brenier formula.
Abstract. 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, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy.Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry-Émery and Otto-Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.
Abstract. We study a class of ergodic quantum Markov semigroups on finitedimensional unital C * -algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.
Let C denote the Clifford algebra over R n , which is the von Neumann algebra generated by n self-adjoint operators Qj , j = 1, . . . , n satisfying the canonical anticommutation relations, QiQj + QjQi = 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.
We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional C *-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein-Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transportentropy inequalities, and spectral gap estimates.
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