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

TL;DR

The paper tackles discriminating BPSK-displaced squeezed vacuum states (S-BPSK) and introduces a displacement-squeeze receiver (DSR) that uses Gaussian preprocessing (displacement and a orthogonal squeeze), followed by photon-number-resolving detection with MAP decision. The DSR converts the S-BPSK pair into an effectively larger separation in Fock space, yielding a closed-form ideal performance $P_{err}^{DSR} = \tfrac{1}{2} e^{-4N(N+1)}$ and guaranteeing $P_{err}^{DSR} \in [P_{HB}^{DSS}, 2P_{HB}^{DSS}]$ for all $N$, while surpassing the S-BPSK SQL in the low-photon-number regime (around $N\approx 0.3$) and beating the C-BPSK HB near $N\approx 0.4$. The authors analyze the impact of nonidealities—PNR inefficiency and dark counts, phase diffusion, and receiver thermal noise—and show that adaptive MAP thresholds preserve a practical advantage over the SQL across realistic operating conditions. The work provides explicit performance trends, threshold prescriptions, and robustness insights, with potential extensions to multi-ary constellations and hardware implementations in quantum communications using squeezed-state carriers.

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.

Figures (10)

• Figure 1: The ratios of $P_{\text{HB}}^{\text{DSS}}\left(N,\beta_{\text{opt}}\right)$, $P_{\text{SQL}}^{\text{DSS}}\left(N,\beta_{\text{opt}}\right)$ and $P_{\text{SQL}}^{\text{CS}}\left(N\right)$ to $P_{\text{HB}}^{\text{CS}}\left(N\right)$ vs. input state energy N. Here, $P_{\text{SQL}}^{\text{CS}}\left(N\right) = \frac{1}{2} \left[ 1-\mathrm{erf}\left( \sqrt{2\cdot N} \right) \right]$ and $P_{\text{HB}}^{\text{CS}}\left(N\right) = \frac{1}{2} \left[ 1- \sqrt{1-\exp\left (-4\cdot N \right )} \right]$.
• Figure 2: Schematic of the proposed DSR. The input state $\rho_i$ is transformed to $\zeta_i$ via a displacement followed by a squeezing operation, $U=S(-r)D(\alpha)$, and is then measured by a photon-number-resolving (PNR) detector with outcome $n$. The final decision is made according to the maximum a posteriori (MAP) criterion.
• Figure 3: Phase-space representations (a--c) and Fock-basis populations (d--f) of the input states $\rho_i$ in the DSR for mean photon number $N=1.0$ and squeezing fraction $\beta=\beta_{\text{opt}}$. Panels (a) and (d) show the input states $\rho_i$; panels (b) and (e) show the states after the displacement operation $D\rho_i D^\dagger$; and panels (c) and (f) show the states after the full DSR transformation $\zeta_i=U\rho_i U^\dagger$.
• Figure 4: Performance of the DSR with an SPD under ideal conditions. (a) Error probability of the DSR. (b) Ratio (in dB) of the benchmarks $P_{\text{HB}}^{\text{CS}}$, $P_{\text{SQL}}^{\text{CS}}$, $P_{\text{HB}}^{\text{DSS}}$, and $P_{\text{SQL}}^{\text{DSS}}$ relative to $P_{\text{err}}^{\text{DSR}}$.
• Figure 5: Performance of the DSR with an non-ideal PNR(M) which quantum efficiency $\eta<1$ and dark count $\nu>0$. (a) and (b) Error probability of the DSR. (c) Optimal threshold $n_{\text{th}}^*$. (d) Ratio (in dB) of the benchmark $P_{\text{SQL}}^{\text{DSS}}$ relative to $P_{\text{err}}^{\text{DSR}}$.