Quantum metrology beyond the quantum Cramér-Rao theorem

A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cramér-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate bound to precision of quantum measurements. We investigate physical situations where there is an intrinsic dependence of the measurement strategy on the parameter and find that quantum-enhanced measurements may be more precise than previously thought.

Figure 1

QFI $J\left(g\right)$ (solid), the FI $F_H\left(g\right)$ for an energy measurement (dashed), and the upper bound to the FI, that is the term ${(\sqrt{J}+\sqrt{{\mathcal{K}}_X})}^2$ of Eq. (15) (dot-dashed). As it is apparent from the plot, the QFI $J$ does not provide the ultimate limits to precision, since $J<F_H$ periodically with period $T=2\pi /\omega$. For this figure the frequency $\omega$, the mass $m$, and the displacement $\delta x$ were set to unity.