A Possibilities of Time-Independent Entangled Photons

[DrChinese]

[Moderator's note: Thread spin off due to topic change. Original thread spin-off point is here.]

You can have entangled photons that do not coexist in time. I would call those biphotons as well, because there is not any particular difference between spatial separation versus temporal separation when it comes to entanglement. Any thoughts?

 

[A. Neumaier]

You can have entangled photons that do not coexist in time. I would call those biphotons as well, because there is not any particular difference between spatial separation versus temporal separation when it comes to entanglement. Any thoughts?

How do you define entanglement of two photons not coexisting in time?
The Schrödinger equation assumes that complete states are given at fixed times. Without this, the terminology becomes ambiguous.

 

[PeterDonis]

there is not any particular difference between spatial separation versus temporal separation when it comes to entanglement

As @A. Neumaier points out, however, the math doesn't really reflect this. The Schrodinger equation is an equation for the wave function of the total system as a function of time. So any mathematical analysis using the Schrodinger equation will not be able to model the two photons you refer to, which are entangled in the sense of showing the appropriate correlations between their measurements, as entangled in the sense of having the appropriate wave function at a particular time--which is the sense the term "entangled" usually has.

For example, in a previous thread I gave a mathematical analysis of the entanglement swapping scenario for the case where the two photons that are entangled at the end of the experiment never coexist (the first is measured before the second is created by preparation). That analysis had to look different for that case than for the case where the two photons do coexist. The final predictions for correlations were the same, but the way to get there was different.

 

[DrChinese]

How do you define entanglement of two photons not coexisting in time?
The Schrödinger equation assumes that complete states are given at fixed times. Without this, the terminology becomes ambiguous.

Agreed, this situation has some difficulty

a. An entanglement swap creates a biphoton (entangled pair) that is "born" separated, usually in different places. But with some clever engineering, a swap can also give "birth" to a biphoton (1 & 4) whose components were created at different times. The trick is syncing laser pulses to create initial biphotons (1 & 2) and (3 & 4) that are in phase but separated in their time of creation (from the same PDC crystal). After the Bell State Measurement (BSM) of the component photons labeled 2 & 3*, we have biphoton (1 & 4).

b. The odd part is: the component photon 1 of the biphoton was already measured before the component photon 4 was created. Yet they demonstrate entanglement the same way any biphoton would after a swap is executed via BSM. In principle, the spatio-temporal separation between the components of an entangled pair of photons can be arbitrarily large in either time or space - or both.

Entanglement Between Photons that have Never Coexisted

So I define the entanglement of biphoton (1 & 4) the same mathematically as any biphoton, it's in one of 4 Bell states. Just prior to the time of the BSM on photons 2 & 3, the photon count is 3. And photon 4 is still entangled with photon 3, and not "yet" with photon 1 (which no longer exists). Not sure how the Schrödinger equation would need to modified to describe the full system evolution. @PeterDonis explains this better in post #53 above. *The 2 & 3 component photons must overlap at a beam splitter in an indistinguishable manner, so they are routed to make that occur. The 2 photon must be delayed precisely for that to happen.

 

[A. Neumaier]

with some clever engineering, a swap can also give "birth" to a biphoton (1 & 4) whose components were created at different times. The trick is syncing laser pulses to create initial biphotons (1 & 2) and (3 & 4) that are in phase but separated in their time of creation (from the same PDC crystal). After the Bell State Measurement (BSM) of the component photons labeled 2 & 3*, we have biphoton (1 & 4).

This means that the biphoton starts to exist only when the Bell State Measurement is completet. Then it is an ordinary fixed-time biphoton.

 

[PeterDonis]

This means that the biphoton starts to exist only when the Bell State Measurement is completet.

But for the case @DrChinese described where one photon is measured before the other is created, the BSM takes place after the first photon is measured (and the measurement destroys the photon). So saying that the biphoton "exists" when the BSM is completed is, at the very least, glossing over a very important detail for this experiment.

 

[PeterDonis]

I define the entanglement of biphoton (1 & 4) the same mathematically as any biphoton, it's in one of 4 Bell states.

But mathematically, this is not the case if we look at the state that appears in the Schrodinger equation; if we analyze the experiment using NRQM and the Schrodinger equation, this Bell state never exists. All we have is a prediction of final correlations between photons 1 & 4 that are the same as for that Bell state. But the actual state that appears in the Schrodinger Equation is never that state at any time during the experiment.

 

[DrChinese]

But mathematically, this is not the case if we look at the state that appears in the Schrodinger equation; if we analyze the experiment using NRQM and the Schrodinger equation, this Bell state never exists. All we have is a prediction of final correlations between photons 1 & 4 that are the same as for that Bell state. But the actual state that appears in the Schrodinger Equation is never that state at any time during the experiment.

