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:

$(\Delta \theta)^2 \ge \frac 1 ,$

where $m$ is the number of independent repetitions, and F_ varrho,H/math> 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)^2_(\Delta B)^2_\ge \frac 1 4 , \langle i ,B\rangle_, ^2.

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

$(\Delta\theta)^2_A \ge \frac.$

Here

$(\Delta\theta)^2_A= \frac=\frac$

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_\left sum_k p_k (\Delta B)_^2\right\frac1 4 F_Q varrho, B

where F_Q varrho, B/math> is the quantum Fisher information.

References

{{DEFAULTSORT:Quantum Cramér-Rao bound
Quantum information science
Quantum optics