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

Each result cited is universally quantified over the data in its own statement.

Preliminaries. Write $\sigma=\sigma_{\Gamma_{t}}:\mathcal{P}_{d}\to\mathcal{P}_{2d}$; by Affine Data and Affine Substitutions of Noncommutative Polynomials §substitution it is the substitution $\sigma_{a}$ of the tuple $a=(x_{1}+\sqrt{t}\,x_{d+1},\dots,x_{d}+\sqrt{t}\,x_{2d})$ of $\Gamma_{t}$ (The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity §datum). Thus $V$ is the operator of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for $\gamma$ and this $a$: $V\widehat{p}^{\,\mu}=\widehat{\sigma(p)}^{\,\gamma}$ for every $p\in\mathcal{P}_{d}$, and $V^{*}V=I$. Its adjoint $V^{*}$ satisfies $\langle V^{*}\zeta,\eta\rangle_{\mathcal{H}_{\mu}}=\langle\zeta,V\eta\rangle_{\mathcal{H}_{\gamma}}$ for $\zeta\in\mathcal{H}_{\gamma}$ and $\eta\in\mathcal{H}_{\mu}$, and inner products are linear in the second and conjugate-linear in the first argument (Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces). Since $\gamma\in\Sigma_{2d}$, there is a real $r>0$ with $\gamma\in\Sigma_{2d,r}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), so Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum gives $\langle\widehat{p}^{\,\gamma},\widehat{q}^{\,\gamma}\rangle_{\mathcal{H}_{\gamma}}=\gamma(p^{*}q)$ for $p,q\in\mathcal{P}_{2d}$. As $t>0$, $\sqrt{t}$ is a real number with $\sqrt{t}\ge0$ and $\sqrt{t}\sqrt{t}=t\neq0$, so $\sqrt{t}>0$ and $1/\sqrt{t}$ is a positive real number.

The Schwinger-Dyson equation for $\gamma$. By The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product, $\gamma=\lambda\star\mathrm{sc}_{d}$ satisfies $\gamma\circ\iota^{1}=\lambda$. Hence Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations §characterisation, applied with $m=n=d$ and $\alpha=\lambda$, shows that $\gamma$ satisfies the Schwinger-Dyson equation in the last $d$ variables:

$$\gamma(x_{d+j}q)=\partial^{\gamma}_{d+j}(q)\qquad(j\in[d],\ q\in\mathcal{P}_{2d}),$$

where $\partial^{\gamma}_{d+j}$ is the free difference quotient of $\gamma$.

Clause 1. Fix $j\in[d]$ and put $\zeta_{j}=\widehat{x_{d+j}}^{\,\gamma}\in\mathcal{H}_{\gamma}$ and $\xi_{\mu,j}=\frac{1}{\sqrt{t}}V^{*}\zeta_{j}\in\mathcal{H}_{\mu}$. Let $p\in\mathcal{P}_{d}$. Because $1/\sqrt{t}$ is real, the inner product is conjugate-linear in its first argument, and by the defining property of $V^{*}$ and of $V$,

$$\langle\xi_{\mu,j},\widehat{p}^{\,\mu}\rangle_{\mathcal{H}_{\mu}}=\frac{1}{\sqrt{t}}\langle V^{*}\zeta_{j},\widehat{p}^{\,\mu}\rangle_{\mathcal{H}_{\mu}}=\frac{1}{\sqrt{t}}\langle\widehat{x_{d+j}}^{\,\gamma},\widehat{\sigma(p)}^{\,\gamma}\rangle_{\mathcal{H}_{\gamma}}=\frac{1}{\sqrt{t}}\,\gamma\bigl(x_{d+j}^{*}\,\sigma(p)\bigr)=\frac{1}{\sqrt{t}}\,\gamma\bigl(x_{d+j}\,\sigma(p)\bigr),$$

using the formula from Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and $x_{d+j}^{*}=x_{d+j}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint). By the Schwinger-Dyson equation for $\gamma$ above, this equals $\frac{1}{\sqrt{t}}\,\partial^{\gamma}_{d+j}(\sigma(p))$. Now $\mu=\gamma\circ\sigma_{\Gamma_{t}}$ by The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity §realisation, so the chain rule, applied to the affine datum $\Gamma_{t}=(A^{t},0)$ from $2d$ to $d$ variables and to $\gamma$, gives

