Proof of The Conjugate Variables of a Free Entropy Penalty are Displacement Monotone along Optimal Couplings

Each result cited is applied with the data of its own statement. Let $\mu,\nu\in\mathcal{D}_{0}$ have conjugate variables $\xi_{\mu}=(\xi_{\mu,1},\dots,\xi_{\mu,d})$ and $\xi_{\nu}=(\xi_{\nu,1},\dots,\xi_{\nu,d})$, and let $\gamma\in\Pi(\mu,\nu)$ be optimal. By definition of a free entropy penalty, $\mathcal{D}_{0}\subseteq\Sigma_{d}$, so $\mu,\nu\in\Sigma_{d}$. As in Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws, $h_{j}=x_{j}-x_{d+j}\in\mathcal{P}_{2d}$ for $j\in[d]$, and every coupling of two laws in $\Sigma_{d}$ lies in $\Sigma_{2d}$.

Step 1 (The swapped coupling is optimal). Let $b=(x_{d+1},\dots,x_{2d},x_{1},\dots,x_{d})$, a $2d$-tuple in $\mathcal{P}_{2d}$, and let $s=\sigma_{b}:\mathcal{P}_{2d}\to\mathcal{P}_{2d}$ be its substitution. The entries of $b$ are variables, hence self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint. Put $\gamma'=\gamma\circ s$. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §swap, $\gamma'\in\Pi(\nu,\mu)$ and $I(\gamma')=I(\gamma)$. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are reals $R_{1},R_{2}>0$ with $\mu\in\Sigma_{d,R_{1}}$ and $\nu\in\Sigma_{d,R_{2}}$; with $R$ the larger of them, $\mu,\nu\in\Sigma_{d,R}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone, and then $W_{2}(\mu,\nu)=W_{2}(\nu,\mu)$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry. Since $\gamma$ is optimal in the sense of The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal,

$$I(\gamma')=I(\gamma)=W_{2}(\mu,\nu)^{2}=W_{2}(\nu,\mu)^{2},$$

so $\gamma'$ is an optimal coupling of $\nu$ and $\mu$. Moreover $\gamma,\gamma'\in\Sigma_{2d}$.

Step 2 (Two tangent inequalities). By Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient §tangent, applied to $\mu$ (which has conjugate variables), $\nu\in\mathcal{D}_{0}$ and the optimal coupling $\gamma\in\Pi(\mu,\nu)$,

$$\mathcal{E}_{0}(\nu)\ge\mathcal{E}_{0}(\mu)+\mathcal{J}^{1}_{\gamma}(\xi_{\mu}).\qquad(1)$$

By the same property, applied to $\nu$ (which has conjugate variables), $\mu\in\mathcal{D}_{0}$ and the optimal coupling $\gamma'\in\Pi(\nu,\mu)$ of Step 1,

$$\mathcal{E}_{0}(\mu)\ge\mathcal{E}_{0}(\nu)+\mathcal{J}^{1}_{\gamma'}(\xi_{\nu}).\qquad(2)$$

Step 3 (The swap isometry). Apply Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry with $m=n=2d$, the law $\gamma\in\Sigma_{2d}$ and the self-adjoint $2d$-tuple $b$; its marginal law is $\gamma\circ\sigma_{b}=\gamma'$. It gives $U\in\mathcal{L}(\mathcal{H}_{\gamma'},\mathcal{H}_{\gamma})$ with

$$U\widehat{p}^{\,\gamma'}=\widehat{s(p)}^{\,\gamma}\quad\text{for every }p\in\mathcal{P}_{2d},\qquad U^{*}U=I.$$

If $A$ is a linear map between complex Hilbert spaces with an adjoint $A^{*}$ satisfying $A^{*}A=I$, then for all $\zeta,\eta$ in its domain, by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint applied to the vectors $A\zeta$ and $\eta$,

$$\langle A\zeta,A\eta\rangle=\langle A^{*}A\zeta,\eta\rangle=\langle\zeta,\eta\rangle.\qquad(3)$$

This applies to $U$ and, since the maps of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries are those of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, to every $V^{i}_{\gamma}$ and $V^{i}_{\gamma'}$. If $A$ and $A'$ satisfy (3) and $A'A$ is defined, then $A'A$ satisfies (3) as well. For a linear $A$ satisfying (3), $\lVert A\zeta-A\eta\rVert=\lVert A(\zeta-\eta)\rVert=\lVert\zeta-\eta\rVert$, so $A$ is continuous.

