Duality for Optimal Couplings in Free Probability
Duality for Optimal Couplings in Free Probability
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DOI：
10.1007/s00220-022-04480-0
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发表时间：
2021-05
影响因子：
2.4
通讯作者：
W. Gangbo;David Jekel;Kyeongsik Nam;D. Shlyakhtenko
中科院分区：
文献类型：
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作者：
W. Gangbo;David Jekel;Kyeongsik Nam;D. Shlyakhtenko
我们研究了二次成本最优耦合的自由概率模拟，其中经典概率空间被tracial von Neumann代数所取代，概率测度被m-元组的非交换律所取代。我们证明了一个模拟的Monge-Kantorovich的对偶性，其特征在于最佳耦合的非交换法律方面的Biane和Voiculescu的非交换Wasserstein距离使用一种新型的凸函数。因此，我们证明了：如果（X，Y）是迹代数中的一对非交换随机变量的最优耦合m-元组，则对所有。最后，我们通过与量子信息理论和算子代数中的结果的联系来说明非交换最优耦合的微妙之处。例如，可以在有限维代数中实现的两个非交换定律可能仍然需要无限维代数来最佳耦合。此外，m-元组的非交换律的空间关于Wasserstein距离是不可分的。
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 onare replaced by non-commutative laws ofm-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-Wasserstein distance using a new type of convex functions. As a consequence, we show that if (X,Y) is a pair of optimally coupledm-tuples of non-commutative random variables in a tracial-algebra, thenfor all. 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 ofm-tuples is not separable with respect to the Wasserstein distance for.