Bounding the quantum limits of precision for phase estimation with loss and thermal noise

We consider the problem of estimating an unknown but constant carrier phase modulation $\theta$ using a general -- possibly entangled -- $n$-mode optical probe through $n$ independent and identical uses of a lossy bosonic channel with additive thermal noise. We find an upper bound to the quantum Fisher information (QFI) of estimating $\theta$ as a function of $n$, the mean and variance of the total number of photons $N_{\rm S}$ in the $n$-mode probe, the transmissivity $\eta$ and mean thermal photon number per mode ${\bar n}_{\rm B}$ of the bosonic channel. Since the inverse of QFI provides a lower bound to the mean-squared error (MSE) of an unbiased estimator $\tilde{\theta}$ of $\theta$, our upper bound to the QFI provides a lower bound to the MSE. It already has found use in proving fundamental limits of covert sensing, and could find other applications requiring bounding the fundamental limits of sensing an unknown parameter embedded in a correlated field.