Graph-theoretic approach to Bell experiments with low detection efficiency

Zhen-Peng Xu1,2, Jonathan Steinberg2, Jaskaran Singh3, Antonio J. López-Tarrida3, José R. Portillo4,5, and Adán Cabello3,6

1School of Physics and Optoelectronics Engineering, Anhui University, 230601 Hefei, People’s Republic of China
2Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany
3Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain
4Departamento de Matemática Aplicada I, Universidad de Sevilla, E-41012 Sevilla, Spain
5Instituto Universitario de Investigación de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, E-41012 Sevilla, Spain
6Instituto Carlos I de Física Teórica y Computacional, Universidad de Sevilla, E-41012 Sevilla, Spain

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Bell inequality tests where the detection efficiency is below a certain threshold $\eta_{\rm{crit}}$ can be simulated with local hidden-variable models. Here, we introduce a method to identify Bell tests requiring low $\eta_{\rm{crit}}$ 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, $\eta_{\rm{crit}}$ can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small $\eta_{\rm{crit}}$. We present examples in which, for maximally entangled states, $\eta_{\rm{crit}} \le 0.516$ for $d=16$, $\eta_{\rm{crit}} \le 0.407$ for $d=28$, and $\eta_{\rm{crit}} \le 0.326$ for $d=32$. We also show evidence that the upper bound for $\eta_{\rm{crit}}$ can be lowered down to $0.415$ for $d=16$ and present a method to make the upper bound of $\eta_{\rm{crit}}$ arbitrarily small by increasing the dimension and the number of settings. All these upper bounds for $\eta_{\rm{crit}}$ 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 $\eta_{\rm{crit}}$ 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 $\eta_{\rm{crit}}$ 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 nonlocality over long distances.

Bell nonlocality, i.e., the existence of correlations that violate Bell inequalities, is a crucial ingredient for many tasks with quantum advantage, including device-independent quantum key distribution. However, when the detection efficiency is below a certain threshold, then the violation of a Bell inequality by the detected pairs does not prove nonlocality, since, then, it is possible to construct local hidden variable models that reproduce the experimental values. Therefore, a fundamental problem is identifying quantum correlations and Bell inequalities in which the threshold detection efficiency is as low as possible.
In this work, we identify quantum correlations and Bell inequalities which require a low threshold detection efficiency and relatively low dimension. For that, we first show a family of Bell inequalities and quantum correlations in which the threshold detection efficiency can be expressed in terms of invariants of some graphs with certain properties. We use this connection to identify cases with low threshold detection efficiency. Then, we optimize the resulting Bell inequalities by developing a version of Gilbert's algorithm that takes advantage of the symmetries. As a result, we obtain threshold detection efficiencies $32$.

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