Chain of Hardy-type local reality constraints for n qubits

Abstract

Nonlocality without inequality is an elegant argument introduced by L. Hardy for two qubit systems, and later generalised to n qubits, to establish contradiction of quantum theory with local realism. Interestingly, for n=2 this argument is actually a corollary of Bell-type inequalities, viz., the CH-Hardy inequality involving Bell correlations, but for n greater than two it involves n-particle probabilities more general than Bell-correlations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for n qubits and then associated with each such inequality, we provide a new Hardy-type local reality constraint without inequalities. Quantum mechanical maximal violations of the chain of inequalities and of the associated constraints are also studied by deriving appropriate Cirel'son-type theorems. These results involving joint probabilities more general than Bell correlations are expected to provide a new systematic tool to investigate entanglement.