Chained Bell Inequality Experiment with High-Efficiency Measurements

Abstract

We report correlation measurements on two 9 Be + ions that violate a chained Bell inequality obeyed by any local-realistic theory. The correlations can be modeled as derived from a mixture of a local-realistic probabilistic distribution and a distribution that violates the inequality. A statistical framework is formulated to quantify the local-realistic fraction allowable in the observed distribution without the fair-sampling or independent-and-identical-distributions assumptions. We exclude models of our experiment whose local-realistic fraction is above 0.327 at the 95 % confidence level. This bound is significantly lower than 0.586, the minimum fraction derived from a perfect ClauserHorne-Shimony-Holt inequality experiment. Furthermore, our data provides a device-independent certification of the deterministically created Bell states.Recently several groups have reported loophole-free tests of local realism with Bell's theorem [1], rejecting with high confidence theories of local realism [2][3][4]. While these experiments falsify the idea that nature obeys local realism, they are limited in the extent to which their data differs from local realism. Chained Bell inequality (CBI) [5] experiments can show greater departures from local realism in the following sense: Elitzur, Popescu, and Rohrlich [6] described a model of the distribution of outcomes measured from a quantum state as a mixture of a local-realistic distribution, which obeys Bell's inequalities, and another distribution that does not. Following their convention, we call these distributions "local" and "non-local." According to Ref. [6], a probability distribution P for the outcomes of an experiment can be written aswhere P L represents a local joint probability distribution (a "local part") and P N L represents a non-local distribution, with p local as the weight of the local component bound by 0 ≤ p local ≤ 1. For an ideal Clauser-HorneShimony-Holt (CHSH) Bell inequality experiment where two physical systems (usually particles) are jointly measured with four different measurement settings [7], the lowest attainable upper bound on the local content p local in any quantum distribution is ∼ 0.586 [8,9]. In principle, this bound can be lowered to zero by using a chained Bell inequality experiment.As indicated in Fig. 1.(a), CBI experiments are a generalization of a CHSH-type experiment. During each trial, a source that may be treated as a "black box" emits two systems labeled a and b, respectively. The experimentalist records the measurement outcomes after choosing a pair of measurements to perform separately on a and b. We use the symbols a k , b l to denote the respective measurement settings and a k b l for the pair. The latter is usually simply referred to as "the settings" or "the setting pairs". There is a hierarchy in which the Bell test SourceMeasurement:Illustration of a Bell inequality experiment. A source emits two systems a and b, here two 9 Be + ions. After choosing measurement settings a k and b l , the experiment implements Hilbert-space ...

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