  For the maximal violation of all Bell inequalities by an arbitrary pure
two-qudit state of any dimension, we derive a new lower bound expressed via the
concurrence of this pure state. This new lower bound and the upper bound on the
maximal Bell violation, found in [J. Phys. A: Math. Theor. 55, 285301 (2022)]
and also expressed via the concurrence, analytically quantify Bell nonlocality
of a pure two-qudit state via its entanglement, in particular, prove explicitly
that entanglement of a pure two-qudit state is necessary and sufficient for its
Bell nonlocality. By re-visiting the pure two-qubit case, we also find and
rigorously prove the new results on the correlation properties of an arbitrary
pure two-qubit state.

Quantum Physics