# What does delayed choice entanglement swapping require?

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StevieTNZ
A follow-up question from a previous thread: https://www.physicsforums.com/threa...-of-a-quantum-operation-called-a-bsm.1047772/.

So my point of view is this: there are systems 1-4 (which refer to photons 1-4 in https://www.nature.com/articles/nphys2294)

There is entanglement between systems 1 and 2 (Alice) and systems 3 and 4 (Bob). Systems 2 and 3 get sent to Victor for a BSM to occur, projecting systems 1 and 4 into an entangled state. However, in the delayed choice experiment, measurement occurs on systems 1 and 4 before the other two systems are projected into a bell-state.

My question is: wouldn't entanglement need to exist between systems 1 and 2 (likewise systems 3 and 4) in order for it be swapped? If a measurement occurs on system 1 and 4 and physical collapse occurs, entanglement is broken and there is essentially nothing to swap. So in light of what the experimental results how, what can we say about physical collapse if entanglement is indeed shown between systems 1 and 4?

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DrChinese, physika and vanhees71

## Answers and Replies

Mentor
Moderator's note: A slight edit has been made to the OP.

Mentor
wouldn't entanglement need to exist between systems 1 and 2 (likewise systems 3 and 4) in order for it be swapped?
Both the predictions of the basic math of QM and the experimental results show that the entanglement between systems 1 and 4, as shown by the statistics of measurements on them, can be controlled by an interaction between systems 2 and 3, regardless of the spacetime relationship between all of the measurements. So whatever is going on, it apparently can go on even if the measurements of systems 1 and 4 are in the past light cone of the interaction between systems 2 and 3. That is just the experimental fact.

What stories can validly be told to "explain" this experimental fact will depend on which interpretation of QM you want to use. The "entanglement swapping" story appears, at least to its proponents, to be consistent with an fairly "realist" interpretation of the quantum state; but such an interpretation is also what one might call "acausal", in that it does not require any kind of "normal" cause and effect relationship between the interaction between systems 2 and 3, which one might think would "cause" the swapping, and the measurement results for systems 1 and 4, which one might think would be "effects" of the swapping.

If a measurement occurs on system 1 and 4 and physical collapse occurs, entanglement is broken
Only in some interpretations, not all.

in light of what the experimental results show, what can we say about physical collapse if entanglement is indeed shown between systems 1 and 4?
I think any claims along these lines would also have to be interpretation dependent.

I don't think there has been a lot of time for proponents of the various QM interpretations to digest these types of experiments, so there might not be much to go on in the literature.

DrChinese, gentzen and berkeman
Gold Member
2022 Award
A follow-up question from a previous thread: https://www.physicsforums.com/threa...-of-a-quantum-operation-called-a-bsm.1047772/.

So my point of view is this: there are systems 1-4 (which refer to photons 1-4 in https://www.nature.com/articles/nphys2294)

There is entanglement between systems 1 and 2 (Alice) and systems 3 and 4 (Bob). Systems 2 and 3 get sent to Victor for a BSM to occur, projecting systems 1 and 4 into an entangled state. However, in the delayed choice experiment, measurement occurs on systems 1 and 4 before the other two systems are projected into a bell-state.

My question is: wouldn't entanglement need to exist between systems 1 and 2 (likewise systems 3 and 4) in order for it be swapped? If a measurement occurs on system 1 and 4 and physical collapse occurs, entanglement is broken and there is essentially nothing to swap. So in light of what the experimental results how, what can we say about physical collapse if entanglement is indeed shown between systems 1 and 4?
Indeed, the entanglement swapping only works if you have an entanglement between 1&2 (Alice) and 3&4 (Bob). You can think of realized these as two entangled photon pairs (all together a four-photon state). As an example take the state
$$|\Psi \rangle = \frac{1}{2} [\hat{a}^{\dagger}(\vec{p}_1,+1) \hat{a}^{\dagger}(\vec{p}_2,-1)-\hat{a}^{\dagger}(\vec{p}_1,-1) \hat{a}^{\dagger}(\vec{p}_2,+1)][\hat{a}^{\dagger}(\vec{p}_3,+1) \hat{a}^{\dagger}(\vec{p}_4,-1)-\hat{a}^{\dagger}(\vec{p}_3,-1) \hat{a}^{\dagger}(\vec{p}_4,+1)]|\Omega \rangle.$$
If you then use the photons with momenta ##\vec{p}_2## and ##\vec{p}_3## (at V's place) to make a Bell measurement and measure only photons with momenta ##\vec{p}_1## and ##\vec{p}_4## where V got one out of four possible Bells states, you'll see that this subensemble is described by a Bell state for photone ##\vec{p}_1## and ##\vec{p}_4##.

