Is there an unentangled electron pair in the universe? 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?


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?

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.
 A: Your concern is unnecessary.  Suppose the universe is divided into two separate regions A and B, and every particle in region A is entangled with a corresponding particle in region B, but not entangled with other particles in region A.  No experiment done solely on the particles in region A can reveal that the particles are entangled with particles in the other region.  The particles in that one region will behave the same, whether entangled or not.  The only way to detect entanglement between the particles is to perform experiments on both members in the pairs - one member residing in A and the other member residing in B.
A: I am answering the title:

Is there an unentangled electron pair in the universe?

Please keep in  mind that entanglement really  means that "there exists a quantum wavefunction that describes the entangled particles". Then one uses the conservation laws to find a way  to check the statement experimentally . i.e. it is a quantum mechanical model.
Assuming that a quantum mechanical model will be the "theory of everything" TOE allows to make a statement that in this theory there exists a wave function of the universe, so, in such a model,  everything is entangled with everything else.  Considering the number of particles involved in the universe, the dimensions etc there is no way to check this hypothesis experimentally, even though there are models that state this, particularly  ones involved in cosmology.
In real life and data, one uses the density matrix formalism to model many particle states quantum mechanically, to distinguish between coherent, particles described by the same wavefunction, (where entanglement can be found), and incoherent ones where the information is lost because of experimental accuracy . See this lecture.
So the answer is : in a quantum mechanical TOE in principle no, everything is entangled by the wave function of the universe. BUT  the size of the coupling constants and the dimensions do not allow to find correlations experimentally, and impose a density matrix formalism which separates the universe  into coherent and incoherent states.
