It makes sense to talk about the split propagating. A subsystem which is spatially located at a distance from the splitting event will not immediately become involved in the superposition, but may do once information about the event has traveled.

When we talk about MWI as the asker is trying to more fully understand, we are explicitly not talking about subsystems. There's just the one big evolving state (of the universe).

That's true, but that global state of the universe can be written in terms of its reduced states on spatial subsystems. From this perspective we can talk meaningfully about propagating superpositions. For example, take a toy example in one dimension with three spatial subsystems A, B and C representing disjoint intervals arranged in order. Immediately after a splitting event the combined wavefunction over all subsytems might be:

(psi_1 + psi_2)_A x phi_b x rho_C

(ignoring normalisation for convenience) then after a certain amount of time it becomes

(psi_1 x phi_1 + psi_2 x phi_2)_AB x rho_C

and then eventually

(psi_1 x phi_1 x rho_1 + psi_2 x phi_2 x rho_2)_ABC .

Now, all of that is certainly happening within the realm of some joint superstate, but it still makes sense to talk about how fast the split propagates, surely?

All depends on the context. Does it make sense to talk about propagating splitting in this context? Surely no - for three reasons:

1. You're describing a time evolution _of the subsystems_, which isn't really a thing in the MWI.

A many-worlder would say, instead, that the Universe has split a bunch more in the interim. He would point to the time evolution of the state of the Universe, and perhaps there have a discussion about how the inseparability of particular subsystems has propagated over time. Put differently, the many-worlder might say the correlations of these particular relative states with one another propagated over time.

What you've done, Everett would call characterizing branches of the universal state in a space-like locality.

2. Split != superposition. Frequently, splitting in MWI is identified with decoherence, so in that sense there is a self-consistent way to describe local splitting - but then you'd really mean, when you referred to the splitting of "an object" or "a system", that Universal splitting had occurred in such a way as to cause the object to exist in some particular multiple new branches.

3. None of this line of discussion helps the parent gain an understanding of how MWI is importantly different from (and the same as) other interpretations of QM. It's far too shallow to amount to any real expert insight and yet too technical to amount to any real layperson insight.

What can a discussion on propagating splitting illuminate here? It seems to me that it is a less than useful idea for the parent and readers like him/her, and many-worlds is more clearly understood without it.

> You're describing a time evolution _of the subsystems_

In my head I'm thinking about the time evolution of the global state, but examining the reduced state over certain subsystems at specific points in time.

> Split != superposition. Frequently, splitting in MWI is identified with decoherence

Decoherence is a superposition effect, is it not? Entanglement with the environment, i.e. a superposition of system-environment states.

> then you'd really mean, when you referred to the splitting of "an object" or "a system", that Universal splitting had occurred in such a way as to cause the object to exist in some particular multiple new branches

Yes, this is what I mean.

> What can a discussion on propagating splitting illuminate here?

Tbh I think it's unlikely that the parent is still following but I'm continuing for the selfish purpose of trying to better understand your point. That said, I believe that considering my toy example of a global quantum state in one dimension would illuminate their question about superpositions propagating from here and alpha centauri and meeting in the middle.

> Decoherence is a superposition effect, is it not? Entanglement with the environment, i.e. a superposition of system-environment states.

The point I'm trying to make is that "splitting", while sometimes identified with decoherence, isn't superposition (or any other well defined traditional QM phenomenon). It's a term peculiar to MWI and it importantly has no clear canonical technical definition. It generally refers to something just considered abstractly: the branching of a single _universe_ into multiple. If you use "split" and "entanglement" or "superposition" or any other QM term interchangeably, you are bound to invite misunderstanding.

> ...illuminate their question about superpositions propagating...

Agreed... if that was their question. But their question didn't reference superposition at all, it was about a split propagating:

> ...a quantum event occurs here and the universes split, that split propagates out at the speed of light...

Which is why I responded as I did. It is understandably confusing to wonder what it means for propagating split universes to meet years later, if you start talking about splits in this way. Propagating superposed particles? Much easier to make sense of.

Thank you for bearing with me for so long. I think I understand the point of contention, i.e. that "split" is a slightly nebulous term which depends not only on splitting but somehow on there being a negligible likelihood of future interference between branches. In this context I agree it doesn't make sense to speak of a split being spatially localised.