# The generalized Gell-Mann representation and violation of the CHSH inequality by a general two-qudit state

## TL;DR

This paper introduces a generalized Gell-Mann representation for qudit observables, analyzes CHSH inequality violations in two-qudit states, and derives new bounds for the maximal CHSH expectation, with explicit results for GHZ states.

## Contribution

It develops a generalized Gell-Mann framework and establishes new bounds for CHSH violation in two-qudit states, including explicit maximum values for GHZ states across dimensions.

## Key findings

- New bounds for CHSH expectation in two-qudit states.
- Exact maximum CHSH expectation for two-qudit GHZ states.
- For two-qubit states, bounds coincide with known Tsirelson results.

## Abstract

We formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and analyze via this representation violation of the CHSH inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $d\geq2$, this allows us to find two new bounds, lower and upper, expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for each two-qudit state. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH\ result of Horodeckis, and also, for the Greenberger-Horne-Zeilinger (GHZ) state with an odd $d\geq2,$ where the new upper bound is less than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension $d\geq2$ and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in $[-1,1]$ is equal to $2\sqrt{2}$ if $d\geq2$ is even and to $\frac{2(d-1)}{d}\sqrt{2}$ if $d>2$ is odd.