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

Abstract

For an even qudit dimension $d\ge 2,$ 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 $d\ge 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the $\frac{3}{2}$ upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound $2\sqrt{2}$ specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound $\frac{3}{2}$ 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.

arXiv comment16 pages

References in corpus (8)

• The Bloch Vector for N-Level Systems
• Bloch vectors for qudits
• Generalizing Tsirelson's bound on Bell inequalities using a min-max principle
• New Bell inequalities for the singlet state: Going beyond the Grothendieck bound
• Bell's nonlocality in a general nonsignaling case: quantitatively and conceptually
• Towards experiments to test violation of the original Bell inequality
• Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations
• Local hidden variable modelling, classicality, quantum separability, and the original Bell inequality

Cited by in corpus (2)

• The Bloch vectors formalism for a finite-dimensional quantum system
• Specifying nonlocality of a pure bipartite state and analytical relations between measures for bipartite nonlocality and entanglement