Quantum Fisher information matrix

Fix a unit reference state $\psi$ ($\langle\psi|\psi\rangle=1$) and write $\langle X\rangle:=\langle\psi|X|\psi\rangle$. For bare, $\theta$-independent Hermitian generators $H_1,\dots,H_M$ ($H_j=H_j^{\dagger}$), define the centred states $|c_j\rangle:=(H_j-\langle H_j\rangle)|\psi\rangle$. The quantum Fisher information matrix $F\in\mathbb R^{M\times M}$ is $$[F]_{jk}=4\operatorname{Re}\!\big(\langle H_jH_k\rangle-\langle H_j\rangle\langle H_k\rangle\big)=4\operatorname{Re}\langle c_j|c_k\rangle.$$ As four times the real part of the Gram matrix $\langle c_j|c_k\rangle$, it is symmetric positive semidefinite, $F=F^{\top}\succeq 0$. If the generators lie in the real span of a Hermitian basis of the dynamical Lie algebra $\mathfrak g$, then $\operatorname{rank}F\le\dim\mathfrak g$, and in every case $\operatorname{rank}F\le M$. The generators are the bare reference-frame $H_j$, not the Heisenberg-rotated $U_j^{\dagger}H_jU_j$; identifying $F$ with the Fubini--Study metric is a named bridge hypothesis, not proved here.