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

[PeterDonis]

Ma-X, not Ma. See @DrChinese's earlier posts. It is his modification.

Which post, specifically? This thread is getting pretty long.

 

[Morbert]

Which post, specifically? This thread is getting pretty long.

https://www.physicsforums.com/threa...ce-entanglement-swapping.1078114/post-7241224

Under this modified setup, the four photons are prepared in the state $\ket{LL}_{23}\ket{RR}_{14}$ result $\phi^+$ is not possible as $\bra{\phi^+}LL\rangle = \bra{\phi^+}RR\rangle = 0$.

[edit] - Actually rereading, the initial state was $\ket{LL}_{14}\ket{RR}_{23}$. Though the same consequences follow from either $\ket{RR}$ or $\ket{LL}$

 

[PeterDonis]

https://www.physicsforums.com/threa...ce-entanglement-swapping.1078114/post-7241224

Under this modified setup, the four photons are prepared in the state $\ket{LL}_{23}\ket{RR}_{14}$ result $\phi^+$ is not possible as $\bra{\phi^+}LL\rangle = \bra{\phi^+}RR\rangle = 0$.

[edit] - Actually rereading, the initial state was $\ket{LL}_{14}\ket{RR}_{23}$. Though the same consequences follow from either $\ket{RR}$ or $\ket{LL}$

This involves a setup in which there is no entanglement prepared; the four-photon state prepared is completely separable. So of course there can't be any entanglement swapping, since there is no entanglement to swap. So of course no Bell state of any pair of the photons can be output.

But of course this is not the same as an entanglement swapping experiment, where the four photons are not prepared in a completely separable state. @DrChinese was obviously not intending this "Ma-X" variation to be relevant to an analysis of entanglement swapping. He was intending it to illustrate what happens where there is no entanglement swapping possible.

 

[eloheim]

This involves a setup in which there is no entanglement prepared; the four-photon state prepared is completely separable. So of course there can't be any entanglement swapping, since there is no entanglement to swap. So of course no Bell state of any pair of the photons can be output.

But of course this is not the same as an entanglement swapping experiment, where the four photons are not prepared in a completely separable state. @DrChinese was obviously not intending this "Ma-X" variation to be relevant to an analysis of entanglement swapping. He was intending it to illustrate what happens where there is no entanglement swapping possible.

I believe it is the point being discussed right now though. You can put 2+3 into a bell state whether they come from entangled pairs or not, and the question is whether the results are the same as the real experiment. I think we all agree that if Alice and Bob are allowed to measure on a different basis then Victor's results won't match, but I think DrChinese is saying the results won't match the real experiment even if Alice and Bob measure in the prepared basis.

I also think DrChinese is saying if you follow Mjelva's math (like his equations 5 or 9) you should get results incompatible with experimental reality but I'm still trying to parse that.

 

[PeterDonis]

You can put 2+3 into a bell state whether they come from entangled pairs or not

You're missing the point. If all four photons are in a completely separable state, which is the premise @Morbert was talking about, then you cannot entangle anything using the configuration in the experiments under discussion. You can only put 2+3 into a Bell state if the pairs 1+2 and 3+4 start out in Bell states and you then do an entanglement swap.

 

[Morbert]

@DrChinese's goal is to demonstrate that Mjelva's forward-in-time analysis - an analysis which eliminates entanglement swapping in the sense that Victor's BSM projects onto a state where photons 1 & 4 are still separable/not entangled - leads to contradictions with what is observed in experiments. The Ma-X variation is intended to bring focus to such contradictions.

The impasse we have reached is that DrChinese believes if the initial 4 photons are prepared in the sate $\ket{LL}_{14}\ket{RR}_{23}$, then "In the Ma-X scenario: there is no correlation between the $\ket{LL}_{14}$ outcomes and the $\ket{\phi^-}_{23}$ outcomes versus the $\ket{\phi^+}_{23}$ outcomes." and " A $\phi^-$ BSM is an actual compatible outcome, but equally frequent is a $\phi^+$ BSM experimental outcome" yet when I work through this variation, I show that $\ket{\phi^+}_{23}$ outcomes cannot actually occur, as $\bra{\phi^+}RR\rangle_{23} = \bra{\phi^+}LL\rangle_{23} = 0$.

 

[PeterDonis]

an analysis which eliminates entanglement swapping in the sense that Victor's BSM projects onto a state where photons 1 & 4 are still separable/not entangled

But this would disagree with the actual experimental results.

 

[Morbert]

But this would disagree with the actual experimental results.

See equations 5a-5d in Mjelva. Projection onto states like these will correctly predict experimental results. These states have no entangled 1 & 4 pairs.

Mjelva's 4.1.1 analysis is quite general. I have applied this to the cases where Alice and Bob measure photon polarization in the H/V or L/R basis, which greatly simplifies things.

 

[DrChinese]

I believe it is the point being discussed right now though. You can put 2+3 into a bell state whether they come from entangled pairs or not, and the question is whether the results are the same as the real experiment. I think we all agree that if Alice and Bob are allowed to measure on a different basis then Victor's results won't match, but I think DrChinese is saying the results won't match the real experiment even if Alice and Bob measure in the prepared basis.

I also think DrChinese is saying if you follow Mjelva's math (like his equations 5 or 9) you should get results incompatible with experimental reality but I'm still trying to parse that.

This is what I am asserting. Specifically:

Mjelva (and Morbert) claim there is a forward in time only evolution of quantum states that provides a viable description of swapping experiments. In the examples such as his formulae (pre-8), (8) and (9), that is presented and superficially appears correct. Note that everyone agrees that the initial 4 photon state is something like:

|Ψ->₁₂ ⊗ |Ψ->₃₄ (1)

Ma & Megidish and everyone else presents the next step of the evolution (swap=on) as being the result of a BSM. That post-swap state is of course one of the 4 Bell states, here per Ma's (2) and Megidish's (3):

|Ψ〉₁₂₃₄= ½(|Ψ+〉₁₄⨂|Ψ+〉₂₃ − |Ψ−〉₁₄⨂|Ψ−〉₂₃ − |Φ+〉₁₄⨂|Φ+〉₂₃ + |Φ−〉₁₄⨂|Φ−〉₂₃) (2)

But that is NOT the same evolution as Mjelva describes at all. The Ma/Megidish evolution goes directly from my (1) to my (2) without Mjelva's intermediate steps for each element of the measurement process. That's because timing/order of the intermediate measurements is not a factor in the final quantum state post-BSM (swap). As a result, Mjelva ends up with an intermediate state like his (8) in which there is an "A" fork and a "B" fork. He states explicitly: "Unlike in ordinary entanglement swapping-experiments, this does not project the pair (1,4) into a Bell state..." Whoops, this assertion is diametrically opposed to what Ma says: "Consequently photons 1 (Alice) and 4 (Bob) also become entangled and entanglement swapping is achieved..." so we have our point of departure.

Mjelva's intermediate presentation pre-BSM (8A) goes to post-BSM (9A), and he claims this evolution is what actually occurs. That's the hypothesis anyway, and it is subject to experimental confirmation. Well, it is easily possible to test that! If you start with a quantum state of his (8A), you simply perform the BSM and see if what you get is his (9A). For example: If you measured photons 1 & 4 as up-up, then Mjelva says that you can see the Φ- BSM signature about half the time, and the Φ+ BSM signature not at all. But in the real world, this does NOT happen.

To prove this, merely start with source photons 1 & 4 being in the up-up state (a la 8A) and perform a BSM on photons 2 & 3. If Mjelva is correct and there is no such thing as 1 & 4 being cast after the fact (as in Ma's experiment) into an entangled state, this should yield the results per Mjelva's (9A). That's can't happen, because a BSM on such a state produces both Φ- BSM signatures and Φ+- BSM signatures equally. That's because photons 1 & 2 must be in a polarization entangled state to start with, which is qualitatively different than Mjelva's (8A) - which actually is NOT an intermediate step in the quantum state evolution.

Note that there are no actual papers demonstrating this exact experimental point as I describe in the paragraph above, you must deduce that result from the Ma/Megidish experiments indirectly. Both of these experiments use Type II PDC to create polarization entangled pairs. But that requires an important element during setup: you must select pairs from a very precise overlapping portion of the cones (see the diagram in various posts above). If they didn't do that, the resulting source photons will have known and separable states. A simple manipulation will then produce the desired up-up results for photons 1 & 4. And according to Mjelva, the results (post measurement of photons 1 & 4 but before the BSM on photons 2 & 3) would be exactly per his (9A). Were that true, we would simply ask the Ma team why they both to capture from the overlapping regions. Why not instead save yourself the time and effort to overlap, since the evolution after a BSM will be identical?

Answer: The evolution from (8A) to (9A) does not occur. Mjelva is simply wrong on this point. It looks good in his paper, but it won't work if (8A) is re-created in practice using unentangled sources for either 1&2 or 3&4. What's good for the goose is good for the gander (i.e. you can't pick and choose when to apply your premise - either 8A goes to 9A or it doesn't). Neither Ma, nor Megidish, nor anyone presenting entanglement swapping experiments presents anything like what Mjelva describes. His justification for a forward in time only hypothesis is flawed.

 

[DrChinese]

See equations 5a-5d in Mjelva. Projection onto states like these will correctly predict experimental results. These states have no entangled 1 & 4 pairs.

Mjelva's 4.1.1 analysis is quite general. I have applied this to the cases where Alice and Bob measure photon polarization in the H/V or L/R basis, which greatly simplifies things.

