Is Quantum Cramer-Rao bound for single parameter always attainable?

Question

First I will give some background of Quantum Cramer-Rao bound. There is an amount called Fisher Information:$F(\lambda)=\sum_x{p\left( x|\lambda \right) \left( \partial _{\lambda}\ln p\left( x|\lambda \right) \right) ^2}$ where $p\left( x|\lambda \right)$ is probability distribution of result $x$ based on a to be estimated parameter $\lambda$. We can translate this into quantum formalism as follows.

In quantum mechanics, according to the Born rule we have $p(x \mid \lambda)=\operatorname{Tr}\left[\Pi_{x} \varrho_{\lambda}\right]$ where $\left\{\Pi_{x}\right\}, \int d x \Pi_{x}=\mathbb{I}$, are the elements of a positive operator-valued measure $(\mathrm{POVM})$ and $\varrho_{\lambda}$ is the density operator parametrized by the quantity we want to estimate. Introducing the Symmetric Logarithmic Derivative (SLD) $L_{\lambda}$ as the selfadjoint operator satistying the equation $$ \frac{L_{\lambda} \varrho_{\lambda}+\varrho_{\lambda} L_{\lambda}}{2}=\frac{\partial \varrho_{\lambda}}{\partial \lambda} $$ we have that $\partial_{\lambda} p(x \mid \lambda)=\operatorname{Tr}\left[\partial_{\lambda} \varrho_{\lambda} \Pi_{x}\right]=\operatorname{Re}\left(\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]\right) .$ The Fisher Information(similar to discrete case given at the beginning) is then rewritten as $$ F(\lambda)=\int d x \frac{\operatorname{Re}\left(\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]\right)^{2}}{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]} $$ For a given quantum measurement, i.e. a POVM $\left\{\Pi_{x}\right\}$, Fisher Information establish the classical bound on precision, which may be achieved by a proper data processing, e.g. by maximum likelihood, which is known to provide an asymptotically efficient estimator. On the other hand, in order to evaluate the ultimate bounds to precision we have now to maximize the Fisher information over the quantum measurements. We have $$ \begin{aligned} F(\lambda) & \leq \int d x\left|\frac{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]}{\sqrt{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}}\right|^{2} \\ &=\int d x\left|\operatorname{Tr}\left[\frac{\sqrt{\varrho_{\lambda}} \sqrt{\Pi_{x}}}{\sqrt{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}} \sqrt{\Pi_{x}} L_{\lambda} \sqrt{\varrho_{\lambda}}\right]\right|^{2} \\ & \leq \int d x \operatorname{Tr}\left[\Pi_{x} L_{\lambda} \varrho_{\lambda} L_{\lambda}\right] \\ &=\operatorname{Tr}\left[L_{\lambda} \varrho_{\lambda} L_{\lambda}\right]\\ &=\operatorname{Tr}\left[\varrho_{\lambda} L_{\lambda}^{2}\right] \end{aligned}\tag{1} $$ On the other hand, the second inequality above is based on the Schwartz inequality $\left|\operatorname{Tr}\left[A^{\dagger} B\right]\right|^{2} \leq \operatorname{Tr}\left[A^{\dagger} A\right] \operatorname{Tr}\left[B^{\dagger} B\right]$ applied to $A^{\dagger}=\sqrt{\varrho_{\lambda}} \sqrt{\Pi_{x}} / \sqrt{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}$ and $B=\sqrt{\Pi_{x}} L_{\lambda} \sqrt{\varrho_{\lambda}}$ and it is saturated when $$ \frac{\sqrt{\Pi_{x}} \sqrt{\varrho_{\lambda}}}{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x}\right]}=\frac{\sqrt{\Pi_{x}} L_{\lambda} \sqrt{\varrho_{\lambda}}}{\operatorname{Tr}\left[\varrho_{\lambda} \Pi_{x} L_{\lambda}\right]} \quad \forall \lambda, $$ This condition is satisfied iff $\left\{\Pi_{x}\right\}$ is made by the set of projectors over the eigenstates of $L_{\lambda}$, which, in turn, represents the optimal POVM to estimate the parameter $\lambda .$

Then I will give a detailed description of my specific question. In this literature, the author states that

For the single-parameter quantum estimations, the quantum Cramér–Rao bound can be attained with a theoretical optimal measurement.

But it seems that the optimal measurement stated in the background might depend on parameter $\lambda$ while in the background part we see that we use $\partial_{\lambda} p(x \mid \lambda)=\operatorname{Tr}\left[\partial_{\lambda} \varrho_{\lambda} \Pi_{x}\right]$ which imply that $\Pi_x$ is independent of $\lambda$. So based on the knowledge of the background, I can't see why we can always find a parameter independent POVM $\Pi_x$ to saturate the inequality(1)?

Answer 1

The answer given in the literature is always yes, this is guaranteed to be possible for single-parameter estimation. What you are noticing eventually gives rise to some cool things that I'll mention at the end here, but first it is important to explain why the literature is allowed to make these claims.

The Cramér-Rao lower bound is saturable in the asymptotic limit. What this means is that you can attain a minimal uncertainty in the situation where you are already pretty certain as to the actual value of the underlying parameter $\lambda$, as quantified by your estimator $\hat{\lambda}$ being sufficiently close to $\lambda$, with $|\lambda-\hat{\lambda}|\ll 1$ in some appropriate units.

In this scenario, we set up a POVM based on $\hat{\lambda}$, not on $\lambda$, because all we actually know is our value of the estimator! With this POVM comprised of the set of projectors over the eigenstates of $L_{\hat{\lambda}}$, we will approximately saturate the Cauchy-Schwarz inequality (no t in this Schwarz but I seldom remember that). Notably, the POVM does not change with the underlying parameters because we set it a priori! Yes, we may update our estimator $\hat{\lambda}$ and then use a new POVM updated to better match the actual eigenstates of $L_\lambda$, but never in the actual probability distribution for a given measurement do the POVM elements change with the underlying parameter $\lambda$, so we can saturate the quantum Cramér-Rao bound as advertised.

---

Now you unearth the fascinating question of what happens when, somehow, the POVM elements do change with the parameter itself, in that you don't personally get to choose the POVM elements but they are in some sense emergent from the dynamical process that you are trying to measure. This is not what one usually thinks of in measurement theory but is definitely possible. In that case, you are correct that the saturation proof falls apart. This was realized rather recently in Quantum metrology beyond the quantum Cramér-Rao theorem (arxiv version), where the authors show you get bounds other than Cramér-Rao when you have parameter-dependent POVM elements. In theory, one can beat the bound in this case for some scenarios.