Forward-in-time analysis of delayed-choice entanglement swapping

[DrChinese]

1. Alice and Bob measure in the +/- basis and get the result ++, projecting the state (rule 7) onto $$\ket{++}_{14}\ket{--}_{23}$$The probability that Victor's BSM gets the result $\phi^-$ for these runs is (rule 6)$$p(\phi^-) = |\bra{\phi^-}--\rangle|^2 = 0$$2. This can be shown by expanding $\ket{--}$ in the Bell basis: $$\ket{--} = (\ket{\phi^+} - \ket{\psi^+})/\sqrt{2}$$Consistent with the Ma experiment, Victor can't get the result $\phi^-$ when Alice's and Bob's measurements are correlated in the +/- basis.

1. Your first state - exactly as you have it - is one which can be prepared for each and every run*. That does not lead to DCES statistics on unbiased bases for a BSM. So either you should be presenting this in some different form*, or it is excluded experimentally. Which?

2. There is no such thing as what you describe here. A Product state of 2 particles is not equivalent to a expression of entangled states of those same particles. But the reverse can be true: An entangled state can lead to a product state of 2 particles.


BTW: your simplistic rule references are useless for these situations. We're discussing entangled systems of 2 and 4 particles, which are not discussed in the "7 rules".


*Presumably not a 4-fold Product state.

 

[PeterDonis]

or equivalently, physicsforum's prescription will get you the right predictions.

@DrChinese is correct that the "7 Rules" are a very simplified presentation and do not cover all possible experiments. In particular, they do not cover cases like the one under discussion, where you are trying to apply the projection postulate without taking into account the full measurement context.

In particular:

Applying it to these experiments In particular, a system prepared in state is evolved until Alice's and Bob's measurements with the time-dependent Schrödinger equation (TDSE) (rule 3). Measurement outcome probabilities are given by the Born rule (rule 6) and upon measurement, the state is updated (rule 7) and is evolved with the TDSE until Victor's measurement.

You are ignoring crucial caveats to the rules you cite:

Rule 3 applies to an isolated quantum system. But the only isolated quantum system in the experiment under discussion is the system of all four photons. You can't pick out just photons 1 & 4 and treat them as an isolated system, because they're entangled with photons 2 & 3. So you can't apply rule 3 the way you are doing it here.

Rule 6 applies to the probabilities of possible outcomes, but that is irrelevant to the analysis you're doing, because you're assuming particular outcomes for the photon 1 & 4 measurements, and then trying to analyze what that means for the photon 2 & 3 measurements. Nothing in your analysis considers probabilities at all. So rule 6 is irrelevant.

Rule 7 says that a measurement with a given outcome can be treated as a state preparation for future measurements on the same quantum system. Which, as above, needs to be an isolated quantum system. Rule 7 does not say you can treat a measurement with a given outcome on one system as a projection of some other system. So you can't apply rule 7 the way you are doing it here.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

After moderator review, the thread will remain closed as all relevant arguments have been thoroughly made. Thanks to all who participated.