The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation

Let $\mathbb{N}$ be the set of natural numbers, let $d\in\mathbb{N}$, let $R>0$ be real, let $\Sigma_{d,R}$ be the set of tracial states on $\mathcal{P}_{d}=\mathbb{C}\langle x_{1},\dots,x_{d}\rangle$ with norm bound $R$, and let weak-star convergence be that of Weak-Star Convergence of Noncommutative Laws §weak-star. Let $\Pi(\mu,\nu)$ and $I$ be the couplings and their cost, $W_{2}$ the noncommutative Wasserstein distance, and $\gamma_{t}$ the displacement interpolants of a coupling $\gamma$. For a tracial state $\lambda$ on $\mathcal{P}_{d}$ write $M(\lambda)=\sum_{j=1}^{d}\lambda(x_{j}^{2})$, where $x_{j}^{2}=x_{j}x_{j}$. Let $\mu,\nu\in\Sigma_{d,R}$.

1. (Optimal couplings exist)¶ There is an optimal coupling $\gamma\in\Pi(\mu,\nu)$.

2. (Symmetry)¶ $W_{2}(\mu,\nu)=W_{2}(\nu,\mu)$.

3. (Separation)¶ $W_{2}(\mu,\mu)=0$, and $W_{2}(\mu,\nu)=0$ holds only if $\mu=\nu$.

4. (Moment bound)¶ $W_{2}(\mu,\nu)^{2}\le2M(\mu)+2M(\nu)$.

5. (Weak-star lower semicontinuity)¶ Let $(\mu_{m})$ and $(\nu_{m})$ be sequences in $\Sigma_{d,R}$ converging weak-star to $\mu$ and $\nu$, and let $c$ be real with $W_{2}(\mu_{m},\nu_{m})\le c$ for every $m\in\mathbb{N}$. Then $W_{2}(\mu,\nu)\le c$.

6. (Displacement interpolation)¶ Let $\gamma\in\Pi(\mu,\nu)$ be optimal and let $t\in\mathbb{R}$ with $0\le t\le1$. Then $W_{2}(\mu,\gamma_{t})\le t\,W_{2}(\mu,\nu)$ and $W_{2}(\gamma_{t},\nu)\le(1-t)\,W_{2}(\mu,\nu)$.