S‑BPSK: Displaced Squeezed Vacuum States

• S-BPSK are binary quantum signals generated by sequential displacement and squeezing operations on the vacuum, enabling enhanced noise reduction in a targeted quadrature.
• The method employs homodyne and displacement-squeeze receivers to optimize state discrimination, achieving error rates below the coherent-state Helstrom bound.
• Optimal squeezing allocation in S-BPSK yields exponential SNR improvements and robust performance even under realistic channel loss and detector noise.

Binary phase-shift keyed (BPSK) displaced squeezed vacuum states—hereafter S-BPSK—encode binary information onto quantum states that combine displacement and squeezing operations applied to the optical vacuum. S-BPSK protocols seek to optimize discrimination of binary-encoded quantum signals by leveraging squeezing-induced noise reduction along a selected quadrature, achieving measurement error rates that can outperform the quantum-limited performance of conventional coherent-state BPSK under certain conditions. Recent experimental and theoretical advances have demonstrated that, when paired with homodyne or photon-number-resolving detection architectures, S-BPSK schemes can surpass the coherent-state Helstrom bound using only Gaussian resources, and exhibit robustness to practical nonidealities (Waslh et al., 23 Oct 2025, Bai et al., 14 Jan 2026).

1. Definition of S-BPSK Signal States

The logical “$+$” and “$-$” symbols in S-BPSK are represented by pure Gaussian states generated by sequential displacement and quadrature squeezing on the vacuum: $$|\psi_{\pm}\rangle = D(\pm\alpha)\; S(r)\; |0\rangle,$$ where $D(\alpha) = \exp(\alpha a^{\dagger} - \alpha^* a)$ is the displacement operator (with $\alpha\in\mathbb R$ for canonical BPSK), $S(r) = \exp(\frac{r}{2}(a^2-a^{\dagger2}))$ is a real squeezing operator with $r > 0$, and $|0\rangle$ denotes the vacuum state.

In phase space, the Wigner function of each state is

$$W_{\pm}(x,p) = \frac{1}{2\pi\sqrt{V_x V_p}}\; \exp \Bigl[-\frac{(x \mp \sqrt{2}\,\beta)^2}{2 V_x} - \frac{p^2}{2 V_p}\Bigr],$$

with quadrature variances $V_x = \frac{1}{2} e^{-2r}$, $V_p = \frac{1}{2} e^{+2r}$, and centroid displacement

$\beta = \sqrt{\alpha^2 - \sinh^2 r},$

for $\alpha^2 > \sinh^2 r$. The total mean photon number is

$\bar n = |\alpha|^2 + \sinh^2 r.$

Parameterization in terms of the energy-splitting fraction $\gamma = \sinh^2 r/\bar n$ yields an optimal squeezing allocation

$$\gamma_{\mathrm{opt}} = \frac{\bar n}{2\bar n + 1},$$

for maximal signal-to-noise ratio (SNR) in the homodyne discrimination setting (Waslh et al., 23 Oct 2025, Bai et al., 14 Jan 2026).

2. State Discrimination and Measurement Architectures

The primary discrimination strategies for S-BPSK signals are homodyne detection and the displacement-squeeze receiver (DSR).

Homodyne Receiver

A homodyne receiver measures a selected quadrature $X_{\theta}$, making a binary decision based on whether the measured value is above or below zero. The optimal discrimination occurs at $\theta=0$ (the squeezed quadrature). The error probability is

$$P_{e}^{\mathrm{(opt)}} = \frac{1}{2}\, \mathrm{Erfc} \left[ \beta\, e^{r} \right] = \frac{1}{2}\, \mathrm{Erfc} \left[ \sqrt{\alpha^2 - \sinh^2 r}\; e^{r} \right].$$

For states optimally split between displacement and squeezing, this reduces to

$$P_e^{\rm (HOM)} = \frac{1}{2}\, \mathrm{Erfc} \left[ \sqrt{2(\bar n^2+\bar n)} \right].$$

This homodyne error rate can fall below the coherent-state Helstrom bound for $\bar n \gtrsim 0.67$ (Waslh et al., 23 Oct 2025).

Displacement–Squeeze Receiver (DSR)

The DSR architecture first applies a matched displacement $D(\alpha)$ to realign the phase-space centroids, then a squeezing $S(-r)$ along an axis orthogonal to the transmitter's squeezing. This sequence maps the S-BPSK signal set to an on-off keying pair $\{|0\rangle, |\gamma\rangle\}$ with enhanced separation $\gamma = 2\alpha\, e^{r}$, resulting in sharply distinguishable photon-number distributions.

Photon-number-resolving (PNR) detection with a maximum a posteriori (MAP) threshold is used. With equal priors and in the idealized limit, the threshold is $n_{\rm th}^* = 1$:

• “0” is declared if zero photons are detected,
• “1” is declared otherwise.

This yields an error probability

$$P_{\mathrm{err}}^{\mathrm{DSR}}(N) = \frac{1}{2}\exp\left[-4N(N+1)\right],$$

where $N = |\alpha|^2 + \sinh^2 r$.

