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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 Voiculescus 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 $mathrm{W}^*$-algebra $mathcal{A}$, then $mathrm{W}^*((1 - t)X + tY) = mathrm{W}^*(X,Y)$ 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 $m > 1$.
Let $(X_1,dots,X_m)$ be self-adjoint non-commutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,dots,S_m)$ be a free semicircular family. We show t
As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a
Regularization in Optimal Transport (OT) problems has been shown to critically affect the associated computational and sample complexities. It also has been observed that regularization effectively helps in handling noisy marginals as well as margina
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study t
With the development of human space exploration, the space environment is gradually filled with abandoned satellite debris and unknown micrometeorites, which will seriously affect capture motion of space robot. Hence, a novel fast collision-avoidance