Mjelva's mistake is the assumption he uses to get his (4). That's where he asserts the system is in 1 of 4 equally probable states. The initial system is actually only in the state:

|Ψ->₁₂ ⊗ |Ψ->₃₄

Which is a product state of 2 entangled states. His leap to his (4) is exactly where his assumption can be tested experimentally. We simply start with any of those, and perform a BSM on 2 & 3. It's that simple, and he simply overlooked this possibility.

 

[Morbert]

Mjelva's mistake is the assumption he uses to get his (4). That's where he asserts the system is in 1 of 4 equally probable states. The initial system is actually only in the state:

|Ψ->₁₂ ⊗ |Ψ->₃₄

Which is a product state of 2 entangled states. His leap to his (4) is exactly where his assumption can be tested experimentally. We simply start with any of those, and perform a BSM on 2 & 3. It's that simple, and he simply overlooked this possibility.

Mjelva's (4) is a mixture of fours states that Alice's and Bob's irreversible measurements can project the initial state onto. If Alice and Bob do not first measure 1 & 4, then (4) would not be correct. But in delayed choice experiments like Ma's, they do, and hence (4) is correct, and Mjelva uses it to represents of a sample of runs. It reproduces all correlations and frequencies observed in the actual experiment, and the same procedure can be used to reproduce all correlations in any modification of the experiment (say, replacing the entangled states in the initial state with separable states).

Similarly, you can apply the standard forward-in-time unitary evolution + state reduction to calculate probabilities for the outcomes of individual runs, and arrive at the same correct numbers. E.g. If Alice and Bob measure in the R/L basis and get the results LL (and hence project photons 2 & 3 onto |RR〉₂₃), then we know Victor cannot obtain the result Φ+, as ⟨Φ+|RR〉₂₃ = 0. This is indeed a correlation observed in the actual experiment, and we can compute any such quantity for any set of possible outcomes recorded by Alice Bob and Victor.

 

[DrChinese]

Mjelva's (4) is a mixture of fours states that Alice's and Bob's irreversible measurements can project the initial state onto. If Alice and Bob do not first measure 1 & 4, then (4) would not be correct. But in delayed choice experiments like Ma's, they do, and hence (4) is correct, and Mjelva uses it to represents of a sample of runs. It reproduces all correlations and frequencies observed in the actual experiment, and the same procedure can be used to reproduce all correlations in any modification of the experiment (say, replacing the entangled states in the initial state with separable states).

Similarly, you can apply the standard forward-in-time unitary evolution + state reduction to calculate probabilities for the outcomes of individual runs, and arrive at the same correct numbers. E.g. If Alice and Bob measure in the R/L basis and get the results LL (and hence project photons 2 & 3 onto |RR〉₂₃), then we know Victor cannot obtain the result Φ+, as ⟨Φ+|RR〉₂₃ = 0. This is indeed a correlation observed in the actual experiment, and we can compute any such quantity for any set of possible outcomes recorded by Alice Bob and Victor.

State evolution, according to you/Mjelva:

1. Initially:
   |Ψ>₁₂₃₄ = |Ψ->₁₂⊗|Ψ->₃₄ is the starting state (also agreed by Ma).
2. |Ψ>₁₂₃₄ = (|LR>₁₂+|RL>₁₂)⊗(|LR>₃₄+|RL>₃₄) also shows the entanglement of pairs 1 & 2 and 3 & 4. Nothing controversial here.
   
   After Alice & Bob measure:
3. |Ψ>₁₂₃₄ = |R>₁⊗|L>₂⊗|L>₃⊗|R>₄ is the state immediately after 1 & 4 are measured and found to be |RR>, per your example. This is simply one of the 4 possibilities per the mixture of Mjelva's (pre-4, showing all four individually) and (4, shown as a "proper mixture"), expressed on the R/L basis.
   This is our point of departure. I say that there is no such |Ψ>₁₂₃₄ state as Mjelva's (4) until and unless all 4 are measured. I.e. there is no mixture of unentangled (4-fold separable) intermediate representations of |Ψ>₁₂₃₄ prior to an executed BSM.
4. |Ψ>₁₂₃₄ = |R>₁⊗(|H>₂+|V>₂)⊗(|H>₃+|V>₃)⊗|R>₄ is equivalent to 3. It shows the 2, 3 photons in a superposition on the H/V basis. At this point, we could be describing any 4 individual photons in the entire universe, as they are all in a Product (separable) State and there is no entanglement between any two.
   
   After BSM performed on 2 & 3 on the H/V basis:
5. The BSM in fact entangles them (2 & 3) in one of 4 Bell states, since they are now indistinguishable on that basis. Per Mjelva: "We observe that each of these joint states is a product state of the state of particle 1, the state of particle 4, and the [entangled] state of the pair 2, 3".
6. |Ψ>₁₂₃₄ = |R>₁⊗(½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃))⊗|R>₄ is therefore the actual post BSM state, in which |Φ+>₂₃ is non-zero.

If you started with 3., which can be arranged by the "Ma-X" technique I described in post #86: You would in fact evolve to 6. The actual experimental result of that yields equal numbers of |Φ+>₂₃ and |Φ->₂₃ instead of your |Φ+>₂₃=0. We know that with certainty for the reasons already explained, see point 3 in post #95. But, if you start with 1.: you never get to 3. in actual entanglement experiments. That step is skipped, and the evolution proceeds as described in Ma's (2). That's why those swapping experiments produce the expected quantum results.

Surely you can see the contradictions here. You are using the EPR reasoning to infer an "element of reality" to the intermediate state of 3. That hypothetical intermediate state does NOT produce the desired entanglement swapping results, therefore it cannot describe reality. A BSM evolves the state from Ma's (1) to Ma's (2) regardless of the timing of measurements by Alice and Bob. Which is a disproof of Mjelva's premise of a forward in time only description of entanglement swapping.

 

[DrChinese]

At this point, @Morbert must be asking something like: "Wait a minute! If DrChinese is correct* about Mjelva's evolution per his post #112, then how does Mjelva get to the same correct prediction for the swapping results as per Ma's experiment?"

Fair question. The answer (I will present specifically in my next post) is that Mjelva makes a second mistake that reverses his first mistake. That first mistake having been identified in point 3. of my post.


*That Mjelva's hypothetical forward in time only evolution is contradicted by experiment.

 

[DrChinese]

As mentioned in my previous post: Mjelva makes a second mistake in his state evolution that reverses his first. I know what his first error is, and I know he fixes it by the end, so the second should be easy to troubleshoot. It's easier to demonstrate this with his state evolution for the Megidish experiment, see diagram in Figure 1. We will contrast that with Mjelva's pre-(8), (8), and (9). His pre-(8) is wrong*, but his (9) is correct and is in line with Megidish's experimental results. So we need only look from pre-(8) to (8), or (8) to (9), to locate the second error.

The issue is that after a BSM on photons 2 & 3, they are cast into one of the 4 Bell states as shown in Mjelva's (8). However, his presentation of photon 4 is incorrect. I will re-cast the bases to match what Morbert and I have been using.

1. After our |R>₁ outcome for Alice:
|Ψ>₁₂₃₄
= |RL>₁₂ ⊗ |Ψ->₃₄
= |R>₁ ⊗ ½(|H>₂+|V>₂) ⊗ ½(|HV>₃₄-|VH>₃₄)
... and making explicit that there is no correlation whatsoever between an |R>₁ result and any future measurement or projection on photon 2's H/V basis. That, as I have said previously, is canonical.

2. And after the BSM:
|Ψ>₁₂₃₄
= |R>₁ ⊗ ¼(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃) ⊗ ½(|L>₄+|R>₄)
... and likewise making explicit that there is no correlation whatsoever between any of the 4 entangled Bell states for photons 2 & 3, and any future measurement on photon 4's R/L basis. Photon 4 is now in a superposition in the R/L basis. This is what he should have been presented; for if he had, Mjevla wouldn't think he had matched the correct quantum prediction.

3. And we can now plainly see photons 1 & 4 have no correlation whatsoever, under a proper rendering of Mjelva's evolution.
So Mjelva deftly inserts that required (but non-existent) correlation when he goes from (8) to (9) by simply dropping all the terms in which we don't get the expected outcome of |RR>₁₄. (That includes dropping the term with |RR>₁₄ and |Φ+>₂₃.) And that is how the earlier error is fixed.** Regardless, (9) does not follow from his pre-(8) states using his own assumptions. So (9) is itself correct, but for the wrong reasons.

I'm sorry to report to Mjelva: In my opinion, a single paper has 2 critical mistakes; they negate the relevant conclusions.


*We know this per my 3. in post #112. After a measurement by Alice, photons 1 & 2 are NOT in a state like Mjelva's (pre-8): |Ψ>₁₂ = |Up Down>₁₂+|Down Up>₁₂ - not unless and until they are both observed as such.

**Or you could simply say he assumes what he seeks to prove.

 

[Sambuco]

Mjelva (and Morbert) claim there is a forward in time only evolution of quantum states that provides a viable description of swapping experiments.

Don't forget me! I'm on the @Morbert (and Mjelva) team. After all, I shared Mijelva's paper for some reason

As mentioned in my previous post: Mjelva makes a second mistake in his state evolution that reverses his first.

@DrChinese I find it difficult to follow your reasoning. In post #112 you critized Mjelva's eq. (4) and, since he finally got a result that agrees with the experiment, you try to find what the second error is that compensates for the first one. However, in post #114, you mentioned eqs. (8) and pre-(8). What confuses me is that eq. (4) correspond to Ma's experiment, while eqs. (8) and pre-(8) correspond to Megidish's experiment. In other words, if eq. (4) was wrong, the second error should be somewhere before eq. (7), because this is the point where the (projection-based) analysis of Ma's experiment ends.

