Quantum analog of the original Bell inequality for two-qudit states with perfect correlations/anticorrelations

Abstract

For an even qudit dimension d ≥ 2, we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value 3/2 the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger Horne Zeilinger (GHZ) state with an arbitrary even d ≥ 2 exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the 3/2 upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound 2√2 specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound 3/2 specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.