Necessary and sufficient condition for hybrid Bell nonlocality

Abstract

Bell nonlocality [1] is the strongest type of quantum correlations. This makes it an indispensable resource for various practical applications. However, even if the quantum state can in principle exhibit this property, not all quantum measurements are applicable to detect it. In this context, necessary and sufficient conditions for a given measurement scheme, such as tight (facet) Bell inequalities, are of special interest. The problem is that the complexity of these conditions increases dramatically with the number of observables involved and with the number of their outcomes. Such inequalities are particularly difficult to formulate in the case of continuous variables. We consider a hybrid scenario in which both discrete and continuous observables are measured [2]. More specifically, one party measures two dichotomic observables, and the other party measures two continuous observables, represented by outcomes of unbalanced and balanced homodyne detection, respectively. Using Fine’s theorem [3] and the idea presented by Pironio in Ref. [4], we have formulated a necessary and sufficient condition for such a measurement scheme in the form of nonlinear inequalities. We have shown that this type of Bell nonlocality can be detected with hybrid entangled states.