Abstract

We formulate and prove the main properties of the generalized Gell–Mann representation for traceless qudit observables with eigenvalues in and analyze via this representation violation of the Clauser–Horne–Shimony–Holt (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 , 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 a class of two-qudit states where the new upper bound improves the upper bound of Tsirelson. 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 where the new upper bound is lower 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 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 is equal to if is even and to if d > 2 is odd.

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