Evaluation of Estimation Precision in Test of Bell-Type Correlations
In quantum systems, there exist correlations counterintuitive to people who are familiar with classical mechanics. Quantum non-locality and quantum contextuality are such kinds of correlations. There are many theoretical proposals of experiments for proving the existence of such correlations. Some of the methods have been checked in several types of systems. In this chapter, we focus on one of the most famous methods for this purpose, the test of the CHSH inequality. This is a test for non-locality. It is an inequality involving a correlation function called the CHSH correlation. A violation of the CHSH inequality guarantees the existence of counterintuitive correlations that imply non-locality. In experiments, we perform a finite number of measurement trials and try to test whether the inequality is violated or not. In principle, our aim is only to test the violation, and it is not necessary to estimate the value of the CHSH correlation itself. In many experiments, however, the test is done by estimating the value of the correlation, and it is therefore important to accurately evaluate the size of the estimation errors. In this chapter, we give a formula for the accurate evaluation of such estimation errors. In Sect. 4.1, we explain a concept of quantum non-locality, which is called entanglement (Unentangled states can exhibit counterintuitive correlations. Quantum discord is one of concepts for characterizing such states). In Sect. 4.2, we present the CHSH inequality and its relation to entanglement. In Sect. 4.3, we analyze estimation errors in a test of the CHSH inequality. We summarize this chapter in Sect. 4.4.