Is the Second Law a consequence of the Many Worlds principle? 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?
 A: Your description of the branches and the information stored seems correct.
Though, there is one thing I would add. You are saying that:


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).


I must disagree. Though I understand your point of view, this would mean that the universe was set up so, that it (spacetime) could be discrete. I think to our current knowledge, spacetime is continuous.
You are describing the wavefunction as if it would need to be stored somewhere (and would need certain size of space to be stored). Actually I have a question on this site about this. Because I had the same idea.
Where is the info of the wavefunction stored at?
Now your idea says:


*

*the wavefunction (at least its information) must be stored somewhere, though the wavefunction itself might just be some selection of information about spacetime itself

*the information is discrete (not continuous), because to store a boolean (1.0) you only need less information, and less space to store as for the probability of 64:1


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.


I agree. What I do not agree with, is that Q sees just the branch it goes into (and so Q will need less and less info as he goes through the branches). Q sees all the tree.
I agree with that there is less and less local info in each branch too. What I do not agree with is that there is need or possibility to store this (less) info in less storage space.
So basically I would just add two things:


*

*Q sees all the tree, and entropy is not evolving, it is constant for the tree. You can select branches for where entropy is increasing.

*You cannot nor need to store less info for a boolean (1,0) then for 64:1 possibility. This is because the wavefunction is just information about spacetime and that is continuous.
It is just us humans who try to use our math and describe in a discrete way spacetime, which to our current knowledge is continuous. 
Now I can set the speed of light as c=1 or c=299 792 458 m/s. The former needs only one bit, the latter needs a few bits. I do understand that the case of probabilities is different, but it is still just convention.
You are saying that as you move through the branches, you make measurements, and that increases entropy, because it decreases the amount of information in each branch as you go forward.
I do not believe that as you make measurements, and go forward in the branches, there is less and less info in each branch. The fact that you are inside that branch contains the complementary info that you lost by moving into that branch. Being inside a branch means not being in the other ones, so the info is globally not decreasing.
I agree that de-branching or as you say going backwards in the tree (joining of multiple branches) is against the Second Law of Thermodynamics.
