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New Journal of Physics

Clauser–Horne inequality for the full counting statistics

Fabio Taddei, Lara Faoro, Elsa Prada, Rosario Fazio

Published Aug 26, 2005• 4 citations
Publisher

Abstract

We discuss the Clauser–Horne (CH) inequality for electrons in mesoscopic systems formulated in terms of the full counting statistics (FCS). A derivation of such an inequality is provided for a prototype setup consisting of an entangler attached to two conducting wires, that brings pairs of spin-entangled electrons into spatially separated counters, where detection takes place. Violation of the CH inequality is analysed as a function of the various parameters characterizing the system. The effect of dephasing, which can occur in realistic wires, is also addressed. As expected, the extent of violation is monotonically suppressed by increasing dephasing strength. The CH inequality is finally applied to a three-arm normal-metallic beam splitter. Our results show that violation takes place even in the absence of interaction.

DOI
10.1088/1367-2630/7/1/183

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We discuss the Clauser–Horne (CH) inequality for electrons in mesoscopic systems formulated in terms of the full counting statistics (FCS). A derivation of such an inequality is provided for a prototype setup consisting of an entangler attached to two conducting wires, that brings pairs of spin-entangled electrons into spatially separated counters, where detection takes place. Violation of the CH inequality is analysed as a function of the various parameters characterizing the system. The effect of dephasing, which can occur in realistic wires, is also addressed. As expected, the extent of violation is monotonically suppressed by increasing dephasing strength. The CH inequality is finally applied to a three-arm normal-metallic beam splitter. Our results show that violation takes place even in the absence of interaction.
In this paper we derive the Clauser-Horne (CH) inequality for the full electron-counting statistics in a mesoscopic multiterminal conductor and discuss its properties. We first consider the idealized situation in which a flux of entangled electrons is generated by an entangler. Given a certain average number of incoming entangled electrons, the CH inequality can be evaluated for different numbers of transmitted particles. Strong violations occur when the number of transmitted charges on the two terminals is the same (Q1=Q2),{(Q}_{1}{=Q}_{2}),(Q1​=Q2​), whereas no violation is found for ${Q}_{1}\ensuremath{\ne}{Q}_{2}.$ We then consider two actual setups that can be realized experimentally. The first one consists of a three terminal normal beam splitter and the second one of a hybrid superconducting structure. Interestingly, we find that the CH inequality is violated for the three terminal normal device. The maximum violation scales as 1/M1/M1/M and 1/M2{1/M}^{2}1/M2 for the entangler and normal beam splitter, respectively, 2M2M2M being the average number of injected electrons. As expected, we find full violation of the CH inequality in the case of the superconducting system.
We analyze the effect of decoherence on the violation of the Clauser-Horne (CH) inequality for the full electron counting statistics in a mesoscopic multiterminal conductor. Our setup consists of an entangler that emits a flux of entangled electrons into two conductors characterized by a scattering matrix and subject to decoherence. Loss of phase memory is modeled phenomenologically by introducing fictitious extra leads. The outgoing electrons are detected using spin-sensitive electron counters. Given a certain average number of incoming entangled electrons, the CH inequality is evaluated as a function of the numbers of detected particles and on the various quantities characterizing the scattering matrix. When decoherence is turned on, we show that the amount of violation of the CH inequality is effectively reduced. Interestingly we find that, by adjusting the parameters of the system, there exists a protected region of QQQ values for which violation holds for arbitrary strong decoherence.
Clauser-Horne (CH) inequality, Eberhard inequality, and Clauser-Horne-Shimony-Holt (CHSH) inequality are used to determine whether quantum entanglement can contradict local realism. However, the loophole is known to allow local realism to violate these inequalities if a sample size is small and not enough [1]. Remarkably though, this paper shows that this loophole can still cause a violation in these inequalities even with a very large sample size, e.g. a 2.4 sigma violation of CH inequality and Eberhard inequality was achieved despite 12,000,000 total trials in a Monte Carlo simulation of a local realist photonic experiment based on Malus' law. In addition, this paper shows how Eberhard inequality is especially vulnerable to this loophole when combined with an improper statistical analysis and incorrect singles counts, e.g. a 13.0 sigma violation was achieved with the same large sample size, and furthermore, a 26.6 sigma violation was produced when a small, acceptable 0.2% production rate loophole was applied. Supplementally, this paper demonstrates how the finite statistics loophole allows a bigger violation in a smaller sample size despite the sample size being enough, e.g. a CHSH violation of 4.4 sigma (2.43 +/- 0.10) was achieved with 280 total trials, and 4.0 sigma (2.16 +/- 0.04) with 3,000 total trials. This paper introduces the aforementioned loopholes as plausible local realist explanations to two observed violations reported by Giustina, et al. [2], and Hensen, et al. [3].
We propose a spin-independent scheme to generate and detect two-particle entanglement in a mesoscopic normal-superconductor system. A superconductor, weakly coupled to the normal conductor, generates an orbitally entangled state by injecting pairs of electrons into different leads of the normal conductor. The entanglement is detected via violation of a Bell inequality, formulated in terms of zero-frequency current cross correlators. It is shown that the Bell inequality can be violated for arbitrary strong dephasing in the normal conductor.
We propose a geometric multiparty extension of Clauser–Horne (CH) inequality. The standard CH inequality can be shown to be an implication of the fact that statistical separation between two events, A and B , defined as P ( A ⊕ B ) , where A ⊕ B = ( A − B ) ∪ ( B − A ) , satisfies the axioms of a distance. Our extension for tripartite case is based on triangle inequalities for the statistical separations of three probabilistic events P ( A ⊕ B ⊕ C ) . We show that Mermin inequality can be retrieved from our extended CH inequality for three subsystems in a particular scenario. With our tripartite CH inequality, we investigate quantum violations by GHZ-type and W-type states. Our inequalities are compared to another type, so-called N -site CH inequality. In addition we argue how to generalize our method for more subsystems and measurement settings. Our method can be used to write down several Bell-type inequalities in a systematic manner.
Violation of Bell inequalities in 2×32\times32×3 dimensional systems