Inverse-Squeezing Kennedy Receiver for Near-Helstrom Discrimination of Displaced-Squeezed BPSK

TL;DR

The IS-Kennedy receiver tackles the problem of discriminating displaced squeezed BPSK states by applying an inverse-squeezing operation after a Kennedy displacement and performing MAP decision with a PNR detector. This approach effectively maps squeezing resources into an amplitude gain, yielding an error probability near the S-BPSK Helstrom bound within a constant ~3 dB across energies, and surpassing coherent-state limits in the low-photon-number regime ($N\approx0.6$). The authors develop a comprehensive non-ideal model including detector efficiency $\eta$, dark counts $\nu$, finite PNR resolution $M$, and inverse-squeezing mismatch, revealing robust thresholding behavior against dark counts and a parity-driven saturation floor under mismatch. Parity effects can be mitigated by higher PNR resolution, making the IS-Kennedy a practical near-optimal squeezed-state receiver with current technology. These results highlight a viable path to high-sensitivity quantum communication using Gaussian operations and photon counting, with potential experimental realizations and extensions to other Gaussian discrimination tasks.

Abstract

To address the discrimination problem of binary phase-shift keyed displaced squeezed vacuum states (S-BPSK), this paper proposes an Inverse-squeezing Kennedy (IS-Kennedy) receiver. This architecture incorporates an inverse-squeezing operator following the displacement operation of a conventional Kennedy receiver, mapping the S-BPSK signals onto equivalent large-amplitude coherent states. Furthermore, it employs a photon-number-resolving (PNR) detector to perform maximum a posteriori (MAP) decision-making. Theoretical analysis demonstrates that, under ideal conditions, the IS-Kennedy receiver effectively translates the transmitter's squeezing resources into a displacement gain at the receiver. Consequently, its error probability approaches the Helstrom bound across the entire energy spectrum, remaining within a constant factor of 3 dB. In the low-photon-number regime ($N \approx 0.6$), the proposed scheme surpasses the coherent-state limit, achieving an error rate below 1\%. Furthermore, this paper provides an in-depth analysis of system performance under non-ideal conditions, revealing the robustness of PNR detection against background dark counts and a characteristic ``parity photon-number step'' saturation effect arising from squeezing parameter mismatch.

Figures (6)

• 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, $P_{\text{SQL}}^{\text{CS}}\left(N\right) = \frac{1}{2} \left[ 1-\text{erf}\left( \sqrt{2\cdot N} \right) \right]$ 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 Inverse-squeezing Kennedy (IS-Kennedy) receiver. The input state $\rho_i$ is first processed by a Kennedy-type nulling displacement $D(\alpha)$, which maps the BPSK displaced squeezed states (S-BPSK) to a squeezed OOK pair (S-OOK). A subsequent inverse-squeezing operation $S(-r)$ converts the two hypotheses from the displaced squeezed alphabet into an amplified coherent-state alphabet (C-OOK), ideally $\{\ket{0},\;\ket{2\alpha e^{r}}\}$ in the matched case. The resulting state $\zeta_i$ is measured by a photon-number-resolving (PNR) detector yielding outcome $n$, and 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 IS-Kennedy 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 full IS-Kennedy transformation $\sigma_i=D(\alpha)\rho_i D^\dagger(\alpha)$. Panels (c) and (f) show the states after inverse-squeezing (IS) $\zeta_i=U\rho_i U^\dagger$ with $U=S(-r)D(\alpha)$.
• Figure 4: Performance of the IS-Kennedy under ideal conditions. (a) Error probability of the IS-Kennedy. (b) Ratio (in dB) of the benchmarks $P_{\text{HB}}^{\text{CS}}$, $P_{\text{SQL}}^{\text{CS}}$, $P_{\text{HB}}^{\text{DSS}}$, $P_{\text{SQL}}^{\text{DSS}}$, and $P_{\text{err}}^{\text{K},\text{ideal}}$ relative to $P_{\text{err}}^{\text{IS-K},\text{ideal}}$.
• Figure 5: Performance analysis of the IS-Kennedy receiver under imperfect detection conditions. (a) Error probability of IS-Kennedy with SPD for varying detection efficiencies $\eta \in \left\{1.0,0.8,0.5\right\}$ and extremely small dark counts ($\nu = 10^{-9}$). (b) Error probability of IS-Kennedy with PNR(2) under different dark count rates $\nu$ (with $\eta = 1.0$). The "step-like" behavior in the error curves arises from the discrete nature of the optimal decision threshold. (c) The optimal detection threshold $n_\text{th}^*$ as a function of signal energy for different dark count rates, calculated via Eq. \ref{['eq:threshold_imperfect_PNR']}. The jumps in threshold correspond exactly to the inflection points observed in panels (b) and (d). (d) The performance ratio (in dB) of the IS-Kennedy with PNR(10) relative to the S-BPSK SQL. The oscillatory behavior reflects the discrete adjustment of $n_\text{th}^*$ to balance signal detection against dark count noise.