How Does MWI Interpret Simultaneous Quantum Measurements?

[Abdullah Naeem]

Suppose there are two independent experiments taking place $\left\vert \Phi\right\rangle =\alpha _{1}\left\vert \phi _{1}\right\rangle +\beta_{1}\left\vert \phi _{2}\right\rangle$ and $\left\vert \Psi \right\rangle=\alpha _{2}\left\vert \psi _{1}\right\rangle +\beta _{2}\left\vert \psi_{2}\right\rangle$. According to MWI, when $\left\vert \Phi \right\rangle$ is measured, there are two "branches" of the world, one for each $\left\vert\phi _{i}\right\rangle$. Similarly, for $\left\vert \Psi \right\rangle$. My question is, what happens when a measurement for each $\left\vert \Phi\right\rangle$ and $\left\vert \Psi \right\rangle$ takes place simultaneously? As I see it, there are two worlds, one for each $\left\vert \Phi \right\rangle$ but for these worlds but in these worlds, what happens to $\left\vert \Psi \right\rangle$? Is it that, in these two worlds, $\left\vert \Psi \right\rangle$ has not taken place?

 

[Demystifier]

In this case the full state of the "multiverse" is $\left\vert \Phi\right\rangle\left\vert \Psi \right\rangle$. If you do the multiplication explicitly, you will see that it contains 2x2=4 branches (4 "worlds").

 

[Abdullah Naeem]

In this case the full state of the "multiverse" is $\left\vert \Phi\right\rangle\left\vert \Psi \right\rangle$. If you do the multiplication explicitly, you will see that it contains 2x2=4 branches (4 "worlds").

Thank you for your reply.
So the state of the world before measurement is $\left\vert \Phi \right\rangle \left\vert \Psi \right\rangle =\alpha _{1}\alpha _{2}\left\vert \phi _{1}\psi _{1}\right\rangle +\alpha _{1}\beta _{2}\left\vert \phi _{2}\psi _{1}\right\rangle +\beta _{1}\alpha _{2}\left\vert \phi _{1}\psi _{2}\right\rangle +\beta _{1}\beta _{2}\left\vert \phi _{2}\psi _{2}\right\rangle$. Four possibilities, right.
So, according to MWI, there will be four branches. I think my confusion
comes in the tensor product. I am thinking of two machines, $\Phi$ and $\Psi$, each of which will enter two worlds. For the first machine $\Phi$,
the outcomes are either $\left\vert \phi _{1}\right\rangle \left( \alpha _{2}\left\vert \psi _{1}\right\rangle +\beta _{2}\left\vert \psi _{2}\right\rangle \right)$ or $\left\vert \phi _{2}\right\rangle \left( \alpha _{2}\left\vert \psi _{1}\right\rangle +\beta _{2}\left\vert \psi _{2}\right\rangle \right)$. How does it know that the other outcome is
either $\left\vert \psi _{1}\right\rangle$ or $\left\vert \psi _{2}\right\rangle$?

 

[Demystifier]

The first machine does not know that the second machine must have one of the two outcomes. But the "world" consists of both machines together. If you are interested in only one machine, then you cannot call it a "world".