Automated generation of photonic circuits for Bell tests with homodyne measurements

TL;DR

This work presents an automated framework for designing photonic implementations of nonlocal realizations using homodyne detections and quantum state heralding, and generates photonic circuits that achieve significant violations of the Clauser-Horne-Shimony-Holt inequality.

Abstract

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.

Figures (7)

• Figure 1: Optical circuit resulting from our automated search. The circuit is composed of two two-mode squeezers ($\hat{S}_2$) acting on modes $(1,2)$ and $(3,4)$ with respective parameters $r = 0.00096$ and $r = 0.44993$, followed by two beam splitters ($\hat{B}$) on modes $(1,3)$ and $(2,4)$, with respective parameters $\theta = 1.50272$ and $\theta = 1.63856$. The state of modes $(1,2)$, shared by Alice and Bob, is heralded on a click in both the threshold detectors of modes $(3,4)$. Alice and Bob measure their modes with homodyne measurements and can obtain an average CHSH score of $\mathcal{B}= 2.068$.
• Figure 2: CHSH score with respect to the distance between Alice, where state preparation occurs, and Bob. We consider an optical fiber with a loss of $0.2$ dB/km, and numerically optimized the circuits' parameters for each step of $0.1$ km. As higher squeezing is beneficial for greater distance, we ensure squeezing of at most $10$dB to respect practical experimental range.
• Figure 3: Evolution of the probability of a successful state heralding with respect to the efficiency of threshold detectors.
• Figure 4: Two alternative ways to herald a mode. In (a), we herald the state on a click in a threshold detector. In (b), we approximate a projection on a single photon by mixing the mode with the vacuum through a beam-splitter with parameter $\theta=0.1$ (the transmittance of the beam-splitter is $\cos(0.1)\approx 0.995$). Then, we herald on a click in the threshold detector at the reflected port and a no-click in the transmitted port. This combined event heralds on the detection of a single photon with high probability.
• Figure 5: The evolution of the CHSH score with the number of completed episodes, for an agent with PPO policy implementing three different strategies to build the optical circuits (c.f. \ref{['app:circuits']}). In both (a) and (b), the increase of the CHSH score with the number of episodes shows that the agent learns from previous interactions with the environment to produce circuits with higher CHSH violations. In particular, the agent finds circuits with CHSH score equal to 2.068 (red dashed line) in (a) and 2.072 (red dashed line) in (b). Conversely, the CHSH score remains constantly below 2.0 (red line) in (c), signifying that the agent could not learn a policy leading to CHSH violations. Samples of the circuits found with the three strategies are reported in \ref{['tab:circuits-agent']}.