DrChinese said:
Something physically changed as a result of the swap. So it couldn't have been merely a selection issue. No experimentalist has ever performed a swapping experiment and concluded they were choosing subsets that show Entanglement that are an artifact of their selection criteria. They all say the same thing: The initial state did not have any correlation between photons 1 and 4 on any basis. And therefore (after a swap) there could be no subset in which 1 and 4 have correlation resembling Entangled State statistics UNLESS the final state was different - as is predicted by QM. That final swapped state being (a la Ma):
(3) |Ψ〉1234 = |Φ−〉14 ⨂ |Φ−〉23
I wish to discuss with you this particular part of your argument, because I believe it can help us to try to understand each other. First, I'll try to summarize how I understand the situation for the the delayed-choice entanglement swapping (Ma's experiment) from a forward-in-time analysis, as in the Mjelva's paper (as
@Morbert, I'll consider the case with the projection postulate, section 4.1.1 in the Mjelva's paper). At some points, my treatment will be much similar to post #21 by
@Morbert. Then, I'll try to make a comparison with what you say.
At time ##t_0##, two photon pairs were created, and the initial state is ##\ket{\Psi(t_0)} = \ket{\psi^-}_{12}\otimes \ket{\psi^-}_{34}##. This is equation (3) in Mjelva's paper. At time ##t_1##, Alice and Bob measure photons 1 and 4, respectively, projecting the state into one of four equally probable states which Mjelva's calls ##\ket{\Psi(t_1)}_A##, ##\ket{\Psi(t_1)}_B##, ##\ket{\Psi(t_1)}_C##, ##\ket{\Psi(t_1)}_D##, in his equations (5a)-(5d). Depending on the outcomes obtained by Alice and Bob, one (and only one) of these is the state of the system between ##t_1## and ##t_2##. Then, Victor decides to perform a BSM, which is represented by a unitary operator that physically changes the states of photons 2&3, allowing to obtain measurement outcomes that were not possible if he performs a SSM. Given the outcomes obtained by Alice and Bob at ##t_1##, the BSM at ##t_2## projected the state into a product of the state of photon 1, the state of photon 4, and the entangled state of photons 2&3. Then, in each run, 1&4 are not in an entangled state. Finally, Alice, Bob and Victor communicate to each other and compare their results, and they realized that, if they grouped the results into subsets depending on the entangled state obtained by Victor for photons 2&3, measurements on photons 1&4 appear to be Bell-correlated, as demonstrated by Mjelva's equation (7). In my opining, the previous analysis shows that the DCES experiment can be interpreted in a forward-in-time way without invoking that the swap remotely changes the state of 1&4.
However, I want to discuss all that from the opposite position, starting from something more akin to what you said. Is it possible to interpret DCES saying that, after Victor performed the swap, the quantum state of photons 1&4 is an entangled state? Well, that is not only what you say, but also what the authors say in the Ma's paper. In fact, that is why they say their experiment is a case of entanglement swapping, i.e. they entangle photons 2&3 and it remotely entangle photons 1&4, i.e. performing a swap and considering some subsets, we could say that the state of the system evolves from ##\ket{\Psi(t_0<t<t_1)} = \ket{\psi^-}_{12}\otimes \ket{\psi^-}_{34}## to ##\ket{\Psi(t>t_2)} = \ket{\phi^-}_{14}\otimes \ket{\phi^-}_{23}##. In that sense, the authors said:
"
If one views the quantum state as a real physical object, one could get the seemingly paradoxical situation that future actions appear as having an influence on past and already irrevocably recorded events. However, there is never a paradox if the quantum state is viewed as to be no more than a “catalogue of our knowledge"".
Anyway, I think that is worth analyzing whether the previous evolution of the state of the system could be regarded as something "real", more in line with ##\Psi\text{-ontic}## interpretations. I believe that you interpreted the results in this way. Am I right?
Well, in this case, if we constrained ourselves to the textbook QM, the short answer is "No". Let me explain why I think that way trying to be a bit "rigorous". In a certain sense, (non-relativistic) QM is a set of rules that, knowing the preparation procedure of a given system at time ##t_1##, allows us to calculate the probabilities of the outcomes of a measurement at ##t_2## (I assume ##t_2>t_1##) by means of (i) something called "the state of the system" which unitarily evolves according to the Schrödinger equation, and (ii) the Born's rule. Because we're considering the non-relativistic version of the theory, the previous statements are true if the number of particles is conserved. This means that, in the case of the DCES, we must apply the rules of QM in two steps: first we solve for times ##t_0<t<t_1##, i.e. from the creation of the two photon pairs until the measurements performed by Alice and Bob, and then for ##t_1<t<t_2##, from the Alice and Bob measurements (which are considered as the preparation procedure for this stage) until the Victor measurement. After ##t_1##, the system is composed of photon 2&3 only because photons 1&4 no longer exist, so that if we are unitarily evolving the state of the system, not state can be assigned to 1&4. Thus, as photons 1&4 only exist for ##t_0<t<t_1##, the only state they have is non-entangled. In fact, regarding the DCES and the state of photons 1&4 after the 2&3 swap,
Peres said: "
(...) thus verifying that the corresponding subset of particles, if it still existed, would have an entangled state".
I want to say that the previous analysis does not disprove your interpretation. It only proves that no backward-in-time change of the state of the system is needed for explaining the measurement outcomes (and the statistics than arises out of them). Furthermore, I think that if we are to interpret the results as being due to a change of the 1&4 state, this cannot be taken as real, in an ontic sense.
For me, there are still two "intriguing" things:
1. As Mjelva's showed, forward-in-time interpretations of the delayed and non-delayed entanglement swapping experiments in terms of the states of the system are very different between them. However, as
@DrChinese mentioned many times, QM predictions of the experimental outcomes are the same regardless of the time order between Alice, Bob and Victor measurements. Maybe, this "symmetry" is calling for an explanation. I don't know.
2. The Hensen's experiment is even more tricky because it is a kind of "space-like entanglement swapping", which makes the previous forward-in-time explanation of the experiment more hard to accept as they don't respect light cones, and even depend on the reference frame.
Lucas.