Lucas.

 

[DrChinese]

Don't forget me! I'm on the @Morbert (and Mjelva) team. After all, I shared Mijelva's paper for some reason


@DrChinese I find it difficult to follow your reasoning. In post #112 you critized Mjelva's eq. (4) and, since he finally got a result that agrees with the experiment, you try to find what the second error is that compensates for the first one. However, in post #114, you mentioned eqs. (8) and pre-(8). What confuses me is that eq. (4) correspond to Ma's experiment, while eqs. (8) and pre-(8) correspond to Megidish's experiment. In other words, if eq. (4) was wrong, the second error should be somewhere before eq. (7), because this is the point where the (projection-based) analysis of Ma's experiment ends.

Lucas.

Wouldn’t want to forget you!

There are 2 problems in the Mjelva paper, and similar in Morberts analysis. Both of them can be seen in either his Ma treatment or Megidish treatment. It’s just convenience as to which is used. They are all equivalent. If you tell me which one to focus on, I’ll be glad to. I didn’t go to the Ma treatment for the second error because Mjelva made an unnecessary trip down Bayes Theorem, and that literally has nothing to do with quantum mechanics. So skip that.

What you have to understand is that a Nobel prize winner wrote a paper with Ma and they present the correct evolution. Morberts is flat out contradictory. So that should be a red flag to anyone. That experiment was done over a decade ago, and still stands. So everything I am saying is simply the same criticism they would be making.

There is no concept similar to Mjelva’s at work. The only thing driving it is a passionate desire to cling to Einsteinian causality. Which obviously QM does not respect.

 

[Morbert]

Segments in bold are my emphasis, and mark disagreements.

State evolution, according to you/Mjelva:

1. Initially:
   |Ψ>₁₂₃₄ = |Ψ->₁₂⊗|Ψ->₃₄ is the starting state (also agreed by Ma).
2. |Ψ>₁₂₃₄ = (|LR>₁₂+|RL>₁₂)⊗(|LR>₃₄+|RL>₃₄) also shows the entanglement of pairs 1 & 2 and 3 & 4. Nothing controversial here.

|R〉 = (|H〉 + i|V〉)/√2
|L〉 = (|H〉 - i|V〉)/√2
so
|Ψ-〉 = i(|RL〉 - |LR〉)/√2
so I work out the initial state to be
|Ψ-〉₁₂⊗|Ψ-〉₃₄ = -(|RL〉₁₂-|LR〉₁₂)⊗(|RL〉₃₄-|LR〉₃₄)/2

After Alice & Bob measure:
3. |Ψ>₁₂₃₄ = |R>₁⊗|L>₂⊗|L>₃⊗|R>₄ is the state immediately after 1 & 4 are measured and found to be |RR>, per your example. This is simply one of the 4 possibilities per the mixture of Mjelva's (pre-4, showing all four individually) and (4, shown as a "proper mixture"), expressed on the R/L basis.
This is our point of departure. I say that there is no such |Ψ>₁₂₃₄ state as Mjelva's (4) until and unless all 4 are measured. I.e. there is no mixture of unentangled (4-fold separable) intermediate representations of |Ψ>₁₂₃₄ prior to an executed BSM.
4. |Ψ>₁₂₃₄ = |R>₁⊗(|H>₂+|V>₂)⊗(|H>₃+|V>₃)⊗|R>₄ is equivalent to 3. It shows the 2, 3 photons in a superposition on the H/V basis. At this point, we could be describing any 4 individual photons in the entire universe, as they are all in a Product (separable) State and there is no entanglement between any two.

|Ψ〉₁₂₃₄
= |R〉₁⊗|L〉₂⊗|L〉₃⊗|R〉₄
= |R〉₁⊗(|H〉₂-i|V〉₂)⊗(|H〉₃-i|V〉₃)⊗|R〉₄/2

After BSM performed on 2 & 3 on the H/V basis:
5. The BSM in fact entangles them (2 & 3) in one of 4 Bell states, since they are now indistinguishable on that basis. Per Mjelva: "We observe that each of these joint states is a product state of the state of particle 1, the state of particle 4, and the [entangled] state of the pair 2, 3".
6. |Ψ>₁₂₃₄ = |R>₁⊗(½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃))⊗|R>₄ is therefore the actual post BSM state, in which |Φ+>₂₃ is non-zero.

If you started with 3., which can be arranged by the "Ma-X" technique I described in post #86: You would in fact evolve to 6. The actual experimental result of that yields equal numbers of |Φ+>₂₃ and |Φ->₂₃ instead of your |Φ+>₂₃=0. We know that with certainty for the reasons already explained, see point 3 in post #95. But, if you start with 1.: you never get to 3. in actual entanglement experiments. That step is skipped, and the evolution proceeds as described in Ma's (2).

|Ψ〉₁₂₃₄
= |R〉₁⊗|L〉₂⊗|L〉₃⊗|R〉₄
= |R〉₁⊗(|H〉₂-i|V〉₂)⊗(|H〉₃-i|V〉₃)⊗|R〉₄/2
= |R〉₁⊗(|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃)⊗|R〉₄/2
= |R〉₁⊗(|Φ-〉₂₃-i|Ψ+〉₂₃)⊗|R〉₄/√2

Note that this expansion only contains the Bell states |Φ-〉₂₃ and |Ψ+〉₂₃, hence |Φ+〉₂₃ is not a possible outcome. This is how the expected correlations are obtained in standard forward-in-time analysis of Ma's experiment. Each possible combination of outcomes for Alice and Bob will project onto a state that will restrict outcomes for Victor.

That's why those swapping experiments produce the expected quantum results.

As an aside: What Ma did in his paper was perfectly correct. It can be the case that both Ma's and Mjelva's approaches get the right results.

[edit] - Fixed math error.

 

[Morbert]

What you have to understand is that a Nobel prize winner wrote a paper with Ma and they present the correct evolution. Morberts is flat out contradictory. So that should be a red flag to anyone. That experiment was done over a decade ago, and still stands. So everything I am saying is simply the same criticism they would be making.

Nothing I have said contradicts the Ma paper. Mjelva's and Ma's analyses obtain the same results. Ma does not present a forward-in-time analysis because that is not the purpose of his paper.

 

[DrChinese]

DrChinese: Great, this is a good representation of the issues. My comments in italics below...

1. Agreed, and your last line's state is an Product State of 2 Entangled ones.

|Ψ-〉₁₂⊗|Ψ-〉₃₄ = -(|RL〉₁₂-|LR〉₁₂)⊗(|RL〉₃₄-|LR〉₃₄)/2

2. Product States below are NOT the state after measurements by Alice and Bob on your 1. After all, it is not entangled! This is error #1. However, it is a viable state that can be created. It is exactly what happens if you use my Ma-X method using Ma's type II PDC. Simply prevent polarization entanglement by failing to perform a key step in the normal process. You will get the Product State exactly as you have written. So we will continue using this as the starting state, instead of 1.

|Ψ〉1234
= |R〉1⊗|L〉2⊗|L〉3⊗|R〉4
= |R〉1⊗(|H〉2-i|V〉2)⊗(|H〉3-i|V〉3)⊗|R〉4/2

3. This does follow from the previous state.

|Ψ〉₁₂₃₄
= |R〉₁⊗(|H〉₂-i|V〉₂)⊗(|H〉₃-i|V〉₃)⊗|R〉₄/2

4. Your final line is error #2. You assume that which you seek to prove - by dropping the "offending" terms without justification.

|Ψ〉₁₂₃₄
= |R〉₁⊗(|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃)⊗|R〉₄/2
= |R〉₁⊗(|Φ-〉₂₃-i|Ψ+〉₂₃)⊗|R〉₄/√2

5. Note that this expansion only contains the Bell states |Φ-〉₂₃ and |Ψ+〉₂₃, hence |Φ+〉₂₃ is not a possible outcome. This is how the expected correlations are obtained in standard forward-in-time analysis of Ma's experiment. Each possible combination of outcomes for Alice and Bob will project onto a state that will restrict outcomes for Victor.

6. It can be the case that both Ma's and Mjelva's approaches get the right results.

1.-4. my comments above.

5. Exactly, you made the answer match the actual results - by using a "cheat" (don't take offence - it's just a word). Ma has nothing like this manipulation in their work on entanglement. Nobody else in the experimental world does either. And neither do accepted theoreticians such as Peres, Zeilinger, Gisin, etc.

The key to understanding error #2: Your 2. is a state that is easily implemented for verification one way or the other. When you start with this state, you actually get results that are diametrically opposite of what you wrongly claim. |Φ+〉₂₃ IS an equal outcome to |Φ-〉₂₃.

How do we know this? It is exactly what happens if you use my Ma-X method using Ma's type II PDC. Simply prevent polarization entanglement by failing to perform a key step in the normal process. You will get the Product State exactly as you have written. The step to skip: overlapping the Vertical and Horizontal cones during PDC. You will get a photon pair with known polarization. That leads to the Product State of your 2. But when you perform a BSM, the final results don't justify your final manipulation. Falsification by experiment.

If what I say weren't true, then there would certainly be no need to overlap the V/H cones in actual experimental production of the |Ψ-> pairs used in Ma (and many others). i) A lot of effort goes into overlapping the cones to make the V and H photons indistinguishable. ii) And that requirement significantly reduces the production rate.

You have yet to explain why they do this, and I have. They do this work because without that overlap, the pairs produce cannot be used for entanglement swapping precisely because they are in the Product State of your 2.

