(This is way beyond my area of expertise so excuse me that this might be a stupid idea.)

I assume the following happens: while a (small) subsystem is in "pure state" (in quantum coherence), no information flows out of this subsystem. Then, when measuring, information flows out and other information flows in, which disturbs the pure state. This collapses of the wave function (quantum decoherence). For all practical purposes, it looks like quantum decoherence is irreversible, but technically this could still be reversible; it's just that the subsystem (that is in coherence) got much, much larger. Sure, for all practical purposes it's then irreversible, but for us most of physics anyway looks irreversible (eg. black holes).