"Proof" of Born rule by principle of indifference

[greypilgrim]

Hi.

In this video of Looking Glass Universe, the host "proves" the Born rule by breaking down states into "finer" ones and then applying the principle of indifference. In the description, she bases this on papers by Deutsch, Hossenfelder, Zurek and Hardy. I have never heard of this argument so far and it seems way too simple, but those are quite respectable names...

How does this "break down" work in Hilbert spaces where the dimension is too small for the number of states needed? E.g. for a qubit in
$$\left|\Psi\right\rangle=\sqrt{\frac{2}{3}}\left|1\right\rangle+\sqrt{\frac{1}{3}}\left|0\right\rangle$$
how would one break down the first state? Or do I need to assume more "hidden" dimensions?

 

[Demystifier]

Or do I need to assume more "hidden" dimensions?

Yes. For example, you can take the spatial part of the wave function, which lives in the infinite dimensional Hilbert space.

 

[greypilgrim]

Shouldn't that be testable? If I have many copies of above state and perform a mutual spin (assuming it's a spin qubit) and position measurement on all of them, shouldn't I get a larger position spread for the $\left|1\right\rangle$ measurement then?

 

[jbergman]

Shouldn't that be testable? If I have many copies of above state and perform a mutual spin (assuming it's a spin qubit) and position measurement on all of them, shouldn't I get a larger position spread for the $\left|1\right\rangle$ measurement then?

My understanding is that this is actually a complicated question and one reason I don't find the principle of indifference that compelling. I believe the story goes something like this, when you perform a measurement the actual system is much more complicated than just the state of the spin itself. It is a tensor product with things like the environment, measurement apparatus, etc. Now, when you perform said experiment may different versions of the environment become entangled with either spin up or down, roughly in proportion to the probability of spin up and down. This is explained by light bouncing off the detector differently in the spin up and spin down configurations.

I have mainly seen this principle invoked in the context of the MWI interpretation. It always bothers me that when invoking the principle of indifference this point that you bring up is just glossed over as somehow obvious when this is a crucial step needed to justify it.