Overparametrization: QFIM-rank bound

For a QNN with $M$ parameters and dynamical Lie algebra $\mathfrak g$, let $F(\theta)\in\mathbb R^{M\times M}$ be the quantum Fisher information matrix and $R(M):=\sup_\theta\operatorname{rank}F(\theta)$ the achievable QFIM rank. Then $R(M)$ is bounded by the algebra dimension and non-decreasing in $M$: $$R(M)\le\dim\mathfrak g,\qquad M\le M'\ \Rightarrow\ R(M)\le R(M').$$ Overparametrization therefore persists: once the achievable rank saturates at $M$, it stays saturated for every $M'\ge M$.