Large-spin measurements in an arbitrary two-qudit state

2024. No. 2412.03470.

Elena R. Loubenets, Louis Hanotel

Violation of the CHSH inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under Alice and Bob spin measurements is still known only in a two-qubit case. In the present article, for a two-qudit state of an arbitrary dimension d=2s+1≥2, we introduce the notion of the spin-s correlation matrix, having dimension 3×3 for all d≥2; establish its relation to the general correlation (d^{2}−1)×(d^{2}−1) matrix of this state within the generalized Pauli representation and derive in terms of the spin-s correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under Alice and Bob spin-s measurements in this state. Specifying this general expression for the two-qudit GHZ state, the nonlocal two-qudit Werner state and some nonseparable pure two-qudit states, we find that, under large spin (s≥1) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin-1/2 measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases with a growth of its entanglement degree, the situation under large-spin measurements is quite different --for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.

Keywords: New upper bound for the CHSH inequality, quantum violation, quantum correlations