|1\rangle$) and momentum ($|R\rangle = |0\rangle, |L\rangle = |1\rangle$) qubits according to $$\begin{aligned}
\lefteqn{c_{HR}|HR\rangle + c_{HL}|HL\rangle + c_{VR}|VR\rangle + c_{VL}|VL\rangle \longrightarrow }
\nonumber \\
&&c_{HR}|HR\rangle + c_{HL}|HL\rangle + c_{VR}|VL\rangle + c_{VL}|VR\rangle,\end{aligned}$$ will transform the four-mode coherent-state input with eigenvalues $$\begin{aligned}
\lefteqn{\hspace*{-.5in}
\left[\begin{array}{cccc} \langle\hat{a}_{HR}\rangle & \langle\hat{a}_{HL}\rangle & \langle\hat{a}_{VR}\rangle & \langle\hat{a}_{VL}\rangle \end{array}\right] = }\nonumber \\
&&\hspace*{.25in}\left[\begin{array}{cccc} \alpha_{HR} & \alpha_{HL} & \alpha_{VR} & \alpha_{VL}\end{array}\right],\end{aligned}$$ into a four-mode coherent-state output with eigenvalues $$\begin{aligned}
\lefteqn{\hspace*{-.5in}\left[\begin{array}{cccc} \langle\hat{a}_{HR}\rangle & \langle\hat{a}_{HL}\rangle & \langle\hat{a}_{VR}\rangle & \langle\hat{a}_{VL}\rangle \end{array}\right] = }\nonumber \\
&&\hspace*{.25in}
\left[\begin{array}{cccc} \alpha_{HR} & \alpha_{HL} & \alpha_{VL} & \alpha_{VR}\end{array}\right],\end{aligned}$$ where the $\hat{a}$’s are annihilation operators for modes labeled by their polarization and beam positions. It follows that the coherent-state $P_E$ and $I_B$ calculations mimic the qubit derivations that we presented in Sec. III, with coherent-state inner products taking the place of qubit-state inner products. At low average photon number, these coherent-state results reduce to the qubit expressions for events which give rise to clicks in the photodetectors shown in Figs. 6–9.

Finally, a word about propagation loss, detector inefficiencies, and extraneous counts from dark current or background light is in order. All of these non-idealities actually help our Eve, in that they lead to a non-zero quantum bit error 