The Distance between Square-Integrable Noncommutative Laws is the Infimum of the Cost over Their Couplings

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$, and let $\Sigma^{2}_{d}$, $\widehat{W}_{2}$, $\Pi^{2}$ and $\mathcal{I}$ be the $L^{2}$ laws of $d$ variables, their metric, the $L^{2}$ couplings and the cost. Let $\mu,\nu\in\Sigma^{2}_{d}$.

1. (Almost optimal couplings)¶ For every real $\varepsilon>0$ there is $\gamma\in\Pi^{2}(\mu,\nu)$ with $\mathcal{I}(\gamma)\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon$.

2. (Infimum)¶ $\Pi^{2}(\mu,\nu)$ is nonempty, and $\widehat{W}_{2}(\mu,\nu)^{2}$ is the infimum of the set $\{\mathcal{I}(\gamma):\ \gamma\in\Pi^{2}(\mu,\nu)\}$.