What is the meaning of divergent quantum fisher information?

Question

I've recently come across this example, and it has me confused. I've computed the quantum fisher information (QFI) for a qubit initialized along the $\hat x$ axis ($|\uparrow_x\rangle=\frac1{\sqrt{2}}\left(|\uparrow\rangle+|\downarrow\rangle\right)$) which then undergoes a depolarizing channel parametrized by $\theta$ such that the state as a function of $\theta$ is \begin{equation} \rho(\theta)=|\uparrow_x\rangle\langle\uparrow_x|(1-\theta)+\frac\theta2I, \end{equation} where $I$ is the identity matrix. The QFI associated with sensing $\theta$ is \begin{equation} \mathcal F_Q[\rho(\theta)]=\frac1{2\theta-\theta^2}, \end{equation} meaning that when the state is initially pure, the QFI for measuring an infinitesimal change in $\theta$ diverges to infinity. Is this physically meaningful? If so, how should I interpret this and where might I expect this to show up?

Answer 1

One way to reason is by approximating the distribution near its maximum by a normal distribution, for which the Fisher information is the inverse of its covariance matrix: $$\mathcal{I}=\Sigma^{-1}.$$ If diagonalized, the Fisher information is a matrix with standard deviations on the diagonal. Then its divergence means that one of the standard deviations tends to infinity, meaning that some components of the measured vector are completely undefined/uncertain.

For more rigorous treatment, one needs to dive into regular parametric models, as stated here:

If the Fisher information matrix is positive definite for all θ, then the corresponding statistical model is said to be regular; otherwise, the statistical model is said to be singular.[19] Examples of singular statistical models include the following: normal mixtures, binomial mixtures, multinomial mixtures, Bayesian networks, neural networks, radial basis functions, hidden Markov models, stochastic context-free grammars, reduced rank regressions, Boltzmann machines.

Remark
I have seen this question (or a very similar one) in the Statistics community and didn't answer it, since my answer admittedly lacks the required statistical rigor. So I post it here FWIW.