Quantum Cramér–Rao bound

The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system:

$(\mathrm{\Delta }\theta {)}^2\ge \frac{1}{m{F}_{\mathrm{Q}}[\varrho ,H]},$

where $m$ is the number of independent repetitions, and $F_{\mathrm{Q}}[\varrho ,H]$ is the quantum Fisher information.

Here, $\varrho$ is the state of the system and $H$ is the Hamiltonian of the system. When considering a unitary dynamics of the type

$\varrho (\theta )=\exp (-iH\theta ){\varrho }_0\exp (+iH\theta ),$

where ${\varrho }_0$ is the initial state of the system, $\theta$ is the parameter to be estimated based on measurements on $\varrho (\theta ).$

Simple derivation from the Heisenberg uncertainty relation

Let us consider the decomposition of the density matrix to pure components as

$\varrho =\sum _k{p}_k|{\mathrm{\Psi }}_k⟩⟨{\mathrm{\Psi }}_k|.$

The Heisenberg uncertainty relation is valid for all $|{\mathrm{\Psi }}_k⟩$

$(\mathrm{\Delta }A{)}_{{\mathrm{\Psi }}_k}^2(\mathrm{\Delta }B{)}_{{\mathrm{\Psi }}_k}^2\ge \frac{1}{4}|⟨i[A,B]{⟩}_{{\mathrm{\Psi }}_k}{|}^2.$

From these, employing the Cauchy-Schwarz inequality we arrive at

$(\mathrm{\Delta }\theta {)}_A^2\ge \frac{1}{4\underset{\{p_k,{\mathrm{\Psi }}_k\}}{\min }[\sum _k{p}_k(\mathrm{\Delta }B{)}_{{\mathrm{\Psi }}_k}^2]}.$

Here

$(\mathrm{\Delta }\theta {)}_A^2=\frac{(\mathrm{\Delta }A{)}^2}{|{\partial }_{\theta }⟨A⟩{|}^2}=\frac{(\mathrm{\Delta }A{)}^2}{|⟨i[A,B]⟩{|}^2}$

is the error propagation formula, which roughly tells us how well $\theta$ can be estimated by measuring $A.$ Moreover, the convex roof of the variance is given as

$\underset{\{p_k,{\mathrm{\Psi }}_k\}}{\min }[\sum _k{p}_k(\mathrm{\Delta }B{)}_{{\mathrm{\Psi }}_k}^2]=\frac{1}{4}{F}_Q[\varrho ,B],$

where $F_Q[\varrho ,B]$ is the quantum Fisher information.