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The three quarks (uud) in a proton are held together by the strong force, and the overall e.m. force between the quarks is repulsive but much weaker, so the quarks stay together. Now two free identical protons can interact by means of the strong interaction, in the high energy regime, after which different particles can emerge. So let's assume that two protons interact at a relatively low energy, after which the spin part of the wavefunction corresponding to the two protons shows a negative correlation between the two proton spins (i.e. the two proton spins are always opposite), no matter how far the protons are apart. The place or momentum are clearly not correlated because the are far apart and thus separable.

Now my question is: how are the spins of the quarks (valence as well as non-valence) in one proton entangled with the quarks in the other proton? I know that (see comment below) the valence quarks carry almost none of the proton spins. In other words, how do the constituent quarks of one proton, become entangled with the constituent quarks of the other proton in such a way that the spin part of the two-proton overall wavefunction shows a negative correlation?

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    $\begingroup$ Actually, the valence quarks carry almost none of the the proton spin. So, even as a first approximation, your hypothesis does not work (proton spin-crisis). Although this wikipedia article is not well-written but it has the relevant references. $\endgroup$ – user154997 Jun 19 '17 at 21:40
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    $\begingroup$ But it does not really matter: you could have made it into a pure though experiment about entanglement of sub-systems, so this was just for information! $\endgroup$ – user154997 Jun 19 '17 at 21:41

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