Testing a Bell inequality in multipair scenarios

To date, most efforts to demonstrate quantum nonlocality have concentrated on systems of two (or very few) particles. It is, however, difficult in many experiments to address individual particles, making it hard to highlight the presence of nonlocality. We show how a natural setup with no access to individual particles allows one to violate the Clauser-Horne-Shimony-Holt inequality with many pairs, including in our analysis effects of noise and losses. We discuss the case of distinguishable and indistinguishable particles. Finally, a comparison of these two situations provides insight into the complex relation between entanglement and nonlocality.

Figure 1

Setup: a source produces $M$ independent pairs (or equivalently $M$ independent sources each produce a pair), the pairing between Alice’s and Bob’s particles is lost during their transmission, and each party measures all their incoming particles in the same basis. The total number $n_{+(-)}$ of particles detected in the +(−) outcome is tallied on both sides.

Figure 2

(Color online) CH violation and resistance to noise for a source producing $M$ independent pairs. The states and settings used are discussed in the text. (a) Maximal CH values for various thresholds: majority voting (solid red line), $3∕4$ voting (dotted green line), and unanimity (dashed blue line). The decrease is as $M^{-1∕2}$ for the first two and exponential for the last one. The highest violation is thus reached using a majority vote. (b) Resistance to noise for the different thresholds (same colors). All curves decrease as $M^{-1}$. The most resistant violation is that achieved by using majority voting.

Figure 3

(Color online) Maximal Bell violation and resistance to noise with a Poissonian source of independent pairs. The red lines represents the majority voting, the green dotted lines the $3∕4$ voting, and the blue dashed lines the unanimity voting. Settings are chose in the form (7); optimal states are discussed in the text. (a) For a large mean photon number, the decrease of ${\mathcal{I}}_{\mathit{CH}}$ goes like ${\mu }^{-1∕2}$ for the majority and $3∕4$ vote, just like for the fixed $M$ case. Concerning the unanimity vote, ${\mathcal{I}}_{\mathit{CH}}$ decreases faster than a polynomial, but slower than an exponential. (b) Resistance to noise is very similar for all strategies, decreasing as ${\mu }^{-1}$ just like with as source of fixed pairs number.

Figure 4

(Color online) CH violation with inefficient detectors as a function of the probability for a photon to be detected. The upper thin curve shows the traditional case $M=1$ with known critical efficiency $\eta =2∕3$ [15]. The two other curves are for $M=5$ pairs with majority (solid red line) and unanimity voting (dashed blue line).

Figure 5

(Color online) Comparison between sources producing independent pairs or indistinguishable photons, using settings of the form (7). (a) Maximal CH violation achieved with a source of indistinguishable photons for various voting procedure (superposed black dots). Compared to the previously calculated violations for independent pairs [same curves as in Fig. 2a], unanimity voting (lower blue dashed line) yields less violation, while majority voting (upper red line) yield the highest ${\mathcal{I}}_{\mathit{CH}}$ values. Note that the maximal violation with indistinguishable photons almost does not depend on the voting procedure used. (b) Maximal resistance to noise in the majority voting scenario (solid red lines) and the unanimity scenario (dashed blue lines) for sources producing independent pairs [thick line, same curves as in Fig. 2b] or indistinguishable photons (thin line). The unanimous vote is more robust for indistinguishable photons, but majority voting on independently produced pairs yields the most persistent violation.