  We demonstrate the achievability of a square root limit on the amount of
information transmitted reliably and with low probability of detection (LPD)
over the single-mode lossy bosonic channel if either the eavesdropper's
measurements or the channel itself is subject to the slightest amount of excess
noise. Specifically, Alice can transmit $\mathcal{O}(\sqrt{n})$ bits to Bob
over $n$ channel uses such that Bob's average codeword error probability is
upper-bounded by an arbitrarily small $\delta>0$ while a passive eavesdropper,
Warden Willie, who is assumed to be able to collect all the transmitted photons
that do not reach Bob, has an average probability of detection error that is
lower-bounded by $1/2-\epsilon$ for an arbitrarily small $\epsilon>0$. We
analyze the thermal noise and pure loss channels. The square root law holds for
the thermal noise channel even if Willie employs a quantum-optimal measurement,
while Bob is equipped with a standard coherent detection receiver. We also show
that LPD communication is not possible on the pure loss channel. However, this
result assumes Willie to possess an ideal receiver that is not subject to
excess noise. If Willie is restricted to a practical receiver with a non-zero
dark current, the square root law is achievable on the pure loss channel.
