If we assume the many-worlds interpretation of quantum physics is true, what exactly happens during decoherence, that makes it impossible for the different worlds to create interference with each other afterwards?
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2 Answers
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There are a couple of ways of looking at this issue.
After decoherence, the relative state the reduced density matrix is no longer pure and that reduces the sharpness of interference see Section 2 of this paper:
https://arxiv.org/abs/1911.06282
Another way of looking at this issue is that in the Heisenberg picture after an information copying interaction the observables of each system have locally inaccessible information about the other system and interference can only take place after all of the copies this information have been combined:
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I'm not sure if this is quite the answer you're looking for but these are my thoughts on the question.
In my understanding, the answer to your question, in principle, is nothing! However, in practice it is essentially impossible. In many worlds, "wavefunction collapse" is described by entanglement of the observer with a given outcome of the system. For example, suppose Alice has a qubit in her lab which she has prepared in the superposition $$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle + |1\rangle \right).$$ To an Everettian, we have actually not fully specified the state. In particular, we must also note that Alice is in the state $|A_i\rangle$ which in words may be described as "the state of not knowing whether the qubit is in the 0 or 1 configuration." The full system is then initially in the state $$|\Psi_i\rangle = |A_i\rangle \otimes |\psi\rangle.$$
After measurement, some apparatus implements a unitary operation on both Alice and the qubit, resulting in the entangled state $$|\Psi_f\rangle = \frac{1}{\sqrt{2}}\left(|A_0\rangle \otimes |0\rangle + |A_1\rangle \otimes |1\rangle\right)$$
where $|A_0\rangle$ and $|A_1\rangle$ are the states corresponding to Alice seeing the qubit in state 0 or in state 1, respectively.
Now for the interference. In principle, there is no obstruction to performing an "intereference experiment" which, for example, returns the state to its initial tensor product form. Indeed, one would literally use the inverse of the unitary which originally formed the super-position! However, Alice, as a quantum system, is not merely another qubit system, but a macroscopic system which many complicated interactions and a very large Hilbert space. So the actual practical implementation of a unitary which inverts the measurement process, or indeed performs any interference experiment on the qubit factor, is exceedingly difficult, as you must have incredibly fine control over Alice's subsystem as well.
In decoherence essentially the same thing is happening. The desired quantum system is becoming increasingly entangled with the macroscopic environment in such a way that performing an interference experiment requires incredibly sensitive control over all of the quantum degrees of freedom not just in your desired system but in the entire environment. However, a "sufficiently powerful" outside observer with access to all the unitaries on the Hilbert space could indeed do such a thing!
This is a rather philosophical answer to the question. I'm not sure if you were looking for a more practical investigation in real-life systems, but I hope this helps!
