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:

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

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 $$\vert\Psi_k\rangle$$

$$(\Delta A)^2_{\Psi_k}(\Delta B)^2_{\Psi_k}\ge \frac 1 4 |\langle i[A,B] \rangle_{\Psi_k}|^2. $$

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

$$(\Delta\theta)^2_A \ge \frac{1}{4\min_{\{p_k,\Psi_k\}}[\sum_k p_k (\Delta B)_{\Psi_k}^2]}. $$

Here

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

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

$$\min_{\{p_k,\Psi_k\}}\left[\sum_k p_k (\Delta B)_{\Psi_k}^2\right]=\frac1 4 F_Q[\varrho, B],$$

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