The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings

Let $d$ be a natural number, let $\Sigma_{d}$ be the set of noncommutative laws of $d$ variables, and for $\mu,\nu\in\Sigma_{d}$ let $\Pi(\mu,\nu)$ be the set of couplings of $\mu$ and $\nu$ and $I(\gamma)$ the cost of $\gamma\in\Pi(\mu,\nu)$. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §tensor and Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost, the set $\{I(\gamma):\ \gamma\in\Pi(\mu,\nu)\}$ is a nonempty set of nonnegative real numbers, so it has an infimum by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, and this infimum is nonnegative because $0$ is a lower bound of the set.

1. (Wasserstein distance)¶ The noncommutative quadratic Wasserstein distance between $\mu$ and $\nu$ is the nonnegative square root

2. (Optimal couplings)¶ A coupling $\gamma\in\Pi(\mu,\nu)$ is optimal if $I(\gamma)=W_{2}(\mu,\nu)^{2}$.