Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\Phi^{*}$ be the free Fisher information, let conjugate variables be as in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate, and let $\mathrm{sc}_{d,s}$ be the semicircular law of variance $s$.

1. (Cramer-Rao inequality)¶ If $\lambda\in\Sigma_{d}$ has conjugate variables, then $M(\lambda)>0$ and $\Phi^{*}(\lambda)\,M(\lambda)\ge d^{2}$.

2. (Semicircular laws)¶ For every real $s>0$, $\mathrm{sc}_{d,s}$ has conjugate variables $\bigl(s^{-1}\widehat{x_{1}},\dots,s^{-1}\widehat{x_{d}}\bigr)$, the classes being taken in $\mathcal{H}_{\mathrm{sc}_{d,s}}$, and $\Phi^{*}(\mathrm{sc}_{d,s})=d/s$.

3. (Closedness under weak-star limits)¶ Let $c\ge0$ be real, and let $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence in $\Sigma_{d}$ that converges weak-star to $\lambda\in\Sigma_{d}$, such that every $\lambda_{k}$ has conjugate variables and $\Phi^{*}(\lambda_{k})\le c$. Then $\lambda$ has conjugate variables and $\Phi^{*}(\lambda)\le c$.