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Optimization of statistical parameters of Eberhard inequality

Polina Titova

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

This work is devoted to mathematical modeling of the parameters of the Eberhard inequality using optimization techniques. One of the main purposes of this study is to consider the more general case when the detectors have different efficiencies. In this paper we consider the optimization of parameters for the Eberhard inequality using coefficient of variation taking into account possible errors in the setup of angles during the experiment.

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This work is devoted to mathematical modeling of the parameters of the Eberhard inequality using optimization techniques. One of the main purposes of this study is to consider the more general case when the detectors have different efficiencies. In this paper we consider the optimization of parameters for the Eberhard inequality using coefficient of variation taking into account possible errors in the setup of angles during the experiment.
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.
Recently, the results of the first experimental test for entangled photons closing the detection loophole (also referred to as the fair sampling loophole) were published (Vienna, 2013). From the theoretical viewpoint the main distinguishing feature of this long-aspired to experiment was that the Eberhard inequality was used. Almost simultaneously another experiment closing this loophole was performed (Urbana-Champaign, 2013) and it was based on the Clauser–Horne inequality (for probabilities). The aim of this note is to analyze the mathematical and experimental equivalence of tests based on the Eberhard inequality and various forms of the Clauser–Horne inequality. The structure of the mathematical equivalence is nontrivial. In particular, it is necessary to distinguish between algebraic and statistical equivalence. Although the tests based on these inequalities are algebraically equivalent, they need not be equivalent statistically, i.e., theoretically the level of statistical significance can drop under transition from one test to another (at least for finite samples). Nevertheless, the data collected in the Vienna test implies not only a statistically significant violation of the Eberhard inequality, but also of the Clauser–Horne inequality (in the ratio-rate form): for both a violation .
Recent experiments using innovative optical detectors and techniques have strongly increased the capacity of testing the violation of the Bell's inequalities in Nature. Most of them have used the Eberhard's inequality (EI) to close the ``detection'' loophole. Closing the ``locality'' loophole has been attempted by spacelike separated detections and fast changes of the bases of observation, driven by random number generators of new design. Also, pulsed pumping and time-resolved data recording to close the ``time-coincidence'' loophole, and sophisticated statistical methods to close the ``memory'' loophole, have been used. In this paper, the meaning of the EI is reviewed. A simple hidden variables theory based on a relaxation of the condition of ``measurement independence,'' which was devised long ago for the Clauser-Horne-Shimony and Holt inequality, is adapted to the EI case. It is used here to evaluate the significance of the results of the recent experiments, which are briefly described. A table summarizes the main results.