d-outcome measurement for a nonlocality test

For the purpose of a 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- Home-Shimony-Holt (CHSH) Bell's inequality for two outcome measurements, leads 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(greater than or equal to2) measurement outcomes when the observables are of SU(d) transformations.

Joint measurements and Bell inequalities

Joint quantum measurements of noncommuting observables are possible, if one accepts an increase in the measured variances. A necessary condition for a joint measurement to be possible is that a joint probability distribution exists for the measurement. This fact suggests that there may be a link with Bell inequalities, as these will be satisfied if and only if a joint probability distribution for all involved observables exists. We investigate the connections between Bell inequalities and conditions for joint quantum measurements to be possible. Mermin's inequality for the three-particle Greenberger-Horne-Zeilinger state turns out to be equivalent to the condition for a joint measurement on two out of the three quantum systems to exist. Gisin's Bell inequality for three coplanar measurement directions, meanwhile, is shown to be less strict than the condition for the corresponding joint measurement.

Violation of Bell inequalities in larger Hilbert spaces: robustness and challenges

We explore the challenges posed by the violation of Bell-like inequalities by d-dimensional systems exposed to imperfect state-preparation and measurement settings. We address, in particular, the limit of high-dimensional systems, naturally arising when exploring the quantum-to-classical transition. We show that, although suitable Bell inequalities can be violated, in principle, for any dimension of given subsystems, it is in practice increasingly challenging to detect such violations, even if the system is prepared in a maximally entangled state. We characterize the effects of random perturbations on the state or on the measurement settings, also quantifying the efforts needed to certify the possible violations in case of complete ignorance on the system state at hand.