Necessary and sufficient state condition for violation of a Bell inequality with multiple measurement settings

High-dimensional quantum entanglement and the advancements in their experimental realization provide a playground for fundamental research and eventually lead to quantum technological developments. The Horodecki criterion determines whether a state violates the Clauser-Horne-Shimony-Holt (CHSH) inequality for a two-qubit entangled state, solely from the state parameters. However, it remains a challenging task to formulate similar necessary and sufficient criteria for a high-dimensional entangled state for the violation of a suitable Bell inequality. Here we develop a Horodecki-like criterion based on the state parameters of arbitrary two-qudit states to violate a two-outcome Bell inequality involving ${2}^{n\ensuremath{-}1}$ and $n$ measurement settings for Alice and Bob, respectively. This inequality reduces to the well-known CHSH and Gisin elegant Bell inequalities for $n=2$ and 3, respectively. While the proposed criterion is sufficient to violate the Bell inequality, it becomes necessary as well for the following cases: (i) $m$ copies of Bell-diagonal states for arbitrary $n$, (ii) nondecomposable states whose correlation matrix is diagonalized by local unitaries, and (iii) for any arbitrary two-qubit state when $n=3$, where the maximal value of the Bell functional is achieved with Bob's measurements being pairwise anticommuting. For any states, we derive the constraints on Alice's measurements in achieving the maximum quantum violation for this inequality.