How can we be sure that there are two electrons in the universe whose spins are uncorrelated (their joint state is the tensor product of their individual projections) but each of them has nonzero magnitude for both + spin and - spin?

Is there are a theoretical foundation requiring this?

More empirically, is there a physical test (a unitary followed by wave function collapse) which would separate qudits

* whose joint state vector is far away from any tensor product in $\ell\_2$ distance versus
* whose joint state vector is within epsilon in $\ell\_2$ of a tensor product?

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What if we repeat the first question with 2 replaced by, say 12? Do we know if there can be 12 spin-uncorrelated electrons at the same time in the universe (say each one has equal magnitude for + spin and - spin)?

What if the 'pilot wave function' of the universe has a fixed dimensionality, say 11, so that we can never find more than 11 particles in the universe whose joint state is the tensor product of the projections.

Such a possibility would say that whatever quantum computer we make, no matter how complex, would be equivalent to a 11 qubit computer.

Has this possibility been ruled out either empirically or theoretically?

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Consider the alternate universe: The `pilot wave function' of the universe is 11 dimensional. We pick 12 electrons which are individually known to have equal magnitude superposition of + spin and - spin. We observe their spins, which forces a wave function collapse and hence we get 12 explicit signs. How would be able to tell if these signs came from a 12-wise independent distribution or (a statistical ensemble of) 11-wise independent distribution(s)? It feels to me that by just plain observations like these there is now way we can tell these apart.