All-Gaussian State Discrimination Beyond the Coherent Helstrom Bound

TL;DR

Problem: discriminating BPSK signals toward the Helstrom limit with coherent states is challenging in practice. Approach: the authors implement an all-Gaussian scheme—displaced squeezed states with homodyne detection—optimizing the energy partition $\gamma$ to maximize the SNR, achieving $\text{SNR}=4(\bar{n}-\bar{n}_s)(\sqrt{\bar{n}_s}+\sqrt{\bar{n}_s+1})^2$ and a maximum $\text{SNR}_{\max}=4(\bar{n}^2+\bar{n})$. Findings: the Gaussian scheme can surpass the coherent-state Helstrom bound for C-BPSK, and experimental demonstration confirms lower error than the coherent Helstrom bound, though channel loss constrains the advantage. Significance: establishes a practical all-Gaussian route to quantum-enhanced state discrimination with implications for high-signal-energy quantum communications and sensing.

Abstract

A core problem in communications is the optimal discrimination of binary-phase-shift-keyed (BPSK) signals. A longstanding goal has been to reach the fundamental quantum limit, known as the Helstrom bound, for BPSK signals encoded in coherent states. However, due to technical constraints, proposals for reaching the bound remain impractical. In this letter we take an alternative approach: using only Gaussian optics - displaced squeezed states and homodyne detection - we achieve discrimination of BPSK signals with error rates below what can be achieved using coherent states and any quantum measurement.

Figures (6)

• Figure 1: Experimental setup for all-Gaussian state discrimination of BPSK signals. A continuous-wave 1550 nm laser output is separated into a pump, signal and local oscillator (LO) by a network of beamsplitters (BS). The pump beam drives a second harmonic generation (SHG) cavity to produce light at $775$ nm which in turn pumps an optical parametric oscillator (OPO) to generate squeezed vacuum at $1550$ nm. Both cavities use periodically-poled potassium titanyl phosphate as a non-linear medium. An electro-optic modulator (EOM) displaces the $3$ MHz sideband of the signal beam which is then mixed with the OPO output on a biased BS to make the S-BPSK signal. Channel loss is implemented using a half-wave plate ($\lambda/2$) and a polarising beamsplitter (PBS) before the signal is measured at a homodyne receiver (HOM).
• Figure 2: (a) Error probability in BPSK communication against the mean photon number of the signal. (b) Error probability in BPSK communication against the proportion of energy allocated to squeezing the signal, for states with mean photon numbers close to $\bar{n}=2.15$. Markers show experimental results from our implementation of a BPSK protocol with displaced squeezed states and homodyne detection. Dashed curves show the expected error given the experimental parameters, and solid curves represent the theoretical limits: the standard quantum limit of coherent states and homodyne detection (SQL), the Helstrom bound for coherent states (HEL), the limit of Gaussian states and measurements (GAUSS), and the Helstrom bound for squeezed states (SQZHEL).
• Figure 3: Error probability as a function of mean photon number for different channel loss. Markers show measured values using a S-BPSK signal with 3.8 dB squeezing. Dashed curves are our model of the experiment, and solid curves show the C-BPSK limits. Error bars are obscured by the marker size.
• Figure 4: Mutual information as a function of mean photon number for different channel loss. Markers show values calculated from the measured probability of error using a S-BPSK signal with 3.8 dB squeezing. Dashed curves are our model of the experiment, and solid curves show the C-BPSK limits. Error bars are obscured by the marker size.
• Figure S1: Probability of error for ternary-phase-shift-keyed signals and dual-homodyne detection for states with a mean photon number of 1, against the proportion of energy allocated to squeezing. Purple and green markers are calculated by numerically integrating over the phase space coordinates or the phase angle respectively. The solid blue line is the coherent state error.