6. It could have been, and it did take me a bit of effort to see why his math seemed to mimic Ma. But now it’s clear why Mjelva is mistaken.

@Sambuco Does this make sense now?

 

[Morbert]

2. Product States below are NOT the state after measurements by Alice and Bob on your 1. After all, it is not entangled! This is error #1. However, it is a viable state that can be created. It is exactly what happens if you use my Ma-X method using Ma's type II PDC. Simply prevent polarization entanglement by failing to perform a key step in the normal process. You will get the Product State exactly as you have written. So we will continue using this as the starting state, instead of 1.

The product states are the states after measurement by Alice and Bob. This follows from standard QM when a forward-in-time analysis is applied, and is in agreement with experiment.

4. Your final line is error #2. You assume that which you seek to prove - by dropping the "offending" terms without justification.

No term is dropped. It follows from textbook QM and basic substitution. The relevant Bell states are

|Φ-〉= (|HH〉-|VV〉)/√2
|Ψ+〉= (|HV〉+|VH〉)/√2

so

|Ψ〉₁₂₃₄
= |R〉₁⊗|L〉₂⊗|L〉₃⊗|R〉₄
= |R〉₁⊗(|H〉₂-i|V〉₂)⊗(|H〉₃-i|V〉₃)⊗|R〉₄/2
= |R〉₁⊗(|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃)⊗|R〉₄/2
= |R〉₁⊗(|Φ-〉₂₃-i|Ψ+〉₂₃)⊗|R〉₄/√2

I invite anyone to check this step to the last line, where the 2 & 3 terms in brackets are:
[|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃]/2 = [ ( |HH〉₂₃-|VV〉₂₃ )/√2 - i( |HV〉₂₃+|VH〉₂₃ )/√2 ]/√2 = [|Φ-〉₂₃-i|Ψ+〉₂₃]/√2

This consistent with experiment, where the Φ- set has correlation in the R/L basis (see Ma's Fig 3) while the Φ+ set has anticorrelation in the R/L basis.

 

[Morbert]

@DrChinese this can be made even more obvious if we consider the +/- basis, as these are anticorrelated in the Φ- set. Fig 3 from Ma.

You can see that, when Victor records Φ-, Alice's and Bob's records will be anticorrelated if they measured in the +/- basis, so let's apply the same forward-in-time analysis as before. Let's assume Alice and Bob both recorded +, so they project the initial state onto

|+〉₁⊗|-〉₂⊗|-〉₃⊗|+〉₄

Like before, we expand 2 & 3 in the Bell basis, via the H/V basis

|+〉 = (|H〉 + |V〉)/√2
|-〉 = (|H〉 - |V〉)/√2

|+〉₁⊗|-〉₂⊗|-〉₃⊗|+〉₄
= |+〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄/2
= |+〉₁⊗(|HH〉₂₃+|VV〉 - |HV〉 - |VH〉₂₃)⊗|+〉₄/2
= |+〉₁⊗(|Φ+〉₂₃ - |Ψ+〉₂₃)⊗|+〉₄/√2

Notice that this time there is no |Φ-〉₂₃ term. Hence, if Alice and Bob record ++ (or --), then the resultant projected state means Victor cannot obtain the result Φ-. Hence, the graph above, showing anticorrelation in the +/- basis when victor records Φ-.

 

[DrChinese]

@DrChinese this can be made even more obvious if we consider the +/- basis, as these are anticorrelated in the Φ- set. Fig 3 from Ma.

1. You can see that, when Victor records Φ-, Alice's and Bob's records will be anticorrelated if they measured in the +/- basis, so let's apply the same forward-in-time analysis as before. Let's assume Alice and Bob both recorded +, so they project the initial state onto

|+〉₁⊗|-〉₂⊗|-〉₃⊗|+〉₄

Like before, we expand 2 & 3 in the Bell basis, via the H/V basis

|+〉 = (|H〉 + |V〉)/√2
|-〉 = (|H〉 - |V〉)/√2

|+〉₁⊗|-〉₂⊗|-〉₃⊗|+〉₄
= |+〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄/2
= |+〉₁⊗(|HH〉₂₃+|VV〉 - |HV〉 - |VH〉₂₃)⊗|+〉₄/2
= |+〉₁⊗(|Φ+〉₂₃ - |Ψ+〉₂₃)⊗|+〉₄/√2

2. Notice that this time there is no |Φ-〉₂₃ term. Hence, if Alice and Bob record ++ (or --), then the resultant projected state means Victor cannot obtain the result Φ-. Hence, the graph above, showing anticorrelation in the +/- basis when victor records Φ-.

1. Ok, here we go:

• i) |+〉₁⊗|-〉₂⊗|-〉₃⊗|+〉₄ and
• ii) |+〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄/2

... are exactly the kind of Product States that will never lead to 4-fold Entanglement Swap Correlations a la Ma. There is no textbook anywhere that says a BSM on 2 & 3 can do that after being in a Product State. You can cast 2 & 3 into a Bell State with a BSM, but of course that is meaningless for anything related to 1 & 4.

I can't seriously believe we are having this discussion, because you are just making statements with no theoretical support whatsoever - and they are contradicted by experiment. I have already explained this, as has @PeterDonis. You simply ignore the fact that the state i), after a BSM, yields both |Φ+〉₂₃ and |Φ-〉₂₃ outcomes.

And... note that according to your logic, after a BSM: All of the following starting states yield a |Φ+〉₂₃ term - but not a |Φ-〉₂₃ term (which of course is wrong).

• iii) |+〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄/2, ... which is per your derivation of ii) above.
• iv) (|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃) which the same thing, without photons 1 & 4: This is what you get when you delete [independent] terms from a Product State.
• v) |-〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄ This is what you get when you delete [independent] terms from a Product State.

You can see that mathematically these yield identical Bell state results after a BSM on photons 2 & 3. I.e. You could not, after a sufficiently large number of runs, distinguish whether you started with iii), iv) or v) by looking at the BSM results. That you can get the same thing from both iii) and v) precisely contradicts your assertion.


2. This is a statement about what actually happens. But you use a cheat to get the correct result from a wrong intermediate state. In the real quantum world, the evolution is from Ma's (1) to Ma's (2) and there are no intermediate states to consider - which is why they aren't shown by Ma or anyone else. Timing and/or ordering of measurements is immaterial, which is why no intermediate states are presented. They simply don't occur as you describe.

 

[DrChinese]

1. The product states are the states after measurement by Alice and Bob. This follows from standard QM when a forward-in-time analysis is applied...

2. No term is dropped. It follows from textbook QM and basic substitution.

1. There is no such thing as a standard "forward-in-time analysis" of entanglement swapping with a 4-fold Product State in the middle. You're just quoting Mjelva, and that won't fly here. We are wondering if there is any support for his novel ideas in other authoritative sources. It's not evident in any of the dozens of swapping experiments I have bookmarked. But maybe you have a good quote from Zeilinger, Wineland or someone like that you could share. Because they don't seem to share your perspective in what I have read.


2. Umm, you're gonna need to provide that textbook quote. Because I already identified that you used a "cheat" in the substitution.


What I really want you to do (pretty please ) is to tell me you believe the following:

1. I set up two PDC sources that emits pairs ONLY in the anti-correlated states |+-〉₁₂ and |-+〉₃₄, which for a 4-fold state is: |+-〉₁₂ ⊗ |-+〉₃₄; or as you might say: |+〉₁ ⊗ |-〉₂ ⊗ |-〉₃ ⊗ |+〉₄.
2. I project photons 2 & 3 into a Bell state via BSM.
3. My resulting Bell state outcomes do not include both |Φ+〉₂₃ and |Φ-〉₂₃ terms.

 

[eloheim]

How do we know this? It is exactly what happens if you use my Ma-X method using Ma's type II PDC. Simply prevent polarization entanglement by failing to perform a key step in the normal process. You will get the Product State exactly as you have written. The step to skip: overlapping the Vertical and Horizontal cones during PDC. You will get a photon pair with known polarization. That leads to the Product State of your 2. But when you perform a BSM, the final results don't justify your final manipulation. Falsification by experiment.



If what I say weren't true, then there would certainly be no need to overlap the V/H cones in actual experimental production of the |Ψ-> pairs used in Ma (and many others). i) A lot of effort goes into overlapping the cones to make the V and H photons indistinguishable. ii) And that requirement significantly reduces the production rate.

You have yet to explain why they do this, and I have. They do this work because without that overlap, the pairs produce cannot be used for entanglement swapping precisely because they are in the Product State of your 2.

Sorry to be back at this again but now I'm again curious. My thought process was: the reason they have to use entangled photons is because they have to account for Alice and Bob changing their measurement angles, at will. In order for Photons 3+4 to get the proper bell states photons 2 and 3 need to know what angles Alice and Bob measure (which 2+3 DO know, nonlocally). However, if we do the Ma-X version and and only measure in the prepared basis, then Morbert's product states give only the correct bell state results, correct?

Don't we know trivially that once 1 and 4 are measured the available bell state results at the end of the experiment are cut in half? That's the whole point of the correlation/anti-correlation results in each basis. If 1 and 4 have already measured R and L then Φ- and Ψ+ can NOT be the result of the 2+3 bell measurement.

What I really want you to do (pretty please ) is to tell me you believe the following:

1. I set up two PDC sources that emits pairs ONLY in the anti-correlated states |+-〉₁₂ and |-+〉₃₄, which for a 4-fold state is: |+-〉₁₂ ⊗ |-+〉₃₄; or as you might say: |+〉₁ ⊗ |-〉₂ ⊗ |-〉₃ ⊗ |+〉₄.
2. I project photons 2 & 3 into a Bell state via BSM.
3. My resulting Bell state outcomes do not include both |Φ+〉₂₃ and |Φ-〉₂₃ terms.

