  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.
