D-outcome measurement for a nonlocality test

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