Yes BUT only if you haven't already measured 1 and 4, which in the delayed-choice version of this experiment, you already have.

 

[Morbert]

1. Ok, here we go:

• i) |+〉₁⊗|-〉₂⊗|-〉₃⊗|+〉₄ and
• ii) |+〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄/2

... are exactly the kind of Product States that will never lead to 4-fold Entanglement Swap Correlations. There is no textbook anywhere that says a BSM on 2 & 3 can do that after being in a Product State. You can cast 2 & 3 into a Bell State with a BSM, but of course that is meaningless for anything related to 1 & 4.

I can't seriously believe we are having this discussion, because you are just making statements with no theoretical support whatsoever - and they are contradicted by experiment. I have already explained this, as has @PeterDonis. You simply ignore the fact that the state i), after a BSM, yields both |Φ+〉₂₃ and |Φ-〉₂₃ outcomes.

I have very carefully explained these projections and the associated mixture are consistent with experiment. And I have very carefully explained the theory behind it.

I have also previously explained why Mjelva's mixture of these states yields the right statistics: Because in delayed choice experiments, Alice and Bob make irreversible measurements before Victor makes any measurement. Hence, it is appropriate to treat the state, post Alice's and Bob's measurements as a projected state or as a mixture representing a sample of runs. In a non-delayed choice variant, where Victor is in the past light cone of Alice and Bob, the mixture is not appropriate as Alice and Bob can choose among complementary measurements. But post-irreversible-measurements, they can't.

And... note that according to your logic, after a BSM: All of the following starting states yield a |Φ+〉₂₃ term - but not a |Φ-〉₂₃ term (which of course is wrong).

• iii) |+〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄/2, ... which is per your derivation of ii) above.
• iv) (|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃) which the same thing, without photons 1 & 4: This is what you get when you delete [independent] terms from a Product State.
• v) |-〉₁⊗(|H〉₂-|V〉₂)⊗(|H〉₃-|V〉₃)⊗|+〉₄ This is what you get when you delete [independent] terms from a Product State.

You can see that mathematically these yield identical Bell state results after a BSM on photons 2 & 3. I.e. You could not, after a sufficiently large number of runs, distinguish whether you started with iii), iv) or v) by looking at the BSM results. That you can get the same thing from both iii) and v) precisely contradicts your assertion.

v) Does not correspond to any of the projected states after Alice's and Bob's measurements.

2. This is a statement about what actually happens. But you use a cheat to get the correct result from a wrong intermediate state. In the real quantum world, the evolution is from Ma's (1) to Ma's (2) and there are no intermediate states to consider - which is why they aren't shown by Ma or anyone else. Timing and/or ordering of measurements is immaterial, which is why no intermediate states are presented. They simply don't occur as you describe.

I'm not a cheat. I am using basic QM, as explored by Mjelva.

 

[DrChinese]

Peres (1999), Delayed choice for entanglement swapping: ...if we attempt to attribute an objective meaning to the [intermediate] quantum state of a single system, curious paradoxes appear: quantum effects mimic not only instantaneous action-at-a-distance but also, as seen here, influence of future actions on past events, even after these events have been irrevocably recorded.

Apparently he never considered the concept of a forward in time only analysis in this seminal paper. Any wonder why?

@Morbert You got any quotes to back up a single criticism I have made? A solid reference? You just keep quoting yourself when challenged.

 

[Morbert]

Peres (1999), Delayed choice for entanglement swapping: ...if we attempt to attribute an objective meaning to the [intermediate] quantum state of a single system, curious paradoxes appear: quantum effects mimic not only instantaneous action-at-a-distance but also, as seen here, influence of future actions on past events, even after these events have been irrevocably recorded.

Apparently he never considered the concept of a forward in time only analysis in this seminal paper. Any wonder why?

@Morbert You got any quotes to back up a single criticism I have made? A solid reference? You just keep quoting yourself when challenged.

Mjelva's paper offers a forward-in-time analysis.

 

[DrChinese]

I'm not a cheat. I am using basic QM, as explored by Mjelva.

I didn't say you were a cheat; I said you are using a cheat. Big difference, I know you are 100% honest. This is what you claim, which is an unsupported cheat:

i) |R>₁⊗(½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃))⊗|R>₄

When you wave your wand like a good magician, you have made 2 terms disappear:

ii) |R>₁⊗(½(|ψ+>₂₃+|Φ->₂₃))⊗|R>₄

Those are not equivalent, so I call that a cheat. And Mjelva et al doesn't count as a reference. So for the Nth time, have you seen this anywhere else... ever? You keep saying it is textbook, standard QM, etc. So where have you seen it?

 

[Morbert]

I didn't say you were a cheat; I said you are using a cheat. Big difference, I know you are 100% honest. This is what you claim, which is an unsupported cheat:

|R>₁⊗(½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃))⊗|R>₄

When you wave your wand like a good magician, you have made 2 terms disappear:

|R>₁⊗(½(|ψ+>₂₃+|Φ->₂₃))⊗|R>₄

Those are not equivalent, so I call that a cheat. And Mjelva et al doesn't count as a reference. So for the Nth time, have you seen this anywhere else... ever? You keep saying it is textbook, standard QM, etc. So where have you seen it?

Your equation

|R>₁⊗(½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃))⊗|R>₄

is not correct, and

|R>₁⊗(½(|ψ+>₂₃+|Φ->₂₃))⊗|R>₄

doesn't match any of my equations (and is also incorrect).

I have given you a simple derivation.

|Φ-〉= (|HH〉-|VV〉)/√2
|Ψ+〉= (|HV〉+|VH〉)/√2

so

|Ψ〉₁₂₃₄
= |R〉₁⊗|L〉₂⊗|L〉₃⊗|R〉₄
= |R〉₁⊗(|H〉₂-i|V〉₂)⊗(|H〉₃-i|V〉₃)⊗|R〉₄/2
= |R〉₁⊗(|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃)⊗|R〉₄/2
= |R〉₁⊗(|Φ-〉₂₃-i|Ψ+〉₂₃)⊗|R〉₄/√2

I invite anyone to check this step to the last line, where the 2 & 3 terms in brackets are:
[|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃]/2 = [ ( |HH〉₂₃-|VV〉₂₃ )/√2 - i( |HV〉₂₃+|VH〉₂₃ )/√2 ]/√2 = [|Φ-〉₂₃-i|Ψ+〉₂₃]/√2

Please either acknowledge this derivation as correct, without any sleight of hand, or specifically point out where it is wrong.

 

[DrChinese]

Mjelva's paper offers a forward-in-time analysis.

Do you not see your circular logic? Seriously? The point of this thread is to debate what he claims, not to accept it. He's wrong, and he will be called out on it soon enough. I don't mean to insult him, as I think scientific work leading to null results are often beneficial. But he made two critical offsetting mistakes in a single paper, not easy to do.

And you think 30 years of dozens of teams exploring Entanglement Swapping theory never tried what he tried?

 

[Morbert]

Do you not see your circular logic? Seriously? The point of this thread is to debate what he claims, not to accept it. He's wrong, and he will be called out on it soon enough. I don't mean to insult him, as I think scientific work leading to null results are often beneficial. But he made two critical offsetting mistakes in a single paper, not easy to do.

And you think 30 years of dozens of teams exploring Entanglement Swapping theory never tried what he tried?

I have defended his claim. You seem to have retreated to saying his paper is wrong because it was not previously published.

 

[Morbert]

@DrChinese The conversation cannot proceed until you address the mistakes in your math, described in post #129. I'm not going to pursue any divergence away from this basic math error that, when corrected, makes agreement between Mjelva and experiment immediately obvious. Either acknowledge it as an error or defend it.

 

[DrChinese]

Your equation

|R>₁⊗(½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃))⊗|R>₄

is not correct, and

|R>₁⊗(½(|ψ+>₂₃+|Φ->₂₃))⊗|R>₄

doesn't match any of my equations (and is also incorrect).

I have given you a simple derivation.Please either acknowledge this derivation as correct, without any sleight of hand, or specifically point out where it is wrong.

Already pointed out the 2 errors, see post here.

I have already give references to actual important papers, such as the Ma paper we are discussing. See their (2). As they present it, a Bell State Measurement leads to: ½(|ψ+>₂₃-|ψ->₂₃-|Φ+>₂₃+|Φ->₂₃, exactly as I say and in complete contradiction to what you say. There is no such thing as what you have done.

 

[Morbert]

Already pointed out the 2 errors, see post here.

In that post you have simply asserted that I have dropped offending terms. I have not.

|Φ-〉= (|HH〉-|VV〉)/√2
|Ψ+〉= (|HV〉+|VH〉)/√2

so

|Ψ〉₁₂₃₄
= |R〉₁⊗|L〉₂⊗|L〉₃⊗|R〉₄
= |R〉₁⊗(|H〉₂-i|V〉₂)⊗(|H〉₃-i|V〉₃)⊗|R〉₄/2
= |R〉₁⊗(|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃)⊗|R〉₄/2
= |R〉₁⊗(|Φ-〉₂₃-i|Ψ+〉₂₃)⊗|R〉₄/√2

I invite anyone to check this step to the last line, where the 2 & 3 terms in brackets are:
[|HH〉₂₃-|VV〉₂₃-i|HV〉₂₃-i|VH〉₂₃]/2 = [ ( |HH〉₂₃-|VV〉₂₃ )/√2 - i( |HV〉₂₃+|VH〉₂₃ )/√2 ]/√2 = [|Φ-〉₂₃-i|Ψ+〉₂₃]/√2

In which line are the offending terms dropped?

