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 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. We consider 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. Here, ${\displaystyle \theta }$ is the parameter to be estimated based on measurements on ${\displaystyle \varrho (\theta ).}$