The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$ and let $W_{2}$ be the noncommutative Wasserstein distance on $\Sigma_{d}$.

1. (Triangle inequality)¶ For all $\mu,\nu,\rho\in\Sigma_{d}$,

2. (Metric)¶ $W_{2}$ is a metric on $\Sigma_{d}$, and for every real $R>0$ its restriction to $\Sigma_{d,R}\times\Sigma_{d,R}$ is a metric on $\Sigma_{d,R}$.