The generalized Gell–Mann representation and violation of the CHSH inequality by a general two-qudit state
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≥2, 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≥2, 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≥2 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√2 if d≥2 is even and to ((2(d-1))/d)√2 if d>2 is odd.