Step 4 ($UV^{1}_{\gamma'}=V^{2}_{\gamma}$). Since $\gamma'\circ\iota^{1}=\nu=\gamma\circ\iota^{2}$, both $UV^{1}_{\gamma'}$ and $V^{2}_{\gamma}$ are linear maps from $\mathcal{H}_{\nu}$ to $\mathcal{H}_{\gamma}$. Let $p\in\mathcal{P}_{d}$. By Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries, $V^{1}_{\gamma'}\widehat{p}^{\,\nu}=\widehat{\iota^{1}(p)}^{\,\gamma'}$, hence $UV^{1}_{\gamma'}\widehat{p}^{\,\nu}=\widehat{s(\iota^{1}(p))}^{\,\gamma}$. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, $\iota^{1}$ is the substitution of the $d$-tuple $(x_{1},\dots,x_{d})$ in $\mathcal{P}_{2d}$; by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition, applied with this $d$-tuple and the $2d$-tuple $b$, one has $s(\iota^{1}(p))=\sigma_{c}(p)$ for the $d$-tuple $c$ with $c_{j}=s(x_{j})$, and $s(x_{j})=b_{j}=x_{d+j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. So $\sigma_{c}=\iota^{2}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, and with Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries again,

$$UV^{1}_{\gamma'}\widehat{p}^{\,\nu}=\widehat{\iota^{2}(p)}^{\,\gamma}=V^{2}_{\gamma}\widehat{p}^{\,\nu}\qquad\text{for every }p\in\mathcal{P}_{d}.$$

Define $T:\mathcal{P}_{d}\to\mathcal{H}_{\gamma}$ by $Tp=\widehat{\iota^{2}(p)}^{\,\gamma}$. It is complex-linear, as $\iota^{2}$ is linear by Substitution of Noncommutative Polynomials into the Variables §substitution and the class map is complex-linear by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry and The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes. By (3) for $V^{2}_{\gamma}$ and by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry for the form $h_{\nu}(p,q)=\nu(p^{*}q)$ of The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns, one has $\lVert Tp\rVert^{2}=\langle\widehat{p}^{\,\nu},\widehat{p}^{\,\nu}\rangle=h_{\nu}(p,p)$. So The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear, applied with the space $\mathcal{P}_{d}$, the form $h_{\nu}$ (whose completion is $\mathcal{H}_{\nu}$), $K=\mathcal{H}_{\gamma}$, $C=1$ and this $T$, gives exactly one continuous map $\mathcal{H}_{\nu}\to\mathcal{H}_{\gamma}$ sending $\widehat{p}^{\,\nu}$ to $Tp$ for every $p$. Both $UV^{1}_{\gamma'}$ and $V^{2}_{\gamma}$ are such maps (continuous by Step 3), so $UV^{1}_{\gamma'}=V^{2}_{\gamma}$.

Step 5 ($U$ reverses $\widehat{h_{j}}$). For $j\in[d]$, $s$ is linear and $s(x_{j})=x_{d+j}$, $s(x_{d+j})=x_{j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, so $s(h_{j})=x_{d+j}-x_{j}=-h_{j}$. Since the class map is complex-linear, $U\widehat{h_{j}}^{\,\gamma'}=\widehat{s(h_{j})}^{\,\gamma}=-\widehat{h_{j}}^{\,\gamma}$.

Step 6 (Transfer of the pairing). By Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing for $\gamma'\in\Pi(\nu,\mu)$, then (3) for $U$, then Steps 4 and 5, and finally linearity of the inner product in its second argument (Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces) together with $\operatorname{Re}(-z)=-\operatorname{Re}z$,

$$\mathcal{J}^{1}_{\gamma'}(\xi_{\nu})=\sum_{j=1}^{d}\operatorname{Re}\bigl\langle V^{1}_{\gamma'}\xi_{\nu,j},\widehat{h_{j}}^{\,\gamma'}\bigr\rangle_{\mathcal{H}_{\gamma'}}=\sum_{j=1}^{d}\operatorname{Re}\bigl\langle UV^{1}_{\gamma'}\xi_{\nu,j},U\widehat{h_{j}}^{\,\gamma'}\bigr\rangle_{\mathcal{H}_{\gamma}}=\sum_{j=1}^{d}\operatorname{Re}\bigl\langle V^{2}_{\gamma}\xi_{\nu,j},-\widehat{h_{j}}^{\,\gamma}\bigr\rangle_{\mathcal{H}_{\gamma}}=-\mathcal{J}^{2}_{\gamma}(\xi_{\nu}),$$

the last equality by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing for $\gamma\in\Pi(\mu,\nu)$.

Step 7 (Conclusion). By Step 6, (2) reads $\mathcal{E}_{0}(\mu)\ge\mathcal{E}_{0}(\nu)-\mathcal{J}^{2}_{\gamma}(\xi_{\nu})$. Adding it to (1), all quantities being real,

$$\mathcal{E}_{0}(\nu)+\mathcal{E}_{0}(\mu)\ge\mathcal{E}_{0}(\mu)+\mathcal{E}_{0}(\nu)+\mathcal{J}^{1}_{\gamma}(\xi_{\mu})-\mathcal{J}^{2}_{\gamma}(\xi_{\nu}),$$

and subtracting $\mathcal{E}_{0}(\mu)+\mathcal{E}_{0}(\nu)$ gives $\mathcal{J}^{1}_{\gamma}(\xi_{\mu})-\mathcal{J}^{2}_{\gamma}(\xi_{\nu})\le0$. $\square$