Optimal conditions for the Bell test using spontaneous parametric down-conversion sources

We theoretically and experimentally investigate the optimal conditions for the Bell experiment using spontaneous parametric down-conversion (SPDC) sources. In theory, we show that a relatively large average photon number (typically ∼0.5) is desirable to observe the maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. Moreover, we show that the violation converges to 0.27 in the limit of large average photon number. In experiment, we perform the Bell experiment without postselection using polarization-entangled photon pairs at 1550-nm telecommunication wavelength generated from SPDC sources. While the violation of the CHSH inequality is not directly observed due to the overall detection efficiencies of our system, the experimental values agree well with those obtained by the theory with experimental imperfections. Furthermore, in the range of small average photon numbers (≤0.1), we propose and demonstrate a method to estimate the ideal CHSH value intrinsically contained in the tested state from the lossy experimental data without assuming the input quantum state.