Some procedures of quantum information technologies, such as quantum teleportation, require two parties to share a pair of entangled particles, i.e., each party has access to one of the particles, and the two particles are entangled.

Suppose I prepare two electrons, and I entangle their spin states. I now send each electron to a different party. I do this by giving each electron an initial velocity along a long tube. I'm at one end of the tube, and the receiving party is at the other end of the tube (There are different tubes for each party). The tubes are straight, and have vacuum.

I want to calculate the probability that when the parties receive the electrons, they will still be entangled as strongly as they were before. Using non-relativistic quantum mechanics, it seems that there is no reason why the electrons should loose the entanglement, since there was vacuum in the tube and therefore there is no decoherence. But I'd like to perform the calculation using the more fundamental theory of quantum electrodynamics (QED).

Typically, in QED we calculate scattering amplitudes, where the number and types of particles may change during the process. The starting point for such calculations is the LSZ reduction formula, where "in" and "out" states are taken, and we calculate their overlap using Feynman diagrams. It seems to me that my problem should also be calculatable in QED, since it seems extremely simple - just two electrons that move along a straight line, without other interacting particles in their way. However, I'm not sure whether the LSZ formula can be applied to this case, for a few reasons: (1) I'm not sure whether it can applied for entangled states. (2) I'm not sure whether the limit of $t\rightarrow -\infty$ can be applied here. (3) The momentum of each electron does not change in the process (i.e. there is only forward scattering). (4) The states in LSZ are not localized (as momentum eigenstates).

My questions are:

  1. Are the tools used for scattering calculations in QED applicable for this calculation?
  2. How would one use QED to perform this calculation (I'm not interested in any explicit calculation, just in the general idea). Are Feynman diagrams involved?
  3. Should we expect to get a trivial result, that entanglement is conserved in the process? Or are there any effects in QED, such as the self energy of the electron, or any other interaction with the electromagnetic field vacuum that may cause decoherence?

In principle, yes you can use QED, but that is going overboard. One can specify the input state as an entangled state (for instance a Bell state) and overlap that with specific output states that would reveal its entanglement (such as the states in a state tomography measurement). In between, you'd just have the two fermion propagators without any interaction. The Feynman diagram for that is just two parallel lines. So it is indeed a severely trivial case from the perspective of QED. You would need to repeat the calculation for the different terms in the input state, resulting from the entanglement, and for the different output states, performing the tomography measurements. But that should not be too onerous, provided you don't go to higher dimensions. We're not expecting any surprices, even if you replace the femion propagators with dressed propagators to take self-energy diagrams into account.

So, to summarize (addressing your three questions)

1.) Yes you can use the tools of QED, but I would call that overkill rather than appropriate.

2.) Specify the input and output states appropriately. The Feynman diagram is just two lines. Repeat calculation for all terms in input state and all output states.

3.) We expecting the same answer one would get from direct quantum mechanics calculations.

QED is really meant for cases where there are interactions.


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