Clauser-Horne inequality for electron counting statistics in multiterminal mesoscopic conductors

TL;DR

This paper derives and analyzes the Clauser-Horne inequality for electron counting in mesoscopic multiterminal conductors, demonstrating violations in specific setups and exploring their scaling with the number of injected electrons.

Contribution

It introduces a new form of the Clauser-Horne inequality applicable to full electron counting statistics and investigates its violation in realistic mesoscopic systems.

Findings

Strong violations occur when transmitted charges are equal ($Q_1=Q_2$).

Violations scale as 1/M and 1/M^2 for different setups, with M being the average number of injected electrons.

Full violation is observed in superconducting systems.

Abstract

In this paper we derive the Clauser-Horne (CH) inequality for the full electron counting statistics in a mesoscopic multiterminal conductor and we 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 ($Q_1=Q_2$), whereas no violation is found for $Q_1\ne Q_2$. 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/M and $1/M^2$ for the entangler and normal beam splitter, respectively, 2$M$ being the average number of injected electrons. As expected, we find full violation of the CH inequality in the case of the superconducting system.