The DSR performance satisfies

$$P_{\mathrm{HB}}^{\mathrm{DSS}} \leq P_{\mathrm{err}}^{\mathrm{DSR}} \leq 2P_{\mathrm{HB}}^{\mathrm{DSS}},$$

where $P_{\mathrm{HB}}^{\mathrm{DSS}}$ is the S-BPSK Helstrom bound,

$$P_{\mathrm{HB}}^{\mathrm{DSS}} = \frac{1}{2}\left(1-\sqrt{1-\exp[-4N(N+1)]}\right).$$

In the energy range $N \gtrsim 0.3$, DSR outperforms the S-BPSK standard quantum limit (SQL), and at $N \gtrsim 0.4$ it beats the coherent-state Helstrom bound (Bai et al., 14 Jan 2026).

3. Theoretical Performance Benchmarks

The S-BPSK framework enables quantum error rates below those achievable for coherent-state BPSK:

• The coherent-state Helstrom bound:

$$P_{\mathrm{HB}}^{\mathrm{C}} = \frac{1}{2}\left(1-\sqrt{1-\exp[-4N]}\right)$$

• S-BPSK homodyne error:

$$P_e^{\rm (HOM)} = \frac{1}{2}\, \mathrm{Erfc} \left[ \sqrt{2N(N+1)} \right]$$

• DSR error:

$$P_{\mathrm{err}}^{\mathrm{DSR}} = \frac{1}{2} \exp[-4N(N+1)]$$

For optimally chosen squeezing, the SNR enhances quadratically with $N$, yielding an exponential improvement in error rates. Homodyne detection outperforms the coherent-state Helstrom bound for $\bar n \gtrsim 0.67$; the DSR surpasses both the S-BPSK SQL and coherent-state Helstrom bound at lower energies ($N \gtrsim 0.3$–$0.4$), achieving sub-percent error near $N \approx 0.6$ (Waslh et al., 23 Oct 2025, Bai et al., 14 Jan 2026).

A numerical benchmark table summarizes key transitions:

N	$P_{\mathrm{HB}}^{\mathrm{C}}$	$P_{\mathrm{SQL}}^{\mathrm{DSS}}$	$P_{\mathrm{err}}^{\mathrm{DSR}}$

0.2 | 0.315 | 0.420 | 0.310

0.3 | 0.290 | 0.295 | 0.187

0.4 | 0.240 | 0.215 | 0.060

0.6 | 0.130 | 0.050 | 0.008

This demonstrates the crossover points and relative performance (Bai et al., 14 Jan 2026).

4. Impact of Channel Loss and Realistic Imperfections

Channel Loss

For optical transmissivity $T<1$, the effective SNR in S-BPSK homodyne scales by $T/(T + (1-T)e^{2r})$, rapidly degrading the squeezing advantage in high-loss regimes. A plausible implication is that strongly squeezed S-BPSK detection is most advantageous in low-loss regimes (Waslh et al., 23 Oct 2025).

Detector and Environmental Noise

Performance under realistic imperfections is captured by explicit models:

• Finite quantum efficiency $\eta$ and dark count rate $\nu$ modify photon-number statistics, but a suitable MAP threshold maintains DSR superiority over SQL for moderate values, e.g., $\eta=0.9$ and $\nu=10^{-2}$.
• Phase diffusion (Gaussian noise in optical phase) of width $\sigma$ degrades discrimination. Simulations at $\sigma=0.1$ show DSR retains performance advantage over SQL for $0.28 \lesssim N \lesssim 0.93$, with peak gain $\approx 3.3$\,dB.
• Receiver thermal noise with average photon number $n_t$ reduces distinguishability, but DSR maintains several-dB advantage for $n_t \lesssim 10^{-3}$ and moderate PNR resolution.

Experimental demonstrations confirm that with squeezing up to $\sim4$\,dB ($r \approx 0.46$) and $\bar n \gtrsim 1$, S-BPSK homodyne error rates fall below the coherent-state Helstrom limit (Waslh et al., 23 Oct 2025). With modern high-efficiency homodyne detection and squeezing beyond $10$\,dB, the regime $\bar n \sim 1$–$10$ becomes accessible for quantum advantage.

5. Key Overlap and Error Expressions

Essential analytical expressions underpinning S-BPSK performance include:

• State overlap:

$$\langle\psi_-|\psi_+\rangle = \exp\left[ -\left(\beta^2 + \sinh^2 r \right) \right]$$

• Signal-to-noise ratio for Gaussian quadratures:

$${\rm SNR} = \frac{(\Delta\langle X\rangle)^2}{\langle (\Delta X)^2 \rangle}$$

• Homodyne error in terms of SNR:

$$P_e = \frac{1}{2}\, {\rm Erfc}\left[ \sqrt{ {\rm SNR}/2 } \right]$$

For optimal S-BPSK splitting, ${\rm SNR} = 4(\bar n^2 + \bar n)$ (Waslh et al., 23 Oct 2025).

6. Significance and Outlook

S-BPSK leverages all-Gaussian resources to overcome fundamental quantum limits set by the Helstrom bound for traditional BPSK under pure-state encoding. Both homodyne and photon-number detection receivers adapted to S-BPSK states deliver exponential reductions in error probability with increasing photon number, making them advantageous for low-energy, high-fidelity quantum communication channels—provided that channel loss and noise are kept within regime boundaries. These findings position S-BPSK as a leading platform for quantum-enhanced binary optical communication, obviating the need for non-Gaussian measurements, photon counting, or feedback-based receivers, and offering robust performance against realistic device imperfections (Waslh et al., 23 Oct 2025, Bai et al., 14 Jan 2026).