The chained Bell inequalities of Braunstein and Caves involving $N$ settings per observer have some interesting applications. Here we obtain the minimum detection efficiency required for a loophole-free violation of the Braunstein-Caves inequalities for any $N\ensuremath{\ge}2$. We discuss both the case in which both particles are detected with the same efficiency and the case in which the particles are detected with different efficiencies.
The chained Bell inequalities of Braunstein and Caves involving N settings per observer have some interesting applications. Here we obtain the minimum detection efficiency required for a loophole-free violation of the Braunstein-Caves inequalities for any N\ensuremath≥2. We discuss both the case in which both particles are detected with the same efficiency and the case in which the particles are detected with different efficiencies.
In many quantum information applications, a minimum detection efficiency must be exceeded to ensure success. Protocols depending on the violation of a Bell inequality, for instance, may be subject to the so-called detection loophole: Imperfect detectors may yield spurious violations, which consequently cannot be used to ensure, say, quantum cryptographic security. Hence, we investigate the possibility of giving lower bounds on detector efficiency even if an adversary has full control over both the source and the detectors. To this end, we present a technique to systematically derive Bell inequalities free from the detection loophole using only the observed measurements statistics. The violation of these inequalities certifies that the detectors used exceed a certain minimal efficiency.
In any Bell test, loopholes can cause issues in the interpretation of the results, since an apparent violation of the inequality may not correspond to a violation of local realism. An important example is the coincidence-time loophole that arises when detector settings might influence the time when detection will occur. This effect can be observed in many experiments where measurement outcomes are to be compared between remote stations because the interpretation of an ostensible Bell violation strongly depends on the method used to decide coincidence. The coincidence-time loophole has previously been studied for the Clauser-Horne-Shimony-Holt and Clauser-Horne inequalities, but recent experiments have shown the need for a generalization. Here, we study the generalized ``chained'' inequality by Pearle, Braunstein, and Caves (PBC) with $N\ensuremath{\ge}2$ settings per observer. This inequality has applications in, for instance, quantum key distribution where it has been used to reestablish security. In this paper we give the minimum coincidence probability for the PBC inequality for all $N\ensuremath{\ge}2$ and show that this bound is tight for a violation free of the fair-coincidence assumption. Thus, if an experiment has a coincidence probability exceeding the critical value derived here, the coincidence-time loophole is eliminated.
Svetlichny's inequality (SI) is a Bell-like inequality, the violation of which can be used to confirm the existence of genuine multipartite correlations. However, poor detector efficiency can possibly cause a so-called detection loophole in actual Svetlichny experiments. We derive an alternative SI to deal with this loophole. If the experimental data can violate this SI, it must result in loophole-free violation of the original SI. We show that the minimum detection efficiency needed for a loophole-free violation of the tripartite SI is about 0.97. For the general case of n particles, we give the analytic expression of the needed detection efficiency, and find that its value monotonically and rapidly approaches 1 as the number of particles increases.
Abstract Bell’s inequality provides a remarkable way to test the consistency between quantum mechanics and classic local realistic theory. However, experimental demonstrations of the loophole-free Bell test are challenging and only recently have been demonstrated with bipartite systems. A central obstacle for the photonic system is that the sampling efficiency, including the collection and detection efficiencies, must be above a certain threshold. We here generalize two-particle Eberhard’s inequality to the n-particle systems and derive a Bell-type inequality for multi-particle systems, which significantly relaxes this threshold. Furthermore, an experimental proposal to achieve a multi-partite Bell test without the fair sampling assumption is presented for the case of three particles. For any given value of the sampling efficiency, we give the optimal configurations for actual implementation, the optimal state, the maximum background noise that the system can tolerate, and the lowest fidelity of the quantum state. We believe our work can serve as a recipe for experimentalists planning to violate local realism using a multi-partite quantum state without the sampling loophole.
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.
The detection loophole is a well-known loophole in Bell nonlocality motivated by the inefficiency of detectors used in Bell experiments. However, this inefficiency is actually a ubiquitous feature of any photonic platform for quantum networks since photons are readily absorbed by the surrounding environment. When the quantum system, usually a photon, is lost, what should the parties output? The most natural choice is to fall back to a deterministic strategy where each party produces an output via a function of her local input. In this paper, we study how to choose such a fallback strategy for general Bell inequalities with respect to different efficiencies η to maximize the possible violation. We mathematically prove that for the CHSH inequality, there exists a fallback strategy that is optimal for all η. However, in general, we find that the optimal fallback strategy can vary with η, sometimes in surprising ways. For example, we find an example of a Bell inequality where the optimal fallback strategy near the threshold efficiency is not an optimal deterministic strategy in the lossless setting. Our results can reduce the efficiency requirements for closing the detection loophole and thus are useful for applications of Bell inequality violation, such as quantum telepathy or device-independent quantum key distribution.