 

[DrChinese]

1. I have defended his claim.

2. You seem to have retreated to saying his paper is wrong because it was not previously published.

1. You can't be the authority for Mjelva's claims, or your own statements.

2. Not at all. I don't criticize his work because he was a PhD. candidate at the time, or because it has received virtually no attention (this is about it, and it isn't agreeing with him). I criticize it because it is flat out wrong, and that is not simply an interpretational difference between him and I.

But normally, if I am taking a position that has no theoretical or experimental support: I question myself because of the lack of authoritative support. It's a red flag, or at the very least a yellow flag. Mjelva's paper has "red flag" written all over it from the start, which is probably why it has received no substantive following.

So to protect myself in cases like this, I look through the work of the top researchers in the field. If they disagree with my position, I go back and study more until I understand where I went off the rails. I guess maybe that isn't something you do? Because I am still waiting for a reference for opposing the errors I pointed out. i) Product states like "|+〉1⊗|-〉2⊗|-〉3⊗|+〉4" (your #121) cannot be used in successful entanglement swapping. ii) A successful entanglement swap, per Ma, is something like ½(|ψ+>23-|ψ->23-|Φ+>23+|Φ->23) and does not drop terms to get the right answer.

 

[Morbert]

1. You can't be the authority for Mjelva's claims, or your own statements.

2. Not at all. I don't criticize his work because he was a PhD. candidate at the time, or because it has received virtually no attention (this is about it, and it isn't agreeing with him). I criticize it because it is flat out wrong, and that is not simply an interpretational difference between him and I.

But normally, if I am taking a position that has no theoretical or experimental support: I question myself because of the lack of authoritative support. It's a red flag, or at the very least a yellow flag. Mjelva's paper has "red flag" written all over it from the start, which is probably why it has received no substantive following.

So to protect myself in cases like this, I look through the work of the top researchers in the field. If they disagree with my position, I go back and study more until I understand where I went off the rails. I guess maybe that isn't something you do? Because I am still waiting for a reference for opposing the errors I pointed out. i) Product states like "|+〉1⊗|-〉2⊗|-〉3⊗|+〉4" (your #121) cannot be used in successful entanglement swapping. ii) A successful entanglement swap, per Ma, is something like ½(|ψ+>23-|ψ->23-|Φ+>23+|Φ->23) and does not drop terms to get the right answer.

So are you going to address #134? The only substantive criticism you have of Mjelva's paper seems to be premised in some simple math mistakes you've made.

 

[DrChinese]

In that post you have simply asserted that I have dropped offending terms. I have not.
In which line are the offending terms dropped?

Easy. When you perform a successful swap on a 2 & 3 pair (just those 2) in any Product state to start with: you get 1 of 4 Bell states, randomly selected. You get 4 terms, not 2. You only inserted 2.

And for the Nth time, you can't use such a 4-fold state for entanglement swapping anyway. How do you not understand this point? I thought that giving you references to seminal work on Type II PDC would cover this. You can't start with a Product State for all 4 photons. You must start with Entangled states, and they cannot become Product states in some intermediate representation and still operate in a swap. This is essential, and is explained in dozens of papers on the subject.

 

[Morbert]

Easy. When you perform a successful swap on a 2 & 3 pair (just those 2) in any Product state to start with: you get 1 of 4 Bell states, randomly selected. You get 4 terms, not 2. You only inserted 2.

And for the Nth time, you can't use such a 4-fold state for entanglement swapping anyway. How do you not understand this point? I thought that giving you references to seminal work on Type II PDC would cover this. You can't start with a Product State for all 4 photons. You must start with Entangled states, and they cannot become Product states in some intermediate representation and still operate in a swap.

Please answer the question posed in post #134. On what line, specifically, are these offending terms dropped? You made an explicit claim that I cheated by dropping terms to get to the right answer. Please back this up specifically.

 

[DrChinese]

I should remind anyone still following why it is called "Entanglement Swapping" in the first place. Initially, pairs 1&2 and 3&4 are entangled. After the swap, pairs 1&4 and 2&3 are entangled. Hence the term "swap", sorta like swapping partners in two couples.

Joking aside: Mjelva flat out denies any swapping occurs, or that photons 1 & 4 are ever entangled. That is the "red flag" I mentioned in an earlier post. He says: "...it should be clear from the analysis that, as each of the states , (9) following the Bell-state measurement can be written as the tensor product of the state of particle 1, the state of particle 4 and the state of the pair (2,3), the particles 1 and 4 are not entangled."

On the other hand, in the words of a top team (that apparently some readers are willing to dismiss at the drop of a hat): "...if Victor subjects his photons 2 and 3 to a Bell-state measurement, they become entangled. Consequently photons 1 (Alice) and 4(Bob) also become entangled and entanglement swapping is achieved."

I don't think the words of Mjelva and those of Ma can be reconciled. Probably any more than the positions of @Morbert and I. If anyone reading wants to collaborate on a comment on the Mjevla paper to be deposited in the arxiv, let me know. I doubt he will be any more likely to acknowledge his errors than Morbert is, but you never know.

Please answer the question posed in post #134. On what line, specifically, are these offending terms dropped?

I already answered, #137. I bolded my response.

---

At this point, I think I've contributed more than anyone would care to know about the 2 critical errors in Mjelva's paper. The justification for my conclusions are well-documented in this thread, no need to further summarize or repeat.

I thank Morbert for his time and assistance in clarifying Mjelva's position. I admit I am disappointed that he was not able to corroborate any of his disputed claims by reference to to suitable published works, and I can only conclude there aren't any.

I will continue to answer questions if it will add something to what's already been presented by myself and Morbert.

Cheers,
-DrC

 

[PeterDonis]

There is no such thing as a standard "forward-in-time analysis" of entanglement swapping with a 4-fold Product State in the middle.

I'd like to expand on this a little.

The idea behind Mjelva's analysis, as I understand it, is that if we start with the state where 1&2 are entangled and 3&4 are entangled, and we then measure 1 & 4, that collapses 2 & 3 as well, so the result is a 4-fold product state.

However, as @DrChinese and I have both pointed out, once you have that 4-fold product state, nothing you can possibly do at the BSM will create any entanglement anywhere. So you can't possibly extract any prediction that shows entanglement.

It seems to me that what Mjelva is doing is ignoring the particular aspects of the experimental results that show entanglement between 1 & 4 when a swap is performed, and just looking at the overall statistics with no post-selection, which of course cannot show any entanglement between 1 & 4, and then saying that, well, the 4-fold product state can produce those same overall statistics, so what's the problem?

The problem is that those overall statistics are not the only things that QM makes predictions about, or the only things that experiments can test. A correct model has to reproduce all of the QM predictions that have been experimentally verified, including the ones that use post-selection to pick out subsets of runs where the 1 & 4 statistics show entanglement. Ignoring those predictions is simply not correct science.

 

[Sambuco]

The idea behind Mjelva's analysis, as I understand it, is that if we start with the state where 1&2 are entangled and 3&4 are entangled, and we then measure 1 & 4, that collapses 2 & 3 as well, so the result is a 4-fold product state.

We should be careful about what we mean by "so the result is a 4-fold product state" because, at least to me, this is part of the misunderstanding here.
For a single run, the answer is yes, the state at the intermediate time (after Alice and Bob measurements on 1&4, but before Victor's BSM measurement on 2&3) is one of the four states that Mjelva's calls $\ket{\Psi(t_1)}_{A,B,C,D}$. Instead, if we are talking about the system formed by a large sample of runs, the state of the system is a mixture, as he shows by the density matrix in eq. (4).

However, as @DrChinese and I have both pointed out, once you have that 4-fold product state, nothing you can possibly do at the BSM will create any entanglement anywhere. So you can't possibly extract any prediction that shows entanglement.

This is obviously true if you are considering only one run out of all the runs of the experiment, but it is not true if you are considering a large sample of runs because in that case you could sort (post-select) the runs according to the results Victor got in the BSM.

It seems to me that what Mjelva is doing is ignoring the particular aspects of the experimental results that show entanglement between 1 & 4 when a swap is performed, and just looking at the overall statistics with no post-selection, which of course cannot show any entanglement between 1 & 4, and then saying that, well, the 4-fold product state can produce those same overall statistics, so what's the problem?

No! Please @PeterDonis, read Mjelva's section 4.1.1. In fact, what Mjelva's proves is that after post-selection according to the results obtained at the 2&3 BSM, every subset of 1&4 photons violate Bell inequalities, "each subset behaves as if it consisted of entangled pairs of distant particles" (the quote is from Peres's seminal work about DCES https://arxiv.org/abs/quant-ph/9904042). In other words, Mjelva's proves the entanglement between photons 1&4 in each one of the four subsets. Mjelva's result IS the theoretical prediction of Ma's experiment!

The only difference between Mjelva's analysis and that of those who argue that temporal entanglement exists is that he could show that entanglement between 1&4 photons in each subset does not imply entanglement between 1&4 photons in each pair. This is the contribution of Mjelva's work!

For more details, see my post #43.

Lucas.

 

[PeterDonis]

if we are talking about the system formed by a large sample of runs, the state of the system is a mixture

That's true, but a mixture is still not an entangled state, so it can't produce an entangled state after the BSM.

in that case you could sort (post-select) the runs according to the results Victor got in the BSM.