$$\partial^{\gamma}_{d+j}(\sigma(p))=\sum_{i=1}^{d}A^{t}_{i,d+j}\,\partial^{\mu}_{i}(p)=\sqrt{t}\,\partial^{\mu}_{j}(p),$$

because, by The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity §datum, $A^{t}_{j,d+j}=\sqrt{t}$, while for $i\in[d]$ with $i\neq j$ the column index $d+j$ differs both from $i$ (as $d+j>d\ge i$) and from $d+i$, so $A^{t}_{i,d+j}=0$. Consequently $\langle\xi_{\mu,j},\widehat{p}^{\,\mu}\rangle_{\mathcal{H}_{\mu}}=\frac{1}{\sqrt{t}}\sqrt{t}\,\partial^{\mu}_{j}(p)=\partial^{\mu}_{j}(p)$ for every $j\in[d]$ and $p\in\mathcal{P}_{d}$. By The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate, $\mu$ has conjugate variables, and by the uniqueness stated there they are $\xi_{\mu}=(\xi_{\mu,1},\dots,\xi_{\mu,d})$ with $\xi_{\mu,j}=\frac{1}{\sqrt{t}}V^{*}\widehat{x_{d+j}}^{\,\gamma}$.

Clause 2. $V^{*}$ does not increase norms. For $\eta\in\mathcal{H}_{\mu}$, the defining property of $V^{*}$ and $V^{*}V=I$ give $\lVert V\eta\rVert^{2}=\langle V\eta,V\eta\rangle_{\mathcal{H}_{\gamma}}=\langle V^{*}V\eta,\eta\rangle_{\mathcal{H}_{\mu}}=\lVert\eta\rVert^{2}$, so $\lVert V\eta\rVert=\lVert\eta\rVert$. Let $\zeta\in\mathcal{H}_{\gamma}$. Then $\lVert V^{*}\zeta\rVert^{2}=\langle V^{*}\zeta,V^{*}\zeta\rangle_{\mathcal{H}_{\mu}}=\langle\zeta,VV^{*}\zeta\rangle_{\mathcal{H}_{\gamma}}$, a nonnegative real number, and the Cauchy-Schwarz inequality (taking nonnegative square roots) gives

$$\lVert V^{*}\zeta\rVert^{2}=\bigl|\langle\zeta,VV^{*}\zeta\rangle_{\mathcal{H}_{\gamma}}\bigr|\le\lVert\zeta\rVert\,\lVert VV^{*}\zeta\rVert=\lVert\zeta\rVert\,\lVert V^{*}\zeta\rVert .$$

If $\lVert V^{*}\zeta\rVert=0$ then $\lVert V^{*}\zeta\rVert\le\lVert\zeta\rVert$ trivially; otherwise dividing by $\lVert V^{*}\zeta\rVert>0$ gives the same inequality.

The norm of $\zeta_{j}$. For $j\in[d]$, 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 §vacuum, Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and the Schwinger-Dyson equation for $\gamma$ with $q=x_{d+j}$,

$$\lVert\zeta_{j}\rVert^{2}=\gamma(x_{d+j}^{*}x_{d+j})=\gamma(x_{d+j}x_{d+j})=\partial^{\gamma}_{d+j}(x_{d+j})=\gamma(x_{\varnothing})\,\gamma(x_{\varnothing})=1,$$

where the fourth equality is the defining formula of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient for the word $(d+j)$ of length $1$, and the last uses $x_{\varnothing}=1$ (The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials) and $\gamma(1)=1$ ((a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state).

The bound. For $j\in[d]$, since $1/\sqrt{t}$ is real, sesquilinearity of the inner product gives $\lVert\xi_{\mu,j}\rVert^{2}=\frac{1}{t}\lVert V^{*}\zeta_{j}\rVert^{2}\le\frac{1}{t}\lVert\zeta_{j}\rVert^{2}=\frac{1}{t}$. By clause 1 and The Free Fisher Information of a Noncommutative Law §fisher,

$$\Phi^{*}(\mu)=\sum_{j=1}^{d}\lVert\xi_{\mu,j}\rVert_{\mathcal{H}_{\mu}}^{2}\le\sum_{j=1}^{d}\frac{1}{t}=\frac{d}{t}.$$