Automated generation of photonic circuits for Bell tests with homodyne measurements

Nonlocal quantum realizations, certified by the violation of a Bell inequality, are core resources for device-independent quantum information processing. Although proof-of-principle experiments demonstrating device-independent quantum information processing have already been reported, identifying physical platforms that are realistically closer to practical, viable devices remains a significant challenge. In this work, we present an automated framework for designing photonic implementations of nonlocal realizations using homodyne detections and quantum state heralding. Combining deep reinforcement learning and efficient simulations of quantum optical processes, our method generates photonic circuits that achieve significant violations of the Clauser-Horne-Shimony-Holt inequality. In particular, we find an experimental setup, robust to losses, that yields a CHSH violation of $2.068$ with $3.9$ dB and $0.008$ dB squeezed light sources and two beam splitters.

💡 Research Summary

The paper tackles a central challenge in device‑independent (DI) quantum information processing: designing a realistic photonic platform that can robustly violate the Clauser‑Horne‑Shimony‑Holt (CHSH) Bell inequality using homodyne detection. While previous proposals have demonstrated CHSH violations with homodyne measurements, they either require highly complex optical states (multiple squeezed modes, high squeezing levels, and many photon‑counting detectors) or achieve only modest violations (B≈2.048) that fall short of thresholds needed for self‑testing or DI‑QKD protocols.

To overcome these limitations, the authors develop an automated design framework that couples a fast Gaussian‑state simulator with deep reinforcement learning (RL). The simulator represents any N‑mode optical circuit by its displacement vector and covariance matrix, allowing unitary Gaussian operations (beam splitters, phase shifters, single‑ and two‑mode squeezers) to be applied via symplectic matrices. Because homodyne measurements on pure Gaussian states admit a local hidden‑variable model, the authors introduce heralding: they attach N‑2 auxiliary modes equipped with threshold detectors. When a photon is detected in each auxiliary mode, the remaining two modes (sent to Alice and Bob) collapse into a non‑Gaussian state whose Wigner function can become negative, thereby enabling Bell violations with homodyne detection after binary sign‑binning of the quadrature outcomes.

The RL environment treats the current optical circuit as the state and the action space as the choice of a gate type and the modes on which to apply it. Gate parameters (angles, squeezing strengths) are discretized to keep the action space tractable. An episode proceeds by sequentially adding gates until a predefined depth (n_circuit) is reached; then the circuit is fully optimized over continuous gate parameters, and the CHSH score is computed. The reward is zero for all intermediate steps and equals the final CHSH value at the episode’s end. The policy is trained with Proximal Policy Optimization (PPO), which balances exploration and exploitation while maintaining training stability.

Experiments focus on a four‑mode system (two signal modes for Alice and Bob, plus two heralding modes) with a maximum circuit depth of four gates. The RL agent discovers a remarkably simple configuration: a single‑mode squeezer (3.9 dB) on one heralding mode, a very weak two‑mode squeezer (0.008 dB) on the other, and two 50:50 beam splitters that mix the signal and heralding modes. This circuit, when simulated with realistic loss models (0.2 dB/km fiber attenuation), yields a CHSH value of 2.068, exceeding the self‑testing threshold (2.051) and the more demanding DI‑QKD threshold (≈2.106). The design is robust: even with detector efficiencies down to 85 % and squeezing efficiencies of 90 %, the violation remains above 2.02, and the setup tolerates up to 8 km of fiber separation.

For comparison, a random‑search baseline exploring the same circuit space achieves an average CHSH of ~2.02 and never exceeds 2.045, confirming that the RL‑driven search efficiently navigates the exponentially large design space and uncovers non‑intuitive gate combinations.

The authors discuss practical implementation: the required components (four modes, two beam splitters, two squeezers) are compatible with current integrated photonic platforms (e.g., silicon nitride or lithium‑niobate waveguides). The modest squeezing levels are well within the capabilities of commercially available optical parametric oscillators. Consequently, the proposed circuit can be realized on a chip, paving the way for fully integrated, room‑temperature DI quantum key distribution or randomness generation devices.

Future directions include scaling the method to larger mode numbers and deeper circuits, incorporating continuous‑parameter optimization after RL pre‑search, and extending the framework to other Bell scenarios (e.g., multipartite inequalities) or to optimize for additional metrics such as key rate or randomness extraction efficiency.

In summary, this work introduces a powerful automated design paradigm that combines efficient Gaussian optics simulation with deep reinforcement learning to produce experimentally feasible photonic circuits achieving record‑high CHSH violations with homodyne detection. The resulting designs are simple, loss‑tolerant, and ready for near‑term experimental demonstration, representing a significant step toward practical device‑independent quantum technologies.