Heisenberg scaling in Gaussian quantum metrology

We address the issue of precisely estimating small parameters encoded in a general linear transformation of the modes of a bosonic quantum field. Such Bogoliubov transformations frequently appear in the context of quantum optics. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires nonclassical but not necessarily entangled states. Our method further allows us to quantify losses in precision arising from being able to monitor only finitely many modes, for which we identify a lower bound.

Figure 1

Parameter estimation scheme. To estimate the parameter $\theta$, a set of modes $k=\{k_1,k_2,...,k_N\}$ is prepared in some initial states ${|\psi \rangle }_k$, while all other modes are left in the vacuum. The unitary transformation $U\left(\theta \right)$ encodes the parameter in the final state $|\tilde{\psi }\rangle$, but measurements to determine $\theta$ can only be performed on the marginal ${\tilde{\rho }}_k$, while information in other modes is lost.