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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)?

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  • $\begingroup$ Just to add something. We can suppose we know the value of $\theta$, and for one $\theta$, we have a set optimal measurement based on $\theta$, for another $\theta$ we also have another set optimal measurement based on $\theta$. Hence it's optimal theoretically while useless experimentally. $\endgroup$
    – Sherlock
    Mar 11 at 6:59
  • $\begingroup$ @Sherlock this is true in classical metrology too. Take interferometry, where you measure a relative phase between two arms by subtracting the intensities at the output ports, the uncertainty depends on $\theta$ (you get the most information when the sinusoid is varying most quickly). A different setup that adds some known additional phase will force you toward the optimal relative phase, so the optimal setup experimentally can depend on you knowing the value of $\theta$, even classically $\endgroup$ Mar 11 at 14:18
  • $\begingroup$ @Sherlock and this is not always true. Let's say you're measuring loss by inputting single photons, the best thing to have is a detector that says you have 0 or 1 photon, regardless of the actual value of the loss parameter $\endgroup$ Mar 11 at 14:20

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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.

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  • $\begingroup$ Thanks! BTW, if I want to learn more theoretical theory of quantum metrology, are there some books that you can advise? $\endgroup$
    – narip
    Mar 11 at 11:01
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    $\begingroup$ @narip I wish I could. There's the older book by Helstrom "Quantum Detection and Estimation Theory" but I tend to prefer review articles... there's the paper by Paris arxiv.org/abs/0804.2981 (published in a journal that most places don't have access to), there's this recent review that I like by Sidhu+Kok doi.org/10.1116/1.5119961, and then there are lots more out there (the modern things tend to be about multiparameter metrology but they quote all the original works/results for single parameter stuff too) $\endgroup$ Mar 11 at 14:26

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