Nonlocality for graph states

The possibility of preparing two-photon entangled states encoding three or more qubits in each photon leads to the following problem: If n quabits were distributed between two parties, which quantum pure states and qubit distributions would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell’s theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs and provide all existing proofs up to n = 7 qubits. On the other hand, the possibility of preparing n-photon n-qubit graph states leads to the following problem: If n qubits were distributed between n parties, which would be the optimal Bell inequalities? We show all optimal n-party Bell inequalities for the perfect correlations of any graph state of n < 6 qubits. Optimal means that the ratio between the quantum violation and the bound for local hidden-variable theories is the maximum over all possible combinations of perfect correlations. This implies that the required detection efficiencies for loophole-free Bell tests are minimal.