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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.

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  • $\begingroup$ Suggestion: You might get better feedback by posting this to the quantum computing stack exchange. $\endgroup$
    – KF Gauss
    Commented Sep 15, 2020 at 5:14
  • $\begingroup$ @KFGauss Thank you, I will try. My understanding is that quantum computing people start from the "unitary transformations in a Hilbert space" abstraction and work from there. I am just trying to convince myself that this abstraction is not a too generous interpolation of the limited experimental results we have. $\endgroup$
    – Noobrador
    Commented Sep 16, 2020 at 16:51

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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.

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  • $\begingroup$ Thanks a lot for the reply. Just to be clear, I am not worried about entanglement; but the opposite: can we have $2^{-6}(|0\rangle-|1\rangle)^{\otimes 12}$ in the universe? What if we can have only up to 11 uncorrelated superpositions in the universe? $\endgroup$
    – Noobrador
    Commented Sep 15, 2020 at 0:38
  • $\begingroup$ My point is that we can construct an isolated system with a very large number of mutually uncorrelated (unentangled) particles. It does not matter that they may be entangled with particles that are outside that isolated system. That is, entanglement with outside particles prior to initiation of a quantum computation will not affect the outcome of the computation. $\endgroup$
    – S. McGrew
    Commented Sep 15, 2020 at 1:25
  • $\begingroup$ @s-mcgrew Thanks. My question is how do we construct the uncorrelated particles in part A? If I pick a bunch of electrons here and there, how do I know they are not entangled in some way. For all I know, all electrons in the universe may be entangled in some ways I would not have any way to detect it. $\endgroup$
    – Noobrador
    Commented Sep 15, 2020 at 1:28
  • $\begingroup$ @s-mcgrew If my worry is correct that the wave function is fixed dimensional, almost all electrons in the universe are entangled, potentially in very complicated ways that is not easily detectable by simple observations $\endgroup$
    – Noobrador
    Commented Sep 15, 2020 at 1:32
  • $\begingroup$ One may say that pick 12 electrons and apply the Hadamard gate one by one to each of them to get to the uncorrelated state. How do I know if I ended up with a uncorrelated state, or the universe fooled me due to the wave function having just 11 dimensions and 1st electron and the 7th got some correlations. How do I test? $\endgroup$
    – Noobrador
    Commented Sep 15, 2020 at 1:37
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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.

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  • $\begingroup$ Thanks a lot. I am wondering about "true unentanglement" defined as follows: How can we be ever sure we have $(a|\mathrm{+}\rangle + b|\mathrm{-}\rangle)(c|\mathrm{+}\rangle + d|\mathrm{-}\rangle)$ in the universe for $0<|a|, |c|<1$. $\endgroup$
    – Noobrador
    Commented Sep 16, 2020 at 15:59
  • $\begingroup$ in my opinion, since mathematics is just a model, the truth depends on experimemental errors , i.e. how well the mathematical model fits the data. Below that , there is incoherence. If you assume that it is the mathematics that generates the data/reality, i.e. the platonic view, then, never, since experiment can continually ( in principle) go to smaller and smaller errors and maybe a different mathematical theory ( maybe deterministic, not quantum mechanical) dominates at very small errors. $\endgroup$
    – anna v
    Commented Sep 17, 2020 at 3:30

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