The Barycentric Decomposition of the Noncommutative Wasserstein Distance

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\mu,\nu\in\Sigma_{d}$. For $k\in\{d,2d\}$ and a law $\rho\in\Sigma_{k}$, $(\mathcal{H}_{\rho},\mathcal{M}_{\rho},\Omega_{\rho})$ is the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star; self-adjoint tuples $s$ in $\mathcal{M}_{\mu}$ and the law $\mathrm{law}(s)$ of such a tuple are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws.

Notation. Let $\gamma\in\Pi(\mu,\nu)$; then $\gamma\in\Sigma_{2d}$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound. Since $\iota^{1}$ is the substitution of $a=(x_{1},\dots,x_{d})$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals and $\gamma\circ\iota^{1}=\mu$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, $\mu=\gamma\circ\sigma_{a}$; let $E_{\gamma}:\mathcal{M}_{\gamma}\to\mathcal{M}_{\mu}$ be the conditional expectation of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation for the law $\gamma\in\Sigma_{2d}$ and this $a$, the operators $L_{p}$ acting on $\mathcal{H}_{\gamma}$ for $p\in\mathcal{P}_{2d}$ and on $\mathcal{H}_{\mu}$ for $p\in\mathcal{P}_{d}$. For $j\in[d]$ let

$$z^{\gamma}_{j}=E_{\gamma}\bigl(L_{x_{d+j}}\bigr)\in\mathcal{M}_{\mu};$$

here $L_{x_{d+j}}\in\mathcal{M}_{\gamma}$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, and $z^{\gamma}_{j}$ is self-adjoint, because $L_{x_{d+j}}^{*}=L_{x_{d+j}}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint and $E_{\gamma}$ commutes with adjoints by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §expectation. The operators $L_{x_{1}},\dots,L_{x_{d}}$ on $\mathcal{H}_{\mu}$ are self-adjoint elements of $\mathcal{M}_{\mu}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star. Write $\zeta_{\gamma}=\mathrm{law}(z^{\gamma})$ for the law of the self-adjoint $d$-tuple $z^{\gamma}=(z^{\gamma}_{1},\dots,z^{\gamma}_{d})$, and $\beta_{\gamma}=\mathrm{law}(L_{x_{1}},\dots,L_{x_{d}},z^{\gamma}_{1},\dots,z^{\gamma}_{d})$ for the law of this self-adjoint $2d$-tuple in $\mathcal{M}_{\mu}$.

1. (Barycentric coupling)¶ For every $\gamma\in\Pi(\mu,\nu)$, $\zeta_{\gamma}\in\Sigma_{d}$ and $\beta_{\gamma}\in\Pi(\mu,\zeta_{\gamma})$. If $\nu\in\Sigma_{d,R}$ for a real $R>0$, then $\zeta_{\gamma}\in\Sigma_{d,R}$.

2. (Orthogonal splitting of the cost)¶ For every $\gamma\in\Pi(\mu,\nu)$, $M(\zeta_{\gamma})\le M(\nu)$ and

$$I(\gamma)=I(\beta_{\gamma})+M(\nu)-M(\zeta_{\gamma}).$$

3. (Upper bound)¶ For every $\gamma\in\Pi(\mu,\nu)$,

$$W_{2}(\mu,\nu)^{2}\le W_{2}(\mu,\zeta_{\gamma})^{2}+M(\nu)-M(\zeta_{\gamma}).$$

4. (Optimal couplings)¶ If $\gamma\in\Pi(\mu,\nu)$ is optimal, then $\beta_{\gamma}$ is an optimal coupling of $\mu$ and $\zeta_{\gamma}$, and

$$W_{2}(\mu,\nu)^{2}=W_{2}(\mu,\zeta_{\gamma})^{2}+M(\nu)-M(\zeta_{\gamma}).$$

5. (Variational formula)¶ The set $\{W_{2}(\mu,\zeta_{\gamma})^{2}+M(\nu)-M(\zeta_{\gamma}):\ \gamma\in\Pi(\mu,\nu)\}$ has least element $W_{2}(\mu,\nu)^{2}$, attained at every optimal coupling $\gamma\in\Pi(\mu,\nu)$; optimal couplings exist by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained.