  We investigate the fundamental limit of quantum-secure covert communication
over the lossy thermal noise bosonic channel, the quantum-mechanical model
underlying many practical channels. We assume that the adversary has unlimited
quantum information processing capabilities as well as access to all
transmitted photons that do not reach the legitimate receiver. Given existence
of noise that is uncontrolled by the adversary, the square root law (SRL)
governs covert communication: up to c*sqrt{n} covert bits can be transmitted
reliably in n channel uses. Attempting to surpass this limit results in
detection with unity probability as n approaches infinity. Here we present the
expression for c, characterizing the SRL for the bosonic channel. We also prove
that discrete-valued coherent state quadrature phase shift keying (QPSK)
constellation achieves the optimal c, which is the same as that achieved by a
circularly-symmetric complex-valued Gaussian prior on coherent state amplitude.
Finally, while binary phase shift keying (BPSK) achieves the Holevo capacity
for non-covert bosonic channels in the low received signal-to-noise ratio
regime, we show that it is strictly sub-optimal for covert communication.
