Plan Jets of Squared Wasserstein Distances at an Optimal Coupling of Bounded Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, bounded plans and the momentum norm $|\cdot|_{\mathrm{mom}}$ are those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, the plan pairing $\mathcal{J}$ that of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing, and the coupling pairings $\mathcal{J}^{1}_{\gamma}$, $\mathcal{J}^{2}_{\gamma}$ those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing; $J^{+}$ and $J^{-}$ are the plan superjet and the plan subjet with slack $0$; sums, differences, real multiples and the $L^{2}$ norm of $L^{2}$ $d$-tuples are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing. Let $\mu,\nu\in\Sigma_{d}$, let $\gamma\in\Pi(\mu,\nu)$ be an optimal coupling, and let $\alpha>0$ and $\beta\ge0$ be real. Let $T$ and $T'$ be the affine data from $2d$ to $2d$ variables with zero shift whose tuples are

$$(x_{1},\dots,x_{d},\ \alpha(x_{1}-x_{d+1}),\dots,\alpha(x_{d}-x_{2d}))\quad\text{and}\quad(x_{d+1},\dots,x_{2d},\ \alpha(x_{1}-x_{d+1}),\dots,\alpha(x_{d}-x_{2d})),$$

and let $\pi=\gamma\circ\sigma_{T}$ and $\pi'=\gamma\circ\sigma_{T'}$. Let $\varphi,\psi:\Sigma^{2}_{d}\to\mathbb{R}$ be

$$\varphi(\lambda)=\tfrac{\alpha}{2}\widehat{W}_{2}\bigl(\lambda,\kappa_{d}(\nu)\bigr)^{2}+\tfrac{\beta}{2}\widehat{W}_{2}\bigl(\lambda,\kappa_{d}(\mu)\bigr)^{2},\qquad\psi(\lambda)=-\tfrac{\alpha}{2}\widehat{W}_{2}\bigl(\kappa_{d}(\mu),\lambda\bigr)^{2}-\tfrac{\beta}{2}\widehat{W}_{2}\bigl(\lambda,\kappa_{d}(\nu)\bigr)^{2}.$$

1. (Plans)¶ $\pi$ is a bounded plan at $\mu$, $\pi'$ is a bounded plan at $\nu$, $|\pi|_{\mathrm{mom}}=|\pi'|_{\mathrm{mom}}=\alpha\,W_{2}(\mu,\nu)$, and $\kappa_{2d}(\pi)$ and $\kappa_{2d}(\pi')$ are plans at $\kappa_{d}(\mu)$ and at $\kappa_{d}(\nu)$ respectively.

2. (Superjet)¶ $\kappa_{2d}(\pi)\in J^{+}\varphi\bigl(\kappa_{d}(\mu)\bigr)$.

3. (Subjet)¶ $\kappa_{2d}(\pi')\in J^{-}\psi\bigl(\kappa_{d}(\nu)\bigr)$.

4. (Pairings)¶ For all $\zeta\in\mathcal{H}_{\mu}^{d}$ and $\zeta'\in\mathcal{H}_{\nu}^{d}$, $\mathcal{J}(\zeta,\pi)=\alpha\,\mathcal{J}^{1}_{\gamma}(\zeta)$ and $\mathcal{J}(\zeta',\pi')=\alpha\,\mathcal{J}^{2}_{\gamma}(\zeta')$.

5. (Realisation)¶ With $M(\lambda)=\sum_{j=1}^{d}\lambda(x_{j}x_{j})$ the second moment of $\lambda\in\Sigma_{d}$ as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, there are a tracial W*-probability space $(K,N,\Psi)$ and $L^{2}$ $d$-tuples $X,Y$ of $(K,N,\Psi)$ with

$$\mathrm{law}\bigl(X,\alpha(X-Y)\bigr)=\kappa_{2d}(\pi),\quad\mathrm{law}\bigl(Y,\alpha(X-Y)\bigr)=\kappa_{2d}(\pi'),\quad\lVert X-Y\rVert_{2}=W_{2}(\mu,\nu),\quad\lVert X\rVert_{2}^{2}=M(\mu),\quad\lVert Y\rVert_{2}^{2}=M(\nu).$$