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:

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

where ${\displaystyle m}$ is the number of independent repetitions, and ${\displaystyle F_{\rm {Q}}[\varrho ,H]}$ is the quantum Fisher information.^[1][2]

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

${\displaystyle \varrho (\theta )=\exp(-iH\theta )\varrho _{0}\exp(+iH\theta ),}$

where ${\displaystyle \varrho _{0}}$ is the initial state of the system, ${\displaystyle \theta }$ is the parameter to be estimated based on measurements on ${\displaystyle \varrho (\theta ).}$

Simple derivation from the Heisenberg uncertainty relation

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

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

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

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

From these, employing the Cauchy–Schwarz inequality we arrive at ^[3]

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

Here ^[4]

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

is the error propagation formula, which roughly tells us how well ${\displaystyle \theta }$ can be estimated by measuring ${\displaystyle A.}$ Moreover, the convex roof of the variance is given as^[5][6]

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

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