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A blog version of this answer of mine is here;

http://motls.blogspot.com/2012/03/energy-measurements-in-two-fermion.html#more

At any rate, you won't find any helpful way to argue that (and why) these two electrons are entangled. The question is either ill-defined or they are not entangled. And even if you found a (contrived) definition that would allow you to say that the simple state is entangled, such an "entanglement" will have no physical consequences. Two highly separated regions (or wells) are independent. In particular, the laws of quantum field theory are exactly local so a measurement or decision done near one well won't immediately influence a spatially separated other well.

1. Finding an electron in the left well ground state means that it has 50% odds to be in the $E_1$ state and 50% to be in the nearby $E_2$ state of the double well problem; we can't simultaneously distinguish left-right as well as $E_1$ vs $E_2$ because the corresponding operators refuse to commute with one another. If we find an electron near the left well, what the antisymmetry allows to tell us is that the second electron is near the right well, and vice versa. But measurements linked to one of the two regions can't tell us about the exact energy of one electron (and therefore it tells us nothing about the energy of the other one, either)

2. In the description of "individual electrons", one can't talk about entanglement because the full Hilbert space is an antisymmetrization (reduced version) of the tensor product, not the full tensor product. In the approximate description of quantum field theory on two regions, the big Hilbert space tensor factorizes and the two-electron state (occupying the two low-lying states) isn't entangled. If the initial state is not entangled and the evolution of the quantum system respects locality (and quantum field theory does), no entanglement may be created by actions done near one well or the other well. Entanglement is always a consequence of the two subsystems' being in contact in the past.

3. Yes, as I said, you're exactly right: if we know that an electron is near the left well, the odds for its being in the $E_1$ two-well state or the nearby $E_2$ two-well state are exactly 50% for both cases. The left-vs-right and $E_1$-vs-$E_2$ can't be measured simultaneously much like $J_z$ and $J_x$ components of the spin cannot; in fact, these two examples are totally mathematically isomorphic.

At any rate, you won't find any helpful way to argue that (and why) these two electrons are entangled: we're not learning any new information (such as the spin) about "the electron in the left well" at all so this "no information" can't be entangled with any information from the right well (which is also empty). The question whether there's entanglement here is either ill-defined or they are not entangled. And even if you found a (contrived) definition that would allow you to say that the simple state is entangled, such an "entanglement" will have no physical consequences. Two highly separated regions (or wells) are independent. In particular, the laws of quantum field theory are exactly local so a measurement or decision done near one well won't immediately influence a spatially separated other well.

1. Finding an electron in the left well ground state means that it has 50% odds to be in the $E_1$ state and 50% to be in the nearby $E_2$ state of the double well problem; we can't simultaneously distinguish left-right as well as $E_1$ vs $E_2$ because the corresponding operators refuse to commute with one another. (I say "refuse", not "fail", because it's a holy right – and the dominant situation – for two operators not to commute. They have no duty to commute in quantum mechanics so a nonzero commutator isn't a failure, isn't bad in any way.) If we find an electron near the left well, what the antisymmetry allows to tell us is that the second electron is near the right well, and vice versa. But measurements linked to one of the two regions can't tell us about the exact energy of one electron (and therefore it tells us nothing about the energy of the other one, either)

2. In the description of "individual electrons", one can't talk about entanglement because the full Hilbert space is an antisymmetrization (reduced version) of the tensor product, not the full tensor product. In the approximate description of quantum field theory on two regions, the big Hilbert space tensor factorizes and the two-electron state (occupying the two low-lying states) isn't entangled. If the initial state is not entangled and the evolution of the quantum system respects locality (and quantum field theory does), no entanglement may be created by actions done near one well or the other well. Entanglement is always a consequence of the two subsystems' being in contact in the past.

3. Yes, as I said, you're exactly right: if we know that an electron is near the left well, the odds for its being in the $E_1$ two-well state or the nearby $E_2$ two-well state are exactly 50% for both cases. The left-vs-right and $E_1$-vs-$E_2$ can't be measured simultaneously much like $J_z$ and $J_x$ components of the spin cannot; in fact, these two examples are totally mathematically isomorphic.

The system is simply composed of two independent systems – two wells in two different regions – that are not correlated or entangled at all. A non-entangled state is defined as one that can be written as a tensor product and that's exactly what we can do here.

The system is simply composed of two independent systems – two wells in two different regions – that are not correlated or entangled at all. In quantum field theory, the tensor product state above could be written as $a^\dagger_\text{left well} a^\dagger_\text{right well}|0\rangle$ where the two creation operators don't carry any labels and they are composed of field operators near the two wells, respectively. A non-entangled state is defined as one that can be written as a tensor product and that's exactly what we can do here (in the two-region approximation).