Can two distinct quantum universes ever have the same configuration, and what does it mean for many-worlds? First, I hear that, on a whiteboard, one may casually invert causality and run time in reverse.
Next, I hear that there are interpretations of QM, like Chaitin's Great Programmer interpretation or de Broglie-Bohm pilot-wave theory, in which we can imagine a universe as being totally described by a configuration, or state, of particles and waves, as well as a set of rules describing how to obtain a configuration's successor. For example, our universe might have a ruleset resembling the Standard Model. As I understand it, the biggest trick with these theories is always remembering that there must be parts of the configuration which aren't fully knowable or measurable. Chaitin uses uncomputability and de Broglie-Bohm uses hidden variables.
Putting these two ideas together, it seems that one must be able to run QM backwards, traversing from "future" configurations to "past" configurations, in a deterministic way. Thermodynamics would appear to be flipped, of course, but everything would still be well-defined, right?
So, now let's consider two pairs of virtual particles. I'd normally call them "Alice" and "Bob", but we'll go with $A^+$ and $A^-$ for the first pair, and $B^+$ and $B^-$ for the second pair. We generally would expect that $A^+$ and $A^-$ will annihilate each other shortly, and similarly for $B^+$ and $B^-$. However, what if we placed our pairs next to each other, and they were pairs of the same sort of particle? Then, it would be legal for $A^+$ to combine with $B^-$ and $B^+$ with $A^-$, right? This suggests to me that there's a way to custom-build situations where two configurations lead to the "same" future, in that the observables of the futures are indistinguishable, but the hidden variables (pilot-wave) or uncomputable real (GP) differ, so that they will diverge again.
We could then imagine that there are actually many configurations for which the hidden parts differ, but the observable parts are the same, based on this idea of overlapping possibilities. And, in particular, we could imagine that perhaps there are situations where the observable parts of two configurations are the same during some span of time. Indeed, perhaps two configurations might have multiple spans of time in which they are observably the same, with intermittent periods of being different.
This reminds me all of Everett's many-worlds interpretation, in which we could imagine many universes that all experienced today (December 7, assuming I finish writing this soon!) roughly the same, apart from some slight differences in Brownian motion in various fluids around the planet, like the upper atmosphere, the Pacific Ocean, etc. These universes, as traditionally described, and as animated by many made-for-television documentaries, form a tree, a basic mathematical structure where parent nodes in the past relate to child nodes in the future via possible evolutions.
However, we can reverse everything, can't we? So surely we can view the many-worlds interpretation, tree and all, in the direction of the past! Parent nodes will be in the future, and child nodes will be in the past. Putting this together, from our current vantage point, we can see a pair of trees branching off into our past and our future.
But some of those possible futures and pasts might overlap, as described earlier. In that case, we don't really have a tree. We do have a graph of all universes (or of the multiverse, if you prefer that term) which seems to provide a foundation for two kinds of analysis. First, path equivalency: when do two different paths lead to the same configuration? And second, meta-universal statistics: are some configurations likelier than others by virtue of having more paths through them or other structural biases? I don't know yet, but I'd first like to know how shaky the ground is beneath these questions!
So, my question is, where did I make a mistake in my thinking? Alternatively, if I didn't, does this imply anything especially interesting? Has anybody else come to this conclusion before?
Edit: A few touchups and also removal of irrelevant wankery. To answer the first question, I was wrong to assume that it's possible to construct universes which alias each other in observable configurations.
 A: I  will answer the title of the question:

Can two distinct quantum universes ever have the same configuration, and what does it mean for many-worlds?

The many worlds interpretation is just that, a different way of interpreting the mathematics of quantum mechanics, with no predictions for any differences that would allow to make a distinction with the Copenhagen  interpretation taught in schools.
The mathematics of quantum mechanics is probabilistic, enclosing the Heisenberg uncertainty principle, HUP, in the algebra of commutators of the quantum mechanical operators.  This means that there exists an inherent uncertainty within the quantum mechanical model of the system, so that for example one may not know accurately the momentum of a particle at a specific location, but only above a limit Δ(p)xΔ(χ) given by the HUP. So identical systems cannot exist.
So the answer is that at our present knowledge, experimental and theoretical, two distinct universes cannot be in an identical state due to the HUP.
