Quantum Metrological Limits via a Variational Approach

The minimum achievable statistical uncertainty in the estimation of physical parameters is determined by the quantum Fisher information. Its computation for noisy systems is still a challenging problem. Using a variational approach, we present an equation for obtaining the quantum Fisher information, which has an explicit dependence on the mathematical description of the noise. This method is applied to obtain a useful analytical bound to the quantum precision in the estimation of phase-shifts under phase diffusion, which shows that the estimation uncertainty cannot be smaller than a noise-dependent constant.

Figure 1

Comparison between upper bound $C_Q^{\max }$ and the maximum quantum Fisher information ${\mathcal{F}}_Q^{\max }$ in Ref. 14 as a function of the average number of photons $N$. The dots stand for the values obtained in Ref. 14, the dashed line corresponds to the noiseless case (${\beta }^2=0$), and the full lines correspond to $C_Q^{\max }$. The inset displays the two quantities up to $N=30$, which was the range considered in Ref. 14. From bottom to top, ${\beta }^2=5\times {10}^{-4};5\times {10}^{-5};5\times {10}^{-6}$.