  We employ quantum relative entropy to establish the relation between the
measurement uncertainty and its disturbance on a state in the presence (and
absence) of quantum memory. For two incompatible observables, we present the
measurement-disturbance relation and the disturbance trade-off relation. We
find that without quantum memory the disturbance induced by the measurement is
never less than the measurement uncertainty and with quantum memory they depend
on the conditional entropy of the measured state. We also generalize these
relations to the case with multiple measurements. These relations are
demonstrated by two examples.
