Displacement-Squeeze receiver for BPSK displaced squeezed vacuum states surpassing the coherent-states Helstrom bound under imperfect conditions

Abstract: We propose a displacement-squeeze receiver (DSR) for discriminating BPSK displaced squeezed vacuum states (S-BPSK). The receiver applies a displacement followed by a squeezing operation with the squeezing axis rotated by $\fracπ{2}$, and performs photon-number-resolving detection with a MAP threshold decision. This processing effectively increases the distinguishability of the input states by elongating their distance in phase space and reducing their population overlap in Fock basis. We show that for all signal energy N, $P_\text{err}^\text{DSR} \in \left[P_\text{HB}^\text{DSS}, 2P_\text{HB}^\text{DSS}\right]$, under equal priors and ideal condition. In the low-energy regime, DSR beats the S-BPSK SQL at $N \approx 0.3$ and drops below the coherent-state BPSK (C-BPSK) Helstrom bound at $N \approx 0.4$, reaching $P_\text{err}^\text{DSR} < 1\%$ near $N \approx 0.6$. Finally, we quantify performance under non-unit efficiency and dark counts, phase diffusion, and receiver thermal noise, with MAP threshold adaptation providing robustness across these nonidealities.