It doesn't matter when you do this choice. You can use a measurement protocol for all 4 photons and post-select each of the four possible projections to Bell states of V's photons.

I think the only consistent interpretation, given the microcausality property of QED and the success in using QED to predict the outcome of this experiment, is that indeed the cause of the correlation between photon 1 and 4 in each of the sub-ensemble is due to the preparation of the four-photon state in the state ##|\Psi \rangle## given above.

A real experiment of this kind is given here:

https://arxiv.org/abs/0809.3991
https://doi.org/10.1103/PhysRevA.79.040302

Gold Member
1. What stories can validly be told to "explain" this experimental fact will depend on which interpretation of QM you want to use. The "entanglement swapping" story appears, at least to its proponents, to be consistent with an fairly "realist" interpretation of the quantum state; but such an interpretation is also what one might call "acausal", in that it does not require any kind of "normal" cause and effect relationship between the interaction between systems 2 and 3, which one might think would "cause" the swapping, and the measurement results for systems 1 and 4, which one might think would be "effects" of the swapping.
2. My question is: wouldn't entanglement need to exist between systems 1 and 2 (likewise systems 3 and 4) in order for it be swapped? If a measurement occurs on system 1 and 4 and physical collapse occurs, entanglement is broken and there is essentially nothing to swap. So in light of what the experimental results how, what can we say about physical collapse if entanglement is indeed shown between systems 1 and 4?

1. Very concise, accurate summary. Yes, this is an acausal interpretation corresponding to viewing the quantum state as "real" (whatever that might mean, obviously not the "real" as in local "realistic" theories).

2. Yes, as already answered, there must be 2 entangled pairs as sources (1 & 2, and 3 & 4).

The problem comes in determining whether a) collapse is physical; and b) if it IS physical, when does it occur? I am going to say: Yes, collapse is physical. (You might expect me to say that, since I also say that entanglement swapping is physical.)

When does it occur? Referencing what PeterDonis said in 1.: such quantum physical processes are acausal and do not follow normal cause and effect rules. The following sequences are equivalent in terms of quantum physical processes:

a. First Victor measures 2 & 3 (the BSM), then Alice measures 1, lastly Bob measures 4.
b. First Alice measures 1, then Victor measures 2 & 3 (the BSM), lastly Bob measures 4.
c. First Alice measures 1, then Bob measures 4, lastly Victor measures 2 & 3 (the BSM).

In addition: 1 can be measured before 3 & 4 come into existence. The Bell test results will be the same in each scenario, indicating successful entanglement swaps.

Our conclusion must be that the collapse does not occur in all places in space at a single point in time. When 1 collapses, that does not cause 2 to collapse at the same "time". I don't know when the collapse occurs, no one does really, because you can't see that operation has occurred until there is a measurement. Only by looking at the total quantum context (measurements on 1, 2, 3, 4) can you make any statements. And when you do, they will not indicate physical action occurred in a past-to-future order.

My preferred mental visual is:
a) draw the full quantum context in spacetime;
b) the full context will always have connections limited by c (although causal direction is ambiguous);
c) apply a random element (perhaps more than one) to the overall context that makes the entire context statistically consistent with quantum predictions.

The ONLY way you get to that (quantum events occur only in past to future order) is by assumption. And in recent experiments, it is becoming clearer and clearer that QM interpretations with that assumption are not looking so good these days...

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Mentor
I think the only consistent interpretation, given the microcausality property of QED and the success in using QED to predict the outcome of this experiment, is that indeed the cause of the correlation between photon 1 and 4 in each of the sub-ensemble is due to the preparation of the four-photon state in the state ##|\Psi \rangle## given above.
But that preparation, by itself, does not lead to any entanglement between photons 1 and 4. If you just do the preparation, but don't do anything to photons 2 and 3 (no BSM or any other measurement that can project those two photons into an entangled state), then photons 1 and 4 are not correlated at all, and measurements on them will show that. Only if you do make a measurement on photons 2 and 3 that can project them into an entangled state, will you see any correlation between photons 1 and 4. So it doesn't seem like the preparation, by itself, can be the cause of any correlation between photons 1 and 4.

vanhees71 and DrChinese
Gold Member
2022 Award
I think, in principle, there is no contradiction between our views. Indeed the preparation alone doesn't entangle photons 1 and 4. To the contrary, it's made precisely such that photons 1 and 4 are not entangled. The important point to understand is that even when doing the measurements on photons 2 and 3 nothing happens to the full ensemble of photons 1 and 4 but that when choosing each of the four subensembles you prepare a specific entangled state of photons 1 and 4 without a causal interaction with either of these photons. This possibility of entanglement swapping is due to the preparation of the 4 photons in the original state, i.e., photons 1 and 2 as well as photons 3 and 4 in a Bell state, but these two pair states in a product state. The point is that, if you accept the validity of the microcausality constraint of relativistic QFT, then there can be no causal effect of the measurement on photons 2 and 3 on the photons 1 and 4. It just enables the (post-)selection of specific subensembles for which photons 1 and 4 are entangled.

