Graph-theoretic approach to Bell experiments with low detection efficiency

Abstract

Bell inequality tests where the detection efficiency is below a certain threshold ηcrit can be simulated with local hidden-variable models. For the Clauser-Horne-Shimony-Holt Bell inequality and maximally entangled states, ηcrit = 0.828. Massar noticed that ηcrit can tend to zero as the dimension d of the local quantum systems grows, but found no advantage except for d > 1600. Vertesi ´ et al. lowered ηcrit down to 0.770 for maximally entangled states using d = 4. Marton ´ et al. studied the case of N copies of the two-qubit maximally entangled state and obtained an upper bound of 0.693 for N = 4 (which is equivalent to d = 16). Recently, Miklin et al. have presented a strategy that allows us to reduce ηcrit down to 0.469 for d = 512. Here, we introduce a method to identify Bell tests requiring low ηcrit and relatively low d. The method has two steps. First, we show a family of bipartite Bell inequalities for which ηcrit is a function of some invariants of a certain type of graphs, and use it to identify correlations that require small ηcrit for maximally entangled states. We present examples in which ηcrit = 0.516 for d = 16, ηcrit = 0.407 for d = 28, and ηcrit = 0.326 for d = 32. We also show evidence that ηcrit can be lowered down to 0.415 for d = 16 and present a method to make ηcrit arbitrarily small by increasing the dimension and the number of settings. All these values for ηcrit are valid (as it is the case in the literature) assuming no noise. The second step is based on the observation that, using the initial state and measurement settings identified in the first step, we can construct Bell inequalities with smaller ηcrit and better noise robustness. For that, we use a modified version of Gilbert’s algorithm that takes advantage of the automorphisms of the graphs used in the first step. We illustrate its power by explicitly developing an example in which ηcrit is 12.38% lower and the required visibility is 14.62% lower than the ones required after the first step. The tools presented here pave the way for high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances