An alternative method of detection-loophole-free Bell test is proposed using local hidden variable (LHV) models with optimal detection efficiencies. A framework for constructing such optimal LHV models is presented. Optimal LHV models for maximally and non-maximally entangled twopartite states are constructed to reproduce the quantum correlations within the critical detection efficiencies. The LHV models are shown to be completely equivalent with the existing twopartite Bell inequalities in their optimized setups, and to have even lower critical efficiencies in the LHV modes' own optimized setups. Applications in Bell tests and in device-independent quantum information processing are discussed.
An alternative method of detection-loophole-free Bell test is proposed using local hidden variable (LHV) models with optimal detection efficiencies. A framework for constructing such optimal LHV models is presented. Optimal LHV models for maximally and non-maximally entangled twopartite states are constructed to reproduce the quantum correlations within the critical detection efficiencies. The LHV models are shown to be completely equivalent with the existing twopartite Bell inequalities in their optimized setups, and to have even lower critical efficiencies in the LHV modes' own optimized setups. Applications in Bell tests and in device-independent quantum information processing are discussed.
Constructing local hidden variable (LHV) models for entangled quantum states is a fundamental problem, with implications for the foundations of quantum theory and for quantum information processing. It is, however, a challenging problem, as the model should reproduce quantum predictions for all possible local measurements. Here we present a simple method for building LHV models, applicable to any entangled state and considering continuous sets of measurements. This leads to a sequence of tests which, in the limit, fully captures the set of quantum states admitting a LHV model. Similar methods are developed for local hidden state models. We illustrate the practical relevance of these methods with several examples.
A local hidden variable model exploiting the detection loophole to reproduce exactly the quantum correlation of the singlet state is presented. The model is shown to be compatible with both the CHSH and the CH Bell inequalities. Moreover, it bears the same rotational symmetry as spins. The reason why the model can reproduce the quantum correlation without violating the Bell theorem is that in the model the efficiency of the detectors depends on the local hidden variable. On average the detector efficiency is limited to 75%.
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.
Let us imagine an experimentalist, who has maximally entangled photons and wants to violate Bell’s inequality in his laboratory. Among many obstacles he has to overcome, two of them are the most significant. Firstly, he has to avoid exchanging information about the measurement settings between the particles. Secondly, he has to detect particles with a high enough probability, because if the detection efficiency is too low, one can reproduce quantum correlations using a LHV (Local Hidden Variables) model. These two problems are known in the literature, respectively, as the locality loophole [1] and the detection loophole [2, 3]. Fortunately, it is possible to close the locality loophole using photons in the experiment [4, 5, 6, 7, 8]. However, closing the detection loophole is still an object of study. The authors of [9] showed that for the CHSH inequality, the lower bound for detector efficiency needed to close the loophole is 82.8% for maximally entangled states. This bound was lowered to 66.7% by Eberhard [10]. Furthermore, the detector efficiency has been studied for other Bell inequalities [11, 12, 13, 14, 15, 16]. In our research, we are trying to answer the following question: If we prepare a Bell experiment and we have two detectors, having different efficiencies both smaller than 100%, which detector should each party receive? And also: Which representation of the chosen Bell inequality should we consider? These are very crucial questions, which arise in laboratories. Hence, we hope that our results will be useful for experimentalists. We focus on the I3322 inequality [17], which is a natural candidate for testing, because of its asymmetry [14, 18]. We consider the case of a post-selection strategy, which consists in rejecting all the rounds in the Bell experiment when one of the detectors does not click. We used numerical optimisation methods to find both the best classical strategy for Alice and Bob and the best detector clicking strategy. Thus, we have found minimal efficiencies of the detectors of Alice and Bob needed to violate I3322. We made the same studies also for symmetric representations of I3322 [19]. Our results show that the minimal efficiencies of Alice and Bob’s detectors of the asymmetric representation of I3322 is lower than in case of the symmetric one [20]. It is work in progress, and we are examining other post-selection methods as well as other Bell inequalities. We want to consider also the case when non-maximally entangled states are used in Bell experiments.
In this Letter, we explore the possibility of developing Bell inequalities predicated on models using a single Local Hidden Variable (s-LHV), a strict subset of general LHV models. Because of the less strenuous constraints imposed by s-LHV models, we were able to derive a contingent Bell inequality in analogy to the CHSH inequality, but which does not require bounding of measurement statistics. Following this, we show by explicit example that there are cases of states that rule out s-LHV models by violating our inequality, but which nonetheless have a multivariate LHV model. Even so, we show how merely ruling out s-LHV models is still sufficient to allow for fully device independent quantum key distribution (QKD) and entanglement witnessing. This being the case, our inequality illustrates two things. First, it makes fully device-independent QKD on continuous variables substantially more straightforward. Second, it shows how the degree of correlation needed to demonstrate device-independent QKD is distinct from both Bell-nonlocality and EPR-steering. Although nonlocality in general requires ruling out all LHV models, s-LHV models are sufficiently useful to warrant further exploration.
Bell's seminal work showed that no local hidden variable (LHV) model can fully reproduce the quantum correlations of a two-qubit singlet state. His argument and later developments by Clauser et al. effectively rely on gaps between the anticorrelations achievable by classical models and quantum theory for projective measurements along randomly chosen axes separated by a fixed angle. However, the size of these gaps has to date remained unknown. Here we numerically determine the LHV models maximizing anticorrelations for random axes separated by any fixed angle, by mapping the problem onto ground state configurations of fixed-range spin models. We identify angles where this gap is largest and thus best suited for Bell tests. These findings enrich the understanding of Bell non-locality as a physical resource in quantum information theory and quantum cryptography.