As far as I know, the discussions on this issue are still ongoing. I thought I'd describe the situation from the viewpoint of my armchair.
Regardless of the discussions regarding whether Brian Cox should perhaps have said “quantum state”, rather than “energy level” in the TV show, this whole discussion has made me try to understand the applicability of the concept of entanglement to a situation such as this. Certainly Cox and Forshaw in their
book did have entanglement in mind in connection with this issue, since they state:
There is only ever one set of energy levels and when anything changes (e.g. an electron changes from one energy level to another) then everything else must instantaneously adjust itself so that no two fermions are ever in the same energy level.
The idea that electrons ‘know’ about each other instantaneously sounds like it has the potential to violate Einstein’s Theory of Relativity. Perhaps we can build some sort of signalling apparatus that exploits this instantaneous communication to transmit information at faster-than-light speeds. This apparently paradoxical feature of quantum theory was first appreciated in 1935 by Einstein in collaboration with Boris Podolsky and Nathan Rosen; Einstein called it ‘spooky action at a distance’ and did not like it. It took some time before people realized that, despite its spookiness, it is impossible to exploit these long range correlations to transfer information faster than light and that means the law of cause and effect can rest safe.
Entanglement does indeed allow quantum measurements to display “instantaneous” influences, but no information can be transmitted using this mechanism. But, how would you go about applying entanglement to the scenarios they're discussing?
The model Cox and Forshaw are using is the double rectangular potential well. This model is described
here. The energy eigenstates of a single rectangular well are split into pairs of energy eigenstates with very closely spaced energy eigenvalues. One member of a pair is a wavefunction with odd reflection symmetry about the origin and the other has even reflection symmetry.
We now populate the double well system with a pair of fermions. For simplicity, they could be spinless electrons, which would have to be in different states to respect their fermionic nature. As an example, they could be in each of the two lowest energy eigenstates, so the system state would be
|\Psi \rangle={1\over{\sqrt{2}}}(|E_1 \rangle |E_2 \rangle-|E_2 \rangle |E_1 \rangle) \ \ \ (0)
The sort of question we would like to ask is whether or not there is entanglement between quantities measured in the left hand well and quantities measured in the right hand well?
Conventionally, entanglement questions would be treated by decomposing the full Hilbert space in the form
{\mathcal{H}=\mathcal{H_{L}}\otimes\mathcal{H_{R}}}
For example, in the "classic" EPR entanglement scenario, this sort of decomposition is clear - \mathcal{H_{L}} is the two dimensional Hilbert space of spin states of a LH-travelling spin 1/2 decay product of a spin 0 singlet state, and \mathcal{H_{R}} the RH-travelling equivalent.
For any pure state |\Psi\rangle\in\mathcal{H}I can choose an orthonormal basis \{|\Psi^L_{i}\rangle\}for \mathcal{H_{L}} and \{|\Psi^R_{i}\rangle\}for \mathcal{H_{R}} such that
|\Psi\rangle=\sum\limits_{i}\alpha_{i}|\Psi^L_{i} \rangle \otimes|\Psi^R_{i}\rangle \ \ \ (1)
here \alpha_{i} are a bunch of coefficients (which can be chosen to be real and positive). This is the Schmidt decomposition. Given this, a good measure of entanglement - namely the entanglement entropy - can be defined as
S_{A}=-\sum\limits_{i} \alpha_{i}^2log \alpha_{i}^2
The higher the entropy of a state, the more entangled it is.
Now trying to apply this to the double well scenario, we immediately run into trouble, because it is not clear how to perform the decomposition \mathcal{H}=\mathcal{H_{L}}\otimes\mathcal{H_{R}}.
If we want to ask the question "is there any entanglement in the double well model?" a key problem is that the two electrons in the system are indistinguishable fermions, so when one tries to construct a two particle state, it must be antisymmetric in the two electron identities. For example, ignoring spins, the position wavefunction representation of a two particle state might be constructed from single particle wavefunctions as:
\Psi(x_1,x_2)={1\over{\sqrt{2}}}(\psi(x_1)\phi(x_2)-\psi(x_2)\phi(x_1)) \ \ \ (2)
An n-particle state would be the same, except it would be a normalised sum over all the even permutations of x_1,x_2,...x_n minus all the odd permutations. Such states/wavefuctions are sometimes called Slater determinants.
Now, there is a fairly large body of literature around which discusses entanglement in multi-fermion systems. However, much of it is concerned with treating entanglement in systems appropriate to quantum computing - for example entanglement between quantum dots. In these cases, the mere fact that you cannot express a two particle state as a product state, but rather a difference of such, like in (2), is deemed *not* to constitute entanglement. For example
Shi defines entanglement in a multi-fermion system to be the inability to express the state (by choosing a suitable single particle basis) as a single Slater determinant (like (2) for the case of 2 particles). In other words, a state is *not* entangled if you *can* express it as a single Slater determinant.
Adopting this definition would immediately rule out the double well energy eigenstate (0) as being entangled – it's a single Slater determinant. But is this criterion really appropriate for the double well discussions? As far as I can tell, the reasoning behind considering (2) as unentangled has immediately made an assumption regarding remote exchange correlations, namely that they can be ignored due to the large separation.
Schliemann, whilst arguing the case for using Slater rank as the entanglement criterion states ( where I've substitued the wavefunctions in (2) for his notation) states:
However, if the moduli of \psi(x_1), \phi(x_2) have only vanishingly small overlap, these exchange correlations will also tend to zero for any physically meaningful operator. This situation is generically realized if the supports of the single-particle wavefunctions are essentially centered around locations being sufficiently apart from each other, or the particles are separated by a sufficiently large energy barrier.
So by construction the double-well electrons will be unentangled if we use Slater rank as the entanglement criterion, so this doesn't really help.
There are other approaches to entanglement of fermions, such as the one discussed by Zanardi et al(
http://arxiv.org/abs/quant-ph/0308043). They state that it is meaningless to discuss entanglement of a state
without specifying the manner in which one can manipulate and probe its constituent physical degrees of freedom. In this sense entanglement is always relative to a particular set of experimental capabilities.
This approach avoids the need to perform the decomposition (1) and instead focuses on the properties of various observables on the state being checked for entanglement. The criterion of Zanardi et al seems quite complex, but its essence is captured in a simpler formulation described in a reference by
Kaplan, to which I was referred by PF user Morberticus. Basically the question of whether or not a state is entangled is asked *with respect to a pair of observables* A, and B. A state \Psi is deemed entangled with respect to A, and B if the covariance function
C_{AB}\equiv \langle\Psi|AB|\Psi\rangle-\langle\Psi|A|\Psi\rangle\langle\Psi|B|\Psi\rangle \ \ \ (3)
is non zero.
However, to apply this to our double well system, we need to be able to define the operators A and B appropriate to "making an energy measurement in the LH well" and "making an energy measurement in the RH well".
The only energy operator I can think of that would be consistent in the two-fermion system would be the total energy operator E_1+E_2. This is symmetric in permutation of the electron identities 1 and 2 as it should be. However, to evaluate (3) to check for entanglement, I'm still left with the job of defining a "left hand well energy operator" E^{A}_1+E^{A}_2 and a "right hand well energy operator" E^{B}_1+E^{B}_2.
I've no idea how to do such a thing, and I'm inclined to agree with the conclusion of
Arnold Neumaier in his answer to my question on physics stackexchange (
http://physics.stackexchange.com/qu...surements-in-a-two-fermion-double-well-system), namely that there is no
simple way to progress this discussion !