But if the state prior to the BSM is a mixture, not an entangled state, no post-selection can produce the statistics of an entangled state for photons 1 & 4. The only way to produce such statistics is for there to have been entanglement between 1 & 2, and 3 & 4, prior to the BSM, that then gets swapped to the pairs 1 & 4, and 2 & 3.

what Mjelva's proves is that after post-selection according to the results obtained at the 2&3 BSM, every subset of 1&4 photons violate Bell inequalities

This is impossible if there is no entangled state prior to the BSM. @DrChinese has already shown where Mjelva makes errors in his analysis.

 

[Sambuco]

But if the state prior to the BSM is a mixture, not an entangled state, no post-selection can produce the statistics of an entangled state for photons 1 & 4.

This is not entirely true. The BSM does not produce statistics of a single entangled state for photons 1&4, but produces four sets of statistics consistent with four different entangled states for photons 1&4.

All that is well-known for more than 25 years! In fact, the first paper where this is discussed (one year before Peres) was by Cohen in 1999 in his paper "Counterfactual entanglement and nonlocal correlations in separable states" (https://arxiv.org/abs/quant-ph/9907109). This paper is cited by Ma and Zeillinger group in their works on entanglement swapping!

As said by Cohen:

"Any separable density matrix may contain “hidden” entanglement in that it can always be rewritten as a sum of projections on entangled states."

Later, he introduced the concept of conterfactual entanglement:

"Remarkably, this analysis can be applied with equal validity to factorable states, with density matrices of the form ρ₁₂ = ρ₁ ρ₂, where the constituent subsystems do not share any entanglement with an extraneous system and need never have interacted with each other. These processes can be seen to give rise to a new kind of postselection-induced Bell inequality violation."

@DrChinese has already shown where Mjelva makes errors in his analysis.

Sorry, but there is no error in Mjelva's projection-based treatment of Ma's experiment. I carefully read the paper.

Lucas.

 

[PeterDonis]

The BSM does not produce statistics of a single entangled state for photons 1&4, but produces four sets of statistics consistent with four different entangled states for photons 1&4.

That's true, but it's also true that the four sets of statistics can easily be separated into four subsets by looking at the photon 2&3 measurement results. That's in an idealized experiment where all four possible results can be distinguished. No actual experiment has achieved that yet. But in actual experiments there is always at least one subset of the results that can be unambiguously picked out, and QM makes definite predictions about the statistics of that particular subset. Ignoring those predictions is simply not good science, and that's what you're doing when you talk as though the statistics of the complete set of data, without any post-selection or any splitting into subsets, is the only thing that matters. It's not.

there is no error in Mjelva's projection-based treatment of Ma's experiment

Then you should point out specifically where you think the analysis @DrChinese has already given in this thread is wrong.

 

[Morbert]

What is meant by a standard forward-in-time analysis is an analysis based on rules like these. Entanglement swapping experiments are a great technical feat, but they do not contain any exotic features like black holes that would make these rules difficult to apply. These rules, when carefully applied, correctly predict all experimental outcomes and all correlations of delayed-choice entanglement swapping experiments. What Mjelva has shown is that, when this approach is adopted, there is no projection onto an entangled 1 & 4 state, even when the measurement is modeled as ideal and nondestructive (this is why he uses massive particle spins rather than photon polarization), and even though all predictions made by Ma et al are recovered with this approach.

That's true, but a mixture is still not an entangled state, so it can't produce an entangled state after the BSM.

But if the state prior to the BSM is a mixture, not an entangled state, no post-selection can produce the statistics of an entangled state for photons 1 & 4. The only way to produce such statistics is for there to have been entanglement between 1 & 2, and 3 & 4, prior to the BSM, that then gets swapped to the pairs 1 & 4, and 2 & 3.

What this signifies is that Alice's and Bob's measurements are irreversible. The projection onto a state after Alice's and Bob's measurement forecloses alternative, complementary measurements Alice and Bob could have done, and so a mixture of these states will get you the right statistics across runs where Alice and Bob made that specific choice. In runs where Alice and Bob made measurements in a different basis, a different set of projected states, and hence a different mixture*, is obtained. Combining all the probabilities and correlations computed from all the alternative projections/mixtures corresponding to the alternative measurement choices Alice and Bob make gets you all predictions concerning the experiment.

If we were considering a non-delayed-choice entanglement swapping experiment, where Alice and Bob make their measurements in the future light cone of Victor, then the state prior to the BSM cannot be modeled with such projected states (or the corresponding mixture).

*[edit] - These mixtures might all be the same. I do not know off the top of my head if the mixture is complete, and would have to check. It makes little difference either way.

This is impossible if there is no entangled state prior to the BSM. @DrChinese has already shown where Mjelva makes errors in his analysis.

Then you should point out specifically where you think the analysis @DrChinese has already given in this thread is wrong.

DrChinese, in post #112, has made a simple math error when expanding the projected states in the Bell basis for 2 & 3 that has lead him to believe Mjelva's results contradict experiment. At that point in the conversation we are deep in the weeds, so it is an easy mistake to make, but I correct his mistake in post #117. This correction recovers all the predictions and correlations observed in the delayed-choice entanglement swapping experiment.

Instead of accepting this correction, DrChinese has begged the question and accused me of using a cheat, secretly dropping offending terms, when in reality those "offending" terms were his mistake, and do not follow from the expansion of the projected 2 & 3 state in a Bell basis.

 

[Sambuco]

That's true, but it's also true that the four sets of statistics can easily be separated into four subsets by looking at the photon 2&3 measurement results.

I completely agree with you. Any QM treatment of the problem must predict the violation of Bell inequalities in each one of the four subsets.

Ignoring those predictions is simply not good science, and that's what you're doing when you talk as though the statistics of the complete set of data, without any post-selection or any splitting into subsets, is the only thing that matters.

I didn't claim something like what you say. To clarify I explicitly said in many post along this thread (#31, #38, #43) that if Alice and Bob sorted their measurement outcomes in four subsets following Victor results at the BSM, each one of the subsets will display non-local correlations (violation of Bell inequalities). And what is extremely important is to remark that to make this post-selection it is mandatory that Victor have performed a BSM. If Victor performs SSM on photons 2&3, there is no way to separate 1&4 measurement outcomes in four separate subsets showing Bell correlations.

Then you should point out specifically where you think the analysis @DrChinese has already given in this thread is wrong.

For example, in his post #112, where he said that, if Alice and Bob both measure R and then Victor perform a BSM, the state of the system would be:

$\ket{\Psi} = \ket{R}_1 \otimes \frac{1}{2} (\ket{\psi^+}_{23} - \ket{\psi^-}_{23} - \ket{\phi^+}_{23} + \ket{\phi^-}_{23}) \otimes \ket{R}_4$

This state is wrong for two reasons:

1. The state that @DrChinese wrote does not follow from QM postulates. If at time $t_1$ Alice and Bob measure 1&4 photons and both obtain R, the projection postulate says that the system after measurement is:
$\ket{\Psi(t_1)} = \ket{R}_1 \otimes \ket{L}_2 \otimes \ket{L}_3 \otimes \ket{R}_4$
Since $\ket{L}_j = \frac{1}{\sqrt{2}} (\ket{H}_j - i \ket{V}_j)$, then,
$\ket{\Psi(t_1)} = \ket{R}_1 \otimes \frac{1}{\sqrt{2}} (\ket{H}_2 - i \ket{V}_2) \otimes \frac{1}{\sqrt{2}} (\ket{H}_3 - i \ket{V}_3) \otimes \ket{R}_4$. Reordening the part of photons 2&3,
$\ket{\Psi(t_1)} = \ket{R}_1 \otimes \frac{1}{2} (\ket{HH}_{2,3} - \ket{VV}_{2,3} - i\ket{HV}_{2,3} - i\ket{VH}_{2,3}) \otimes \ket{R}_4$.
Taking into account that $\ket{\phi^-}_{j,k} =\ket{HH}_{j,k} - \ket{VV}_{j,k}$ and $\ket{\psi^+}_{j,k} =\ket{HV}_{j,k} + \ket{VH}_{j,k}$, the state of the system after Alice and Bob both measured R on 1&4 photons is:
$\ket{\Psi(t_1)} = \ket{R}_1 \otimes \frac{1}{\sqrt{2}} (\ket{\phi^-}_{2,3} - i \ket{\psi^+}_{2,3}) \otimes \ket{R}_4$.

This state is not the same as the one @DrChinese wrote. As the state that I wrote strictly follows from the application of QM axioms, it is demonstrated that the state @DrChinese wrote is wrong.

2. It is contradicted by experimental results from Ma's paper. Following @DrChinese argument, the state of the 2&3 pair is always prior to the BSM $\frac{1}{2} (\ket{\psi^+}_{23} - \ket{\psi^-}_{23} - \ket{\phi^+}_{23} + \ket{\phi^-}_{23})$ irrespective of the measurement outcomes Alice and Bob obtained on photons 1&4. As @Morbert clearly explained in post #121, this assumption predicts that for the set of runs where Alice measured $\ket{+}_1$ and Bob also measured $\ket{+}_4$, there is a 25% chance that Victor obtains $\ket{\phi^-}_{2,3}$, which contradicts the result in Fig. 3(a) in the Ma's paper, where the measurement by Victor of the state $\ket{\phi^-}_{2,3}$ implies an anticorrelation between 1&4 photons in the $+/-$ basis.

Lucas.

