The quantitative adiabatic condition (QAC), or quantitative condition, is a
convenient (a priori) tool for estimating the adiabaticity of quantum evolutions.
However, the range of the applicability of QAC is not well understood. It has
been shown that QAC can become insufficient for guaranteeing the validity of
the adiabatic approximation, but under what conditions the QAC would become
necessary has become controversial. Furthermore, it is believed that the inability
for the QAC to reveal quantum adiabaticity is due to induced resonant transitions.
However, it is not clear how to quantify these transitions in general. Here
we present a progress to this problem by finding an exact relation that can reveal
how transition amplitudes are related to QAC directly. As a posteriori condition
for quantum adiabaticity, our result is universally applicable to any (nondegenerate)
quantum system and gives a clear picture on how QAC could
become insufficient or unnecessary for the adiabatic approximation, which is a
problem that has gained considerable interest in the literature in recent years