  We present a square root limit on the amount of information transmitted
reliably and with low probability of detection (LPD) over additive white
Gaussian noise (AWGN) channels. Specifically, if the transmitter has AWGN
channels to an intended receiver and a warden, both with non-zero noise power,
we prove that $o(\sqrt{n})$ bits can be sent from the transmitter to the
receiver in $n$ channel uses while lower-bounding $\alpha+\beta\geq1-\epsilon$
for any $\epsilon>0$, where $\alpha$ and $\beta$ respectively denote the
warden's probabilities of a false alarm when the sender is not transmitting and
a missed detection when the sender is transmitting. Moreover, in most practical
scenarios, a lower bound on the noise power on the channel between the
transmitter and the warden is known and $O(\sqrt{n})$ bits can be sent in $n$
LPD channel uses. Conversely, attempting to transmit more than $O(\sqrt{n})$
bits either results in detection by the warden with probability one or a
non-zero probability of decoding error at the receiver as $n\rightarrow\infty$.
