  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.
