This question is related but my question here is much more elementary than discussions of the Pauli principle across the universe.
There has been a fair amount of discussion around at the moment on the double well potential model in quantum mechanics. The question I have here is just concerned with some very simple basic schoolboy quantum mechanics that you need to get right in order to progress with the discussions (I think these questions would be too elementary for a physicist to ask - fortunately I'm not a physicist, so my embarrassment is under control).
The scenario is this: I have a double square finite depth potential well with a large distance separating the wells. Call them left hand and right hand wells. The model I want to use is elementary nonrelativistic quantum mechanics. The energy eigenstates of such a system are similar to those of a single well, but what was one energy eigenstate now becomes two with incredibly closely spaced energy levels (ref).
Suppose I have two electrons in this system in the two very closely spaced lowest levels $E_1$ and $E_2$ . The two electrons are indistinguishable, so the state is $${{1}\over{\sqrt{2}}}(|E_1\rangle|E_2\rangle-|E_2\rangle|E_1\rangle)$$ The statement that has often been made in these discussions is something like "an observer in the left hand well now makes an energy measurement of an electron and gets the value $E_1$ ".
In talking about this scenario, in the link I gave above Sean Carroll stated
"Imagine that we have such a situation with two electrons in two atoms, in a mutually entangled state. We measure our electron to be in energy level 1. Is it true that we instantly know that our far-away friend will measure their electron to be in energy level 2? Yes, absolutely true."
This is where I'm confusing myself: By assumption, our system was in an energy eigenstate, and we've made an ideal energy measurement, so by the rules of quantum mechanics after the measurement it ends up still in an energy eigenstate after the measurement. But the observer knows that we have a double well two-fermion system and the correct system state is still the antisymmetrized one $${{1}\over{\sqrt{2}}}(|E_1\rangle|E_2\rangle-|E_2\rangle|E_1\rangle)$$. So for an idealized measurement, the system started in this state and remained in this state. So I ask
question (1): If the measurement proceeds like this, without any extra entanglement-introducing mechanism, won't the observer think "if my far-away friend now makes an energy measurement, he'll have a 50/50 chance of getting $E_1$ or $E_2$" ?
In other words: since this is non-relativistic QM, the particle could have moved to the other well immediately after the first measurement, so the state is still antisymmetrized over the particle identities.
In other other words, the state I described isn't mutually entangled - just antisymmetrized over the electron identities.
question (2): if this correct then how, typically, would I introduce some entanglement between the electrons, as in Sean's statement ?
Finally, when we say "an observer in the left well makes an energy measurement", doesn't this introduce some locality into the picture for the following reason: What it implies is that the observer's measuring equipment is spatially localized in the left well, so that when an energy measurement is made, there is also an implied acquisition of information about the measured electron's location at that time. So I come to:
question (3) Wouldn't this mean that after such a local measurement, the measured electron cannot be in one of the system's energy eigenstates ? So the system state would be something like $${1\over{\sqrt{2}}}(|\Psi\rangle|E_2\rangle-|E_2\rangle|\Psi\rangle)$$.
where $\Psi$ is a spatially localized state, concentrated in the LH well.
This doesn't sound right. Where is my reasoning going wrong ?
Answer
Your confusion is fully appropriate. It is far from schoolboy QM, most wouldn't even notice the discrepancy and live happily with ensuing inconsistencies.
Entanglement is not applicable to this situation, as the Hilbert space in question is not a tensor product.
You can entangle only properties of a composite system that are described by a full tensor product. But states of identical particles only make up the antisymmetric part of a tensor product; so all the common machinery goes awry.
The situation is explained in detail in the section ''Indistinguishable particles and entanglement'' of Chapter B3: Basics on quantum fields of A theoretical physics FAQ, which ends with:
''But if, on a more practical level, one knows already that (by preparation) there are two electrons or two photons, one moving to the left and one moving to the right, one can use this information to distinguish the electrons or photons: The particles are now described by states in the tensor product of two independent, much smaller effective Hilbert spaces with a few local degrees of freedom each (rather than a single antisymmetrized 2-particle Hilbert space with degrees of freedom for every pair of positions), which makes them distinguishable effective objects. Therefore, one can assign separate state information to each of them, and construct arbitrary superpositions of the resulting tensor product states. These effective, distinguishable particles can be (and typically are) entangled.''
[edit after discussion below] I updated my FAQ to give some more details, using spins in place of your energies, which makes the situation more realistic. I quote the relevant passages:
''For example, two electrons are on the fundamental level indistinguishable; their joint wave function is proportional to |12>-|21>; no other form of superposition is allowed. [...] Thus it is impossible to project it by measurement to a separable state, as discussions about entanglement always assume.''
''Realistic discussions about entanglement in quantum information theory usually consider the entanglement of spin/polarization degrees of freedom of photons transversally localized in different rays, electrons localized in quantum dots, etc.. For example, two electrons in two quantum dots $dot_1$ and $dot_2$ are in the antisymmetrized state
$|\psi_1,\psi_2\rangle := |\psi_1,dot_1 \rangle \otimes |\psi_2,dot_2 \rangle- |\psi_2,dot_2 \rangle\otimes|\psi_1,dot_1\rangle$
in 2-particle space (depending on 6 coordinates apart from the spins, the latter represented by $\psi_1$ and $\psi_2$), but as the positions are ignored (being just used to identify which electron is where), they are treated as effective electrons in the 2-particle state $\psi_1 \otimes\psi_2$ residing in the tiny 4-dimensional tensor product $C^2\otimes C^2$. If spin up is measured at $dot_1$, the spin part $\psi_1$ collapses to up, etc.. Note that there is no position measurement involved in observing the spin, as the quantum dots already (and permanently) have measured the presence of a particle in each dot.
Thus in this small effective space, talking about electron entanglement is meaningful, as disregarding position (responsible for the antisymmetrizing) reduces the state space to a tensor product. But in the full 2-particle space, the attempt to reason with entanglement fails on the formal level and is at best paradoxical if just considered informally.
Edit2 (March 13, 2012): The approximate representation mentioned in the answer by Lubos is essentially the same as the exact reduced representation in my answer, if we identify left well = dot_1 and right well = dot_2. It is a reduced description that accounts for the knowledge that one electron is in each well and has to stay there because of the walls of the well. (If the walls were infinitely high, his representation would be exactly the same.) This means having a way to distinguish the electrons, so that one can use a smaller tensor product Hilbert space inside the big antisymmetric Hilbert space to (approximately or exactly, depending on the form of the well) describe the degrees of freedom left.
In my answer above, I took as only dof left the spins, as this is the situation that prevails in quantum information theory. However, the reduced Hilbert space can take other degrees of freedom into account. As long as these additional degrees of freedom are local to each well/dot, one still has the tensor product structure, and hence a setting where entanglement makes sense, and in which one can consider to prepare an entangled 2-electron state.
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