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

## TL;DR

This paper introduces a class of two-qudit states with perfect correlations and proves that their maximal Bell inequality violation is bounded by 3/2, extending understanding of quantum nonlocality bounds beyond qubits.

## Contribution

The paper defines a new class of two-qudit states with perfect correlations and establishes an upper bound of 3/2 on their Bell inequality violations, generalizing previous results.

## Key findings

- Maximal violation of Bell inequality by these states is bounded by 3/2.
- Two-qudit GHZ states with even dimension exhibit perfect correlations.
- The results support the conjecture of a universal 3/2 upper bound for such violations.

## Abstract

For an even qudit dimension $d\geq 2,$ 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 $d\geq 2$ 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.