A dual formula for the noncommutative transport distance

Abstract

In this article we study the noncommutative transport distance introduced by Carlen and Maas and its entropic regularization defined by Becker and Li. We prove a duality formula that can be understood as a quantum version of the dual Benamou-Brenier formulation of the Wasserstein distance in terms of subsolutions of Hamilton-Jacobi-Bellmann equation.

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

“…This infimum is an analog of the Benamou-Brenier formula for the classical Wasserstein 2-distance [1]. More recently, Wirth [68] provided a dual formulation of this distance (albeit in the case of a primitive quantum Markov semigroup) : for any two states…”

“…One can observe that for each j, the multiplication operator [ρ j ] is an operator mean in section 2.3. Indeed, [23,68] studied the quantum transport metric by replacing [ρ] ω by a family of general operator means…”

“…for a family of operator mean Λ j , j ∈ J . The simpler setting (39) is sufficient for symmetric quantum Markov semigroup on finite von Neumann algebra [67] and the second setting ( 40) is needed for finite dimensional GNS-symmetric semigroups as in [22,23,68] . In both cases, we define the following weighted norm on H:…”

“…The Connes distance T L can then be recovered as a special case after taking B = {(x, x)| x ∈ M R , L(x) ≤ 1}. Interestingly, due to a recent result by Wirth [68], the quantum Wasserstein 2-distance of Carlen and Maas also fits into this framework. This allows us to take a unified approach to Ricci curvature on both quantum Wasserstein 1 and 2-distances: we say that a triple (M, B, P) satisfies a coarse Ricci curvature lower bound κ ∈ R if for any two states ω 1 , ω 2 ,…”

A Non-Commutative Entropic Optimal Transport Approach to Quantum Composite Systems at Positive Temperature

“…In the work [8], the authors study the case of ε = 0 temperature and prove a duality result for the non-commutative problem in the very same spirit of the Kantorovich duality for the classical Monge problem. The recent work [53] studies the entropic quantum optimal transport problem as well, adopting, in constrast to our static approach, a dynamical formulation. Therein, the author proves a dynamical duality result at positive and zero temperature.…”