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:

$(\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,H]}},$

where $m$ is the number of independent repetitions, and $F_{\rm {Q}}[\varrho ,H]$ 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

$\varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .$

The Heisenberg uncertainty relation is valid for all $\vert \Psi _{k}\rangle$

$(\Delta A)_{\Psi _{k}}^{2}(\Delta B)_{\Psi _{k}}^{2}\geq {\frac {1}{4}}|\langle i[A,B]\rangle _{\Psi _{k}}|^{2}.$

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

$(\Delta \theta )_{A}^{2}\geq {\frac {1}{4\min _{\{p_{k},\Psi _{k}\}}[\sum _{k}p_{k}(\Delta B)_{\Psi _{k}}^{2}]}}.$

Here

$(\Delta \theta )_{A}^{2}={\frac {(\Delta A)^{2}}{|\partial _{\theta }\langle A\rangle |^{2}}}={\frac {(\Delta A)^{2}}{|\langle i[A,B]\rangle |^{2}}}$

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

$\min _{\{p_{k},\Psi _{k}\}}\left[\sum _{k}p_{k}(\Delta B)_{\Psi _{k}}^{2}\right]={\frac {1}{4}}F_{Q}[\varrho ,B],$

where $F_{Q}[\varrho ,B]$ is the quantum Fisher information.