deduced from ; see also Proposition \[derivatives\_relations\] in the appendix.

Following [@Voi-Entropy-V], we may proceed now by defining (non-microstates) free Fisher information.

Let $(M,\tau)$ be a tracial $W^\ast$-probability space and let self-adjoint elements $X_1,\dots,X_n \in M$ be given. We define their *(non-microstates) free Fisher information $\Phi^\ast(X_1,\dots,X_n)$* by $$\Phi^\ast(X_1,\dots,X_n) := \sum^n_{j=1} \|\xi_j\|_2^2$$ if a conjugate system $(\xi_1,\dots,\xi_n)$ for $(X_1,\dots,X_n)$ in the sense of Definition \[conjugate\_system\] exists, and we put $\Phi^\ast(X_1,\dots,X_n) := \infty$ if no such conjugate system for $(X_1,\dots,X_n)$ exists.

This $\Phi^\ast(X_1,\dots,X_n)$ is just the usual non-microstates free Fisher information as defined in [@Voi-Entropy-V]. However, we have now the advantage that it can be defined even without assuming the algebraic freeness of $X_1,\dots,X_n$ right from the beginning. Actually, our result can now be stated as follows: $\Phi^*(X_1,\dots,X_n)<\infty$ implies the absence of algebraic relations between $X_1,\dots,X_n$.

Let $(M,\tau)$ be a $W^\ast$-probability space and let self-adjoint elements $X_1,\dots,X_n\in M$ be given such that the condition $\Phi^\ast(X_1,\dots,X_n) < \infty$ is fulfilled. Part (a) of Theorem \[main1\] tells us then that $X_1,\dots,X_n$ do not satisfy any algebraic relation, which in other words means that the evaluation homomorphism ${\operatorname{ev}}_X$ induces an isomorphism between the abstract polynomial algebra ${\mathbb{C}}\langle Z_1,\dots, Z_n\rangle$ and the subalgebra ${\mathbb{C}}\langle X_1,\dots,X_n\rangle$ of 