Modeling the Eberhard inequality based tests

TL;DR

This paper enhances the modeling and optimization of parameters for Eberhard inequality tests in quantum experiments, improving previous results and considering detector efficiencies and experimental fluctuations.

Contribution

It introduces advanced numerical optimization for Eberhard inequality tests, accounting for detector efficiencies and experimental variability, improving prior models.

Findings

Optimized parameters for Eberhard inequality tests using Nelder-Mead method.

Improved results over previous models and experiments.

Analyzed the impact of detector efficiency differences and setup fluctuations.

Abstract

Last year the first experimental tests closing the detection loophole (also referred to as the fair sampling loophole) were performed by two experimental groups \cite{Zeilinger}, \cite{Kwiat}. To violate Bell-type inequalities (the Eberhard inequality in the first test and the Clauser-Horne inequality in the second test), one has to optimize a number of parameters involved in the experiment (angles of polarization beam splitters and quantum state parameters). Although these are technicalities, their optimal determination plays an important role in approaching statistically significant violations of the inequalities. In this paper we study this problem for the Eberhard inequality in very detail by using the advanced method of numerical optimization, the Nelder-Mead method. First of all, we improve the the results of optimization for the original Eberhard model \cite{Eberhard} and the Gustina et al. work \cite{Zeilinger} ("Vienna-13 experiment") by using the model of this experiment presented in Kofler et al. \cite{Zeilinger1}. We also take into account the well known fact that detectors can have different efficiencies and perform the corresponding optimization. In previous studies the objective function had the meaning of the mathematical expectation. However, it is also useful to investigate the possible level of variability of the results, expressed in terms of standard deviation. In this paper we consider the optimization of parameters for the Eberhard inequality using coefficient of variation taking into account possible random fluctuations in the setup of angles during the experiment.