# Change of basis in the many worlds interpretation?

Say we have two orthogonal states $$|A\rangle$$ and $$|B\rangle$$. In the many worlds interpretation, we can imagine two parallel universes in which we are either in state $$A$$ or $$B$$.

But now if we change the basis so that we have:

$$|C\rangle = \frac{1}{\sqrt{2}}(|A\rangle+|B\rangle)$$ and $$|D\rangle = \frac{1}{\sqrt{2}}(|A\rangle-|B\rangle)$$

It seems we have to then adjust our interpretation of the parallel universes. The only way out of seems to be if the universes $$A$$ and $$B$$ correspond to a universe with person with a brain beleiving or disbelieving a true/false statement. Since one cannot both believe something is true and false. But then don't we arrive back at the copenhegen interpretation with an observer collapsing a wave function?

What is the answer?

• There seem to be three unrelated ideas here: (1) the phase relationship between A and B (which ends up being irrelevant because of decoherence), (2) the preferred basis problem, and (3) the fact that an observer can't consciously experience the fact of being in a superposition of states. One could certainly draw connections among these ideas, but it's not obvious to me from reading the question what connection you have in mind. – user4552 Jun 18 '19 at 13:30

## 1 Answer

A measurement is an interaction that results in a record that can be copied and examined with arbitrarily high accuracy. In quantum physics, the information that can be copied in this way in a particular measurement consists of the eigenvalues of an observable:

https://arxiv.org/abs/1212.3245

In the many worlds interpretation (MWI) the whole of physical reality consists of a structure in which to a good approximation information flows in channels each of which resembles the universe as described by classical physics:

https://arxiv.org/abs/quant-ph/0104033

So the way that the multiverse is actually divided into universes is constrained by quantum physics so that each universe corresponds to eigenvalues of particular measured observables. We can understand this process to any precision you like in the MWI because there is a particular set of equations of motion that can be understood and modeled. This will coincide with the states of observers because observers have to be able to copy information to create knowledge, do observations and so on, but those observers are not put in a separate category from other physical processes.

The Copenhagen interpretation (CI) avoids making any clear statement about what exists in reality. There is no prospect of the MWI exactly agreeing with the CI because the CI doesn't make clear statements or predictions.