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arXiv:1108.0263

Upper bounds on violation of Bell-type inequalities by a multipartite quantum state

Elena R. Loubenets

Published Aug 1, 2011• 0 citations
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Abstract

We present the new exact upper bounds on the maximal Bell violation for the generalized N-qubit GHZ state, the N-qudit GHZ state and, in general, for an arbitrary N-partite quantum state, possibly infinite-dimensional. Our results indicate that, for an N-partite quantum state of any Hilbert space dimension, violation of any Bell-type inequality (either on correlation functions or on joint probabilities) with S settings and any number of outcomes at each site cannot exceed (2S-1)^{N-1}.

DOI
10.48550/arxiv.1108.0263

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Abstract For an even qudit dimension 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 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 specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound 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.