rate between Alice and Bob in the absence of the FPB attack. If Eve’s $P_E$ value is set below that baseline error rate, then her presence should be undetectable.

The authors acknowledge useful technical discussions with Howard Brandt, Jonathan Smith and Stewart Personick. This work was supported by the Department of Defense Multidisciplinary University Research Initiative program under Army Research Office grant DAAD-19-00-1-0177 and by MIT Lincoln Laboratory.

QND Measurement
===============

Here we show that it is possible, in principle, to use cross-phase modulation between a strong coherent-state probe beam and an arbitrarily-polarized signal beam to make a QND measurement of the signal beam’s total photon number. Let $\{\hat{a}_H, \hat{a}_V, \hat{a}_P\}$ be the annihilation operators of the horizontal and vertical polarizations of the signal beam and the (single-polarization) probe beam, respectively at the input to a cross-phase modulation interaction. We shall take that interaction to transform these annihilation operators according to the following commutator-preserving unitary operation, $$\begin{aligned}
\hat{a}_H &\longrightarrow& \hat{a}'_H \equiv \exp(i\kappa \hat{a}_P^\dagger\hat{a}_P)\hat{a}_H \\ 
\hat{a}_V &\longrightarrow& \hat{a}'_V \equiv \exp(i\kappa \hat{a}_P^\dagger\hat{a}_P)\hat{a}_V\\
\hat{a}_P &\longrightarrow&\hat{a}'_P \equiv  \exp[i\kappa(\hat{a}_H^\dagger\hat{a}_H + \hat{a}_V^\dagger\hat{a}_V)]\hat{a}_P,  \end{aligned}$$ where $0 < \kappa \ll 1$ is the cross-phase modulation coupling coefficient. When the probe beam is in a strong coherent state, $|\sqrt{N}_P\rangle$ 