Graph-theoretic approach to Bell experiments with low detection efficiency

Bell inequality tests where the detec-tion efficiency is below a certain threshold eta crit can be simulated with local hidden-variable models. Here, we introduce a method to identify Bell tests requiring low eta crit and relatively low dimension d of the local quantum systems. The method has two steps. First, we show a family of bipar-tite Bell inequalities for which, for corre-lations produced by maximally entangled states, eta crit can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small eta crit. We present examples in which, for maximally entangled states, eta crit < 0.516 for d = 16, eta crit < 0.407 for d = 28, and eta crit < 0.326 for d = 32. We also show evidence that the upper bound for eta crit can be lowered down to 0.415 for d = 16 and present a method to make the upper bound of eta crit arbitrarily small by increas-ing the dimension and the number of set-tings. All these upper bounds for eta crit are valid (as it is the case in the literature) as-suming 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 eta crit and better noise robustness. For that, we use a mod-ified 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 eta crit is 12.38% lower and the required visibility is 14.62% lower than the upper bounds obtained in the first step. The tools presented here may allow for developing high-dimensional loophole-free Bell tests and loophole-free Bell non-locality over long distances.