I've done a bit of browsing on this subject, and haven't found any papers that directly address this question. Here's the idea:
In the Many Worlds View (MWV), there is no loss of information from the global perspective. An external god-like observer "Q" might add up all the information that's present in all the branches of the universal wavefunction, and find that it never changes. However, from the perspective of an observer "B" who is himself a component of the wavefunction, it should seem that information is steadily leaking out of his world. Every time an event impinges on "B's" state, there is what a Copenhagen convention advocate would call a wavefunction collapse. What Everett would say is that B's world "splits", reducing the uncertainty from B's perspective in each of the "new worlds".
The loss of uncertainty amounts to an increase in the correlation among the components of B's world but a loss of information. For example, B's measuring instrument says "spin up" in one world and says "spin down" in the other world immediately following measurement of a particle's spin. The particle's spin is no longer uncertain in either of the worlds.
Observer Q has no problems with this: he's got Everett's perspective. From B's perspective, though, information has been lost. Before measurement, the wavefunction might need multiple bits to describe it (e.g., the ratio of "up" to "down" probabilities might be 64:1 which needs 6 bits). The wavefunction after measurement consists of one bit: 1 or 0 (up or down).
So, from Q's perspective, it would seem that the universal wavefunction is steadily evolving in such a way that individual branches contain less and less information- so entropy is necessarily increasing in each branch. The Second Law of Thermodynamics, then, would be tantamount to a statement that although branching can occur in the universal wavefunction, "de-branching" or joining of multiple branches to form one branch cannot occur.
Does this make sense? Are there published papers that address the question?