\|Q_{k,1}\| \|Q_{k,2}\|\bigg|\ Q = \sum^N_{k=1} Q_{k,1} \otimes Q_{k,2}\bigg\}$$ for any $Q\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle^{\otimes 2}$, where the infimum is taken over all possible decompositions of $Q$ with $Q_{k,1},Q_{k,2}\in{\mathbb{C}}\langle X_1,\dots,X_n\rangle$ for $k=1,\dots,N$ and some $N\in{\mathbb{N}}$.

Whenever it becomes necessary to mention explicitly the dependence of $\|\cdot\|_\pi$ on the underlying set of generators $X=(X_1,\dots,X_n)$, we will also write $\|\cdot\|_{\pi,X}$ instead of $\|\cdot\|_\pi$.

\[Fisher-bound\] Let $(M,\tau)$ be a tracial $W^\ast$-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 satisfied. Let $(\xi_1,\dots,\xi_n)$ be the conjugate system for $(X_1,\dots,X_n)$. Then, for all $P\in{\mathbb{C}}\langle X_1,\dots,X_n \rangle$ and all $u,v\in{\operatorname{vN}}(X_1,\dots,X_n)$, we have $$\label{eq:Fisher-bound}
|\langle v^\ast (\partial_i P) u, Q\rangle| \leq 4 \|\xi_i\|_2 \bigl(\|Pu\|_2 \|v\| + \|u\| \|P^\ast v\|_2\bigr) \|Q\|_\pi$$ for all $Q\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle^{\otimes 2}$ and $i=1,\dots,n$. In particular, we have $$\begin{aligned}
\lefteqn{\sum^n_{i=1} |\langle v^\ast (\partial_i P) u, Q\rangle|^2}\\
 & \qquad \leq 16 \bigl(\|Pu\|_2 \|v\| + \|u\| \|P^\ast v\|_2\bigr)^2 \Phi^\ast(X_1,\dots,X_n) \|Q\|_\pi^2
\end{aligned}$$ for all $Q\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle^{\otimes 2}$.

Firstly, let us consider $u,v\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle$. In this particular case, we observe that for all $Q_1,Q_2\in{\mathbb{C}}\langle X_1,\dots,X_n\rangle$ $$\begin{aligned}
\langle Pu, & \partial_i^\ast(vQ_1 \otimes Q_2)\rangle\\
&= \langle \partial_i (Pu), vQ_1 \otimes Q_2\rangle\\
&= \langle (\partial_i P) u, vQ_1 \otimes Q_2\rangle + 