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. Here, we introduce a method to identify Bell tests requiring low ηcrit and relatively low dimension d of the local quantum systems. The method has two steps. First, we show a family of bipartite Bell inequalities for which, for correlations produced by maximally entangled states, ηcrit can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small ηcrit. We present examples in which, for maximally entangled states, ηcrit≤0.516 for d=16, ηcrit≤0.407 for d=28, and ηcrit≤0.326 for d=32. We also show evidence that the upper bound for ηcrit can be lowered down to 0.415 for d=16 and present a method to make the upper bound of ηcrit arbitrarily small by increasing the dimension and the number of settings. All these upper bounds 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 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 nonlocality over long distances.

Advances in device-independent quantum key distribution

“…On the theory side, the analysis of the studied schemes is tight when considering CHSH-based protocols. To improve the performance, more sophisticated protocols including e.g., twoway classical communication in the post-processing stage 146 or exploiting higher-dimensional Bell inequalities 90,91 might be an option, though they seem to be challenging at the moment.…”

“…81 ). In this respect, it has been shown that higher-dimensional Bell inequalities offer lower minimum efficiencies and better robustness to noise than the CHSH inequality (see for instance 85,90,91 ). Note, however, that solutions like these rely on the possibility of entangling two particles in higher-dimensional degrees of freedom, which is a more complex task from an experimental point of view.…”

Loophole-free Bell tests with randomly chosen subsets of measurement settings

“…This demands a detection efficiency above a certain threshold η crit [14] that depends on the nonlocal correlations. Recently [15,16], several bipartite quantum correlations between highdimensional quantum systems have been identified requiring η crit smaller than those needed for qubits [17,18] and ququarts [19], which are the systems used in the detection-loopholefree Bell tests performed so far [20][21][22][23][24][25][26][27]. However, for all these cases, achieving a small detection efficiency requires a large number of settings.…”

“…[15], each party has to measure 2 8 settings; to achieve η crit = 0.324 in Ref. [16], each party has to measure 2 30 settings. These correlations need too many settings to be useful for actual experiments.…”

“…Reciprocally, for a given detection efficiency, we want to know the minimum fraction of contexts needed to detect loophole-free Bell nonlocality. In particular, we want to elucidate whether this approach is useful when applied to the correlations in [15,16] and similar examples that may appear in the future.…”