Duality for optimal couplings in free probability

Abstract

We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on ${\mathbb{R}}^m$ are replaced by non-commutative laws of $m$-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu's non-commutative $L^2$-Wasserstein distance using a new type of convex functions. As a consequence, we show that if $(X,Y)$ is a pair of optimally coupled $m$-tuples of non-commutative random variables in a tracial ${\mathrm{W}}^{\ast }$-algebra $\mathcal{A}$, then ${\mathrm{W}}^{\ast }((1-t)X+tY)={\mathrm{W}}^{\ast }(X,Y)$ for all $t\in (0,1)$. Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of $m$-tuples is not separable with respect to the Wasserstein distance for $m>1$.

arXiv comment60 pages, revised

References in corpus (18)

• Factorization and dilation problems for completely positive maps on von Neumann algebras
• A Mini-Introduction To Information Theory
• An asymptotic property of factorizable completely positive maps and the Connes embedding problem
• Topometric spaces and perturbations of metric structures
• Factoriality, type classification and fullness for free product von Neumann algebras
• Transport information geometry I: Riemannian calculus on probability simplex
• Free monotone transport without a trace
• Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla
• Unitary and orthogonal equivalence of sets of matrices
• A non-commutative Path Space approach to stationary free Stochastic Differential Equations
• Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity
• Enforceable operator algebras
• An Elementary Approach to Free Entropy Theory for Convex Potentials
• Lecture Notes on Noncommutative Lp-Spaces
• Free transportation cost inequalities for non-commutative multi-variables
• A Laplace Principle for Hermitian Brownian Motion and Free Entropy I: the convex functional case
• A Survey on Connes' Embedding Conjecture
• The Dyson and Coulomb games

Cited by in corpus (3)

• Wasserstein distance between noncommutative dynamical systems
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• Vanishing first cohomology and strong 1-boundedness for von Neumann algebras