Phase estimation without a priori phase knowledge in the presence of loss

We find the optimal scheme for quantum phase estimation in the presence of loss when no a priori knowledge on the estimated phase is available. We prove analytically an explicit lower bound on estimation uncertainty, which shows that, as a function of the number of probes, quantum precision enhancement amounts at most to a constant factor improvement over classical strategies.

Figure 1

Phase estimation setup. Channel $a$ acquires a phase $\phi$ relative to channel $b$. Losses are modeled by two beam splitters with power transmissions ${\eta }_a$ and ${\eta }_b$.

Figure 2

Log-log plot of optimal phase estimation uncertainty as a function of the number of photons used for three different levels of loss (equal in both arms): $\eta =1$ (solid line), $\eta =0.8$ (dashed line), $\eta =0.6$ (dotted line). The white area in the middle of the picture corresponds to $1/N<\delta \phi <1/\sqrt{N}$. Gray lines represent asymptotic bounds given by Eq. (12) for $\eta =0.8$ and $\eta =0.6$. The inset depicts the structure of the optimal states for the three levels of loss for $N=100$.