Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\star$ be the free product (for $m=n=d$), $\boxplus$ the free additive convolution, $\mathrm{sc}_{d}$ and $\mathrm{sc}_{d,t}$ the semicircular law and the semicircular law of variance $t$, $\Phi^{*}$ the free Fisher information, and $\Gamma_{t}$ the affine datum of the free heat flow. Let $\lambda\in\Sigma_{d}$ and let $t>0$ be real. Put $\gamma=\lambda\star\mathrm{sc}_{d}\in\Sigma_{2d}$ and $\mu=\lambda\boxplus\mathrm{sc}_{d,t}$, so that $\mu=\gamma\circ\sigma_{\Gamma_{t}}$ by The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity §realisation. The tuple $(x_{1}+\sqrt{t}\,x_{d+1},\dots,x_{d}+\sqrt{t}\,x_{2d})$ of $\Gamma_{t}$ is self-adjoint by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint; let $V\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ be the isometry for it (here, as there, $V$ denotes an operator).

1. (Conjugate variables)¶ $\mu$ has conjugate variables $\xi_{\mu}=(\xi_{\mu,1},\dots,\xi_{\mu,d})$, namely

$$\xi_{\mu,j}=\frac{1}{\sqrt{t}}\,V^{*}\,\widehat{x_{d+j}}^{\,\gamma}\qquad(j\in[d]).$$

2. (Fisher bound)¶ $\Phi^{*}(\mu)\le d/t$.