More nonlocality with less entanglement in Clauser-Horne-Shimony-Holt experiments using inefficient detectors

It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\kappa \ge 0$. When $\kappa =0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches $50\%$ from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.

Figure 1

For a given detection efficiency $\eta$, a state with concurrence-squared at or above the solid line will never violate the CHSH inequality. On the other hand, for $\eta >1/2$, there always exists a state that can violate the CHSH inequality whose concurrence-squared is given by the dashed line.