Lower semicontinuity of relative entropy disturbance and its consequences

It is proved that the decrease of quantum relative entropy under the action of a quantum operation is a lower semicontinuous function of the pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under the action of quantum operations. It also implies the lower semicontinuity of the modulus of the joint convexity of quantum relative entropy (as a function of ensembles of quantum states).