Challenging preconceptions about Bell tests with photon pairs

Motivated by very recent experiments, we consider a scenario “à la Bell” in which two protagonists test the Clauser-Horne-Shimony-Holt (CHSH) inequality using a photon-pair source based on spontaneous parametric down conversion and imperfect photon detectors. The conventional wisdom says that (i) if the detectors have unit efficiency, the CHSH violation can reach its maximum quantum value of $2\sqrt{2}$. To obtain the maximal possible violation, it suffices that the source emits (ii) maximally entangled photon pairs (iii) in two well-defined single modes. Through a nonperturabive calculation of nonlocal correlations, we show that none of these statements are true. By providing the optimal pump parameters, measurement settings and state structure for any detection efficiency and dark count probability, our results give the recipe to close all the loopholes in a Bell test using photon pairs.

Figure 1

A source (star) based on spontaneous parametric down conversion is excited, e.g., by a pulsed pump and produced photon pairs entangled, e.g., in polarization. The photons are emitted in correlated spatial modes $a \left(b\right)$. Each of them includes several temporal, frequency, and spatial modes $a_k-b_k$, the number of temporal modes in the pulsed regime being given by the ratio between the pump duration and the photon coherence time for example. The photons emitted in $a \left(b\right)$ are sent to Alice's (Bob's) location where they are projected along an arbitrary direction of the Bloch sphere using a set of wave plates, a polarization beam splitter, and two detectors. Each pump pulse triggers the choice of a measurement setting. The detectors are assumed to be nonphoton number resolving with nonunit efficiency and dark counts.

Figure 2

CHSH values as a function of the detection efficiency $\eta .$ The solid curve is optimized over the structure of the states produced by a spontaneous parametric down-conversion source (the squeezing parameters and the number of modes), the measurement settings, and the local strategy with which the outcomes are assigned to results $\pm 1.$ The dark count probability is set to zero. The optimal violation (solid curve) is compared to the CHSH value obtained by restricting the emission to a single mode, i.e., thermal photon number statistics (dashed line). A zoom on the region of high efficiencies is given in the inset (a). The CHSH value is also compared to the one obtained by focusing on the single mode case with maximally entangled states ($g=\overline{g}$, see dash-dotted line). A zoom on the region of high efficiencies is given in the inset (b) where the monomode case is compared to the monomode case restricted to maximally entangled states. Note that the CH value can be deduced from the here shown CHSH value through $\frac{S-2}{4}.$