I don't see, whether we still disagree or not and if so, about which detail.

Nullstein
Mentor
even when doing the measurements on photons 2 and 3 nothing happens to the full ensemble of photons 1 and 4
This is not correct. Doing the BSM does change the overall statistics for photons 1 and 4. It has to, because if you don't do the measurement, no pairs of photons 1 and 4 are entangled, but if you do the measurement, some pairs of photons 1 and 4 are entangled. So the overall statistics on photons 1 and 4 change from being all "not entangled" to a mixture of "not entangled" and "entangled".

DrChinese
Mentor
when choosing each of the four subensembles you prepare a specific entangled state of photons 1 and 4
There aren't four subensembles; there are only two, the "entangled" and "not entangled" subensembles. The BSM on photons 2 and 3 only has two possible results: either "event ready", meaning a Bell state was produced and photons 1 and 4 are entangled, or "not", meaning no Bell state was produced and photons 1 and 4 are not entangled.

Gold Member
2022 Award
The full ensemble of photons 1&4 is always that of uncorrelated photons. Only the subensembles when projecting 2&3 to one of the Bell states for the 2&3 pair are also Bell states. Quantum states don't depend on which basis is used to express them. It's very simple to see in this paper (Eqs. 2 and 3):

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.3891

martinbn and Nullstein
Mentor
It's very simple to see in this paper (Eqs. 2 and 3):

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.3891
This paper describes a different experiment from the one in the reference given in the OP. In the experiment described in the paper in the quote above, there are four possible Bell states. In the experiment referenced in the OP, there is only one.

Gold Member
1. Indeed the preparation alone doesn't entangle photons 1 and 4. To the contrary, it's made precisely such that photons 1 and 4 are not entangled. The important point to understand is that even when doing the measurements on photons 2 and 3 nothing happens to the full ensemble of photons 1 and 4 but that when choosing each of the four subensembles you prepare a specific entangled state of photons 1 and 4 without a causal interaction with either of these photons. This possibility of entanglement swapping is due to the preparation of the 4 photons in the original state, i.e., photons 1 and 2 as well as photons 3 and 4 in a Bell state, but these two pair states in a product state.

2. The point is that, if you accept the validity of the microcausality constraint of relativistic QFT, then there can be no causal effect of the measurement on photons 2 and 3 on the photons 1 and 4. It just enables the (post-)selection of specific subensembles for which photons 1 and 4 are entangled.

3. I don't see, whether we still disagree or not and if so, about which detail.
1. All of this is correct, and I agree. Preparation alone does not entangle 1 & 4. When a successful swap operation is executed (one of the 4 Bell states), "you prepare a specific entangled state of photons 1 and 4 without a causal interaction with either of these photons." Couldn't have said it better myself.

2. This basically states the exact opposite of 1, so of course I disagree.

There is no "(post-)selection of specific subensembles for which photons 1 and 4 are entangled." No subensembles of 1 & 4 are ever entangled UNLESS the swap occurs. You know this because of monogamy of entanglement (and hopefully I don't need to convince you about that). If a successful the swap occurs, 1's monogamous entanglement with 2 ceases and its monogamous entanglement with 4 begins.

In sum: If you don't perform any swapping at all, there is no sub-ensemble of 1 & 4 pairs that are ever entangled. (And again, hopefully I don't need to convince you about that.)

Also: Your use of the word "causal" in this section is ambiguous. Since quantum nonlocality defies Einsteinian causality, it is not a valid descriptor for the swapping action. Time ordering of events within a full quantum context is not a factor in any case.

3. I split the sections up as requested.

Gold Member
2022 Award
I don't know, how else I can reformulate 2) to make clear what I mean, but I hope you agree that this experiment is described correctly by standard QED. Then there cannot be a causal connection between the measurement on 2+3 on 1 and/or 4. It also doesn't matter, whether the measurement on 2+3 occurs before or after the measurement on 1+4 nor if these measurement events are spacelike separated.

That's why I don't see a contradiction between 1) and 2) let alone any contradiction of this experiment with standard microcausal QED.

Nullstein