Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs

In the setting of Square-Integrable Noncommutative Laws: Standing Notation, let $d\in\mathbb{N}$. Tracial W*-probability spaces are written $(H,M,\Omega)$, the letter $M$ naming a set of operators while $M(\lambda)$ remains the second moment of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws; $L^{2}$ tuples, their differences and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$, pairs $(X,Y)$ and laws $\mathrm{law}(X)$ and $\mathrm{law}(X,Y)$ are those of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws.

1. (Laws)¶ For every $\mu\in\Sigma^{2}_{d}$ there are a tracial W*-probability space $(H,M,\Omega)$ and an $L^{2}$ $d$-tuple $X$ of it with $\mathrm{law}(X)=\mu$.

2. (Couplings)¶ For every $\gamma\in\Sigma^{2}_{2d}$ there are a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,Y$ of it with $\mathrm{law}(X,Y)=\gamma$. For all such $X$ and $Y$, $\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\gamma$, $\mathrm{law}(Y)=\mathrm{pr}^{2}_{\#}\gamma$ and $\lVert X-Y\rVert_{2}^{2}=\mathcal{I}(\gamma)$.

3. (Almost optimal pairs)¶ For all $\mu,\nu\in\Sigma^{2}_{d}$ and every real $\varepsilon>0$ there are a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,Y$ of it with $\mathrm{law}(X)=\mu$, $\mathrm{law}(Y)=\nu$ and $\lVert X-Y\rVert_{2}^{2}\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon$.