d-outcome measurement for a nonlocality test

Abstract

For the purpose of the nonlocality test, we propose a general correlation\nobservable of two parties by utilizing local $d$-outcome measurements with\nSU($d$) transformations and classical communications. Generic symmetries of the\nSU($d$) transformations and correlation observables are found for the test of\nnonlocality. It is shown that these symmetries dramatically reduce the number\nof numerical variables, which is important for numerical analysis of\nnonlocality. A linear combination of the correlation observables, which is\nreduced to the Clauser-Horne-Shimony-Holt (CHSH) Bell's inequality for two\noutcome measurements, is led to the Collins-Gisin-Linden-Massar-Popescu (CGLMP)\nnonlocality test for $d$-outcome measurement. As a system to be tested for its\nnonlocality, we investigate a continuous-variable (CV) entangled state with $d$\nmeasurement outcomes. It allows the comparison of nonlocality based on\ndifferent numbers of measurement outcomes on one physical system. In our\nexample of the CV state, we find that a pure entangled state of any degree\nviolates Bell's inequality for $d(\\ge 2)$ measurement outcomes when the\nobservables are of SU($d$) transformations.\n