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 $d\ge 2$, 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 $d\ge 2,$ 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 $d\ge 2$ 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 $2\sqrt{2}$ if $d\ge 2$ is even and to $\frac{2(d-1)}{d}\sqrt{2}$ if $d>2$ is odd.