The Free Fisher Information of a Noncommutative Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\lambda\in\Sigma_{d}$ have conjugate variables $\xi_{\lambda}=(\xi_{1},\dots,\xi_{d})\in\mathcal{H}_{\lambda}^{d}$, which are unique by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate.

¶The free Fisher information of $\lambda$ is the real number

$$\Phi^{*}(\lambda)=\sum_{j=1}^{d}\lVert\xi_{j}\rVert_{\mathcal{H}_{\lambda}}^{2}.$$