 

[Sambuco]

I would like to say a few things I believe are important to try to disentangle () the discussion:

1. As I previously mentioned, the argument that postselection could lead to the violation of Bell inequalities is completely mainstream. In that sense, I want to share a recent paper by Bacciagaluppi & Hermens (https://arxiv.org/abs/2002.03935) where they deepen into the implications of entanglement swapping for the relativity of pre- and post-selection. In the first paragraph of the introduction, they said:
"Here we treat the case where postselection can give rise to violations of the Bell inequalities as proposed in [2, 3] and realized experimentally in [4] (see also [5]). This should be distinguished from the standard Bell inequality violations due to entanglement."
References [2] and [3] are Cohen's and Peres's works, respectively, while [4] is the Ma's paper on DCES experimental realization ([5] is another paper from Ma, Zeillinger and others on this subject).

2. The Mjelva's forward-in-time treatment (which is in agreement with Cohen's "counterfactual entanglement" and Bacciagaluppi's post-selection interpretations of the DCES experiments) does not contradict the analysis by Ma et al. They are complementary to each other.
If we strictly follow the QM rules of forward-in-time evolution, we will conclude that ordinary (non-delayed) entanglement swapping involves genuine entanglement between 1&4 photons, while DCES does not, because no 1&4 pair have a non-separable state at any moment of time. However, as @DrChinese said many times, the fourfold QM predictions does not depend on the order in which the measurements were performed. As clearly stated by Ma et al., what they wanted to demonstrate is that some kind of entanglement-separability duality arises, in analogy with the wave-particle duality of Wheeler's delayed-choiced experiments. They denied that the quantum state is a "real physical object", favoring the view that the state is "no more than catalogue of our knowledge". Under this interpretation, nobody could prohibit us from giving priority to the Victor BSM measurement and apply the projection postulate on the initial state even when we certainly know that Alice and Bob already measured photons 1&4. Then, the non-separable (entangled) state obtained for 1&4 photons will not coincide with the one that Alice and Bob have after they recorded each one of their measurement results, but the fourfold measurement outcomes are equally well predicted by both approaches. This is the entanglement-separability duality that Ma et al. convincingly showed.

Lucas.

 

[DrChinese]

I would like to say a few things I believe are important to try to disentangle () the discussion:

1. As I previously mentioned, the argument that postselection could lead to the violation of Bell inequalities is completely mainstream.

2. In that sense, I want to share a recent paper by Bacciagaluppi & Hermens (https://arxiv.org/abs/2002.03935) where they deepen into the implications of entanglement swapping for the relativity of pre- and post-selection. In the first paragraph of the introduction, they said:
"Here we treat the case where postselection can give rise to violations of the Bell inequalities as proposed in [2, 3] and realized experimentally in [4] (see also [5]). This should be distinguished from the standard Bell inequality violations due to entanglement."
References [2] and [3] are Cohen's and Peres's works, respectively, while [4] is the Ma's paper on DCES experimental realization ([5] is another paper from Ma, Zeillinger and others on this subject).

Good one.

1. This is hardly what I would call mainstream (since it isn't mainstream). But a mainstream reference would still be welcome, I'd be happy to have judged this too harshly. Where's something from Ma, Megidish on his ideas? Note that the use of the word "post-selection" (or similar) itself is common in the literature, and in no way is equivalent to the forward in time only premise of Mjelva.


2. Thanks for this reference. Note that the novel (and subject to being disputed) ideas of Bacciagaluppi & Hermens in no way follow the referenced papers of Peres, Ma, Zeilinger, etc., as might be implied per your quote.

Obviously, this paper is relied upon by Mjelva, but has otherwise been ignored. And for good reason: As has been common with those attempting to deny quantum nonlocality (or its non-temporal sibling), they propose an experimental version of Ma et al that we already know the answer to.

They basically want to add sufficient distance to the Ma delayed choice swapping version such that no signal containing information about Alice and Bob's choice of settings can get to Vicky in time to influence the results. OK, they are simply adding more experimental hoops, imagining that there could exist a lightspeed effect that could be discerned in swapping. Let's see:

a. We already know that Swapping results are invariant as to whether Vicky's swap occurs before vs. after the measurements of Alice and Bob. (Ma et al)
b. We already know that Swapping results are invariant as to whether Alice's photon ceases to exist before Bob's photon is created. (Megidish et al)
c. We already know that the Swapping results are invariant when the settings of Alice and Bob are changed mid-flight. (Hensen et al)

Are Bacciagaluppi and Hermens proposing there is something different (a hypothetical classical effect) occurring that will be detected if all 3 of the above are tested simultaneously? Of course not; there is not the slightest indication of such effect, and they make no mention of how that might work and have gone unnoticed previously. Would their proposed test be feasible and beneficial? Well, I am a fan of experiments that simply confirm what we already know (thousands of these have been performed with entanglement). So yes, let's hope this happens - assuming it hasn't already been executed...

Because it probably already has. See Wu et al, (2022), Figure 3 in which Alice and Bob's settings are changed mid-flight with Vicky's BSM operating both before and after measurements of Alice/Bob. As suggested by Bacciagaluppi & Hermens, "all three measurements are at spacelike separation from each other as indeed shown in Figure 1". Note that in my reference, the names of the 3 testing stations are Alice, Claire and Bob instead of (respectively) Alice, Bob and Vicky/Victor as is common to many experiments. Although the purpose of this experiment is different than Ma's, it confirms the expected predictions of Entanglement Swapping experiments. Had the Bacciagaluppi & Hermens concept led to a novel effect, that would have been detected. Nothing to see here.

But none of that is what we really care about, we always knew their proposed experiment would not yield a surprise. And we don't care about claims that there are classical examples that yield quantum outcomes, these are a dime a dozen and have absolutely nothing to do with actual quantum experiments despite the pains they go through to draw an analogy. And in fact, such examples merely serve to reveal that the authors have a definite view that Bell's Theorem is flawed in some way. Bacciagaluppi & Hermens: "Although the experiment by Ma et al. [4] was accordingly set up to ensure timelike separation between Alice and Bob’s and Vicky’s measurements, it is precisely this feature that provides the loophole for a classical explanation [i.e. in contradiction to Bell] of the results." We don't case about that either.

---

What we do care about: How this paper relates to Mjelva's, and does it support it in any way. It does relate, and it does repeat some of the debunked claims of Mjelva. Let's get specific, from Bacciagaluppi & Hermens:

i) "If we imagine that Alice’s and Bob’s measurements actually collapse the state at a distance also at Vicky’s site, then the individual pairs of qubits on which Vicky performs the Bell measurements are in definite product states."

As I have already explained ad nauseum: Those 4-fold product states can be easily created (my "Ma-X" example is just one of many that accomplish the same result, a product state). They can be used as inputs to the Ma setup. And they simply do NOT reproduce same entangled state statistics per Ma. Therefore, such measurements do not collapse the state (at a distance or otherwise) as they contemplate. What matters is the initial context (Bell entangled pairs 1&2 and 3&4 in a product state) and the final swapped context (Bell entangled pairs 1&4 and 2&3 in a product state). There are no real intermediate 4-fold definite product states.

To be crystal clear: In delayed choice scenarios, photons 2 & 3 (prior to Vicky's BSM/swap) do not have definite polarizations and should still be considered entangled with their already measured partners. This characterization -"a seemingly paradoxical situation" - violates the spirit of causal norms, but agrees with both theory and experiment.


ii) "The quantum-mechanical predictions are invariant under change of foliation, because measurements at spacelike separation commute. Because of the relativity of pre- and post-selection, instead, the difference between Bell inequality violations due to entanglement and due to post-selection is no longer invariant. What in the case of timelike separation appear as physically different effects, in the case of spacelike separation turn out to be one and the same physical effect."

It is not commonly accepted that phenomena featuring time-like (delayed choice variations) and space-like separation (with settings changing mid-flight) are qualitatively different in Quantum Mechanics. The ground-breaking Entanglement Swapping experiments of Jennewein et al, Kaltenbaek et al, Ma et al, Megidish et al all serve to follow the pioneering theoretical work of Zeilinger and Peres of the 1990's. Their quote doesn't actually tell us anything new, as all of this can be - and was - expected from that theoretical work.

 

[PeterDonis]

If Victor performs SSM on photons 2&3, there is no way to separate 1&4 measurement outcomes in four separate subsets showing Bell correlations.

That's correct, but it misses a key point: it is possible to separate the 1&4 measurement outcomes into subsets according to the photon 2&3 measurement outcomes. It's just that those subsets do not show Bell correlations in the SSM case, but they do in the BSM case.

In other words, if you do what we normally do in any other area of science, and start with the experimental data, and apply a simple, well-defined procedure to that data in both cases, you get different results: BSM -> correlations; SSM -> no correlations. And in any other area of science, this kind of thing is taken to show that the choice BSM vs. SSM has some kind of real effect on photons 1 & 4. But somehow, when it's this particular QM experiment, people make strenuous efforts to avoid this obvious conclusion that in any other area of science would be commonplace.

 

[DrChinese]

The state that @DrChinese wrote does not follow from QM postulates. If at time $t_1$ Alice and Bob measure 1&4 photons and both obtain R, the projection postulate says that the system after measurement is:
$\ket{\Psi(t_1)} = \ket{R}_1 \otimes \ket{L}_2 \otimes \ket{L}_3 \otimes \ket{R}_4$

The projection postulate does not say that at all, and you have misapplied the idea of projection. What is says (as applies here): IF photons 2 & 3 are measured on the same basis as photons 1 & 4, THEN the results will be certainly |LL>₂₃. No one is disputing this. But that is a far cry from what happens - because those measurements were not performed. There statements by many authors in the literature (and actually 866,000 links with this exact quote, per Google) similar to the following famous quote from Peres (1978):

"Unperformed measurements have no results."