Not so sure that I entirely agree, because every alternative ordering of the events in the setup yields the same outcomes. I would say the mathematical descriptions must be equivalent in some reasonable manner. I agree with you in that the Bell state in this particular case does not exist at a single point in time. But I don’t see that as an absolute requirement for a Bell state to be said to exist for (1 & 4). The cited paper says:

"In this work we demonstrate how the time at which quantum measurements are taken and their order, has no effect on the outcome of a quantum mechanical experiment, by entangling two photons that exist at separate times. ... When the two photons of time τ (photons 2 and 3) are projected onto any Bell state, the first and last photons (1 and 4) collapse also into the same [Bell] state and entanglement is swapped."

"The density matrix of the first and last photons was constructed, conditioned on the outcome of the projection of the two photons of time τ. If the projected photons were measured in the |φ+>[τ,τ] state, the first and last photons were entangled in the |φ+>[0,2τ] state (see Fig. 3a). Alternatively, if the projected photons were measured in the |φ−>[τ,τ] state, the first and last photons were entangled in the |φ−>[0,2τ] state (see Fig. 3b)."

Agreeing with both you and @A. Neumaier, I think the "normal" descriptions - mathematical or textual - have difficulty here. It is akin, frankly, to trying to describe entanglement swapping when the swap (BSM) is performed AFTER both photons 1 & 4 were already detected - i.e. the delayed choice scenario (which has also been performed in experiments). In that case, the Bell State for photons (1 & 4) is analogously just as real as when the BSM is executed prior to their detection. The swap casts (2 & 3) into a Bell state, and it casts (1 & 4) into a Bell state... regardless of order. I don't see where there are any published theoretical or experimental treatments saying otherwise.

But again, you can pick a preferred way to describe these situations if you see some significant difference between them. But I don't see anything significantly different because the same rules apply regardless of order, and the result is always identical. In that respect, somewhat akin to an interpretation.

 

[PeterDonis]

Not so sure that I entirely agree, because every alternative ordering of the events in the setup yields the same outcomes. I would say the mathematical descriptions must be equivalent in some reasonable manner.

They are equivalent in that they all give the same probabilities and correlations for the measurement results. That certainly seems to me to qualify as reasonable.

I don’t see that as an absolute requirement for a Bell state to be said to exist

This depends on one's opinion about how words like "exist" should be used, which is a matter of words, not math or physics. I think we agree on the math and physics (meaning the predictions). I personally would rather avoid words like "exist" altogether in cases like this, but of course many papers in the literature do not share that preference.

 

[A. Neumaier]

I don’t see that as an absolute requirement for a Bell state to be said to exist for (1 & 4).

In nonrelativistic QM, an absolute requirement to call something a pure state is that there is an associate wave function. This is not the case in your situation.

The cited paper says:

"In this work we demonstrate how the time at which quantum measurements are taken and their order, has no effect on the outcome of a quantum mechanical experiment, by entangling two photons that exist at separate times. ... When the two photons of time τ (photons 2 and 3) are projected onto any Bell state, the first and last photons (1 and 4) collapse also into the same [Bell] state and entanglement is swapped."

The cited paper talks nonsense. What they mean is that the same statistics is obtained, it is not even attempted to construct a Bell state to justify their statement.

But for the case @DrChinese described where one photon is measured before the other is created, the BSM takes place after the first photon is measured (and the measurement destroys the photon). So saying that the biphoton "exists" when the BSM is completed is, at the very least, glossing over a very important detail for this experiment.

Indeed. I hadn't looked at the details.

 

[DrChinese]

1. In nonrelativistic QM, an absolute requirement to call something a pure state is that there is an associate wave function. This is not the case in your situation

2. The cited paper talks nonsense. What they mean is that the same statistics is obtained, it is not even attempted to construct a Bell state to justify their statement.

1. And yet, the Bell state statistics appear. So there must be an associated wave function.

2. Agreed that the same statistics are obtained. But they did indicate the associated Bell states, and I quoted those in the post. We have the same issues when we discuss the delayed choice version of entanglement swapping: Photons (1 & 4) are measured and their Bell state presumably exists prior to the measurement casting them into that Bell state - by a later choice to execute the swap on (2 & 3).

Experimental delayed-choice entanglement swapping
Zeilinger et al (2012)
"When Victor performs a Bell-state measurement and photons 2&3 are found in the state |Φ−〉23, then photons 1&4 were in the entangled state |Φ−〉14, i.e. the entanglement was swapped. When Victor performs a separable-state measurement, projecting photons 2&3 on the mixture of |𝐻𝐻〉23 or |𝑉𝑉〉23, correlations between measurement results on photon pairs 1&2 and 3&4 show that these pairs were entangled in the states |Ψ−〉12 and |Ψ−〉34, i.e. the entanglement was not swapped."

Again, we're recognizing the difficulty of trying to assign consistent descriptions to measurement timing and outcomes when order is not a factor in QM or QFT. While we want to place events into a chronological order and describe evolution accordingly, there is a subjective element to this process. In one case we have entangled photons that never co-existed. And in another, we have entanglement of photons that no longer exist at all - by a measurement performed later. (It has of course been well established that violations of Bell inequalities require the existence of entanglement.)

And in fact all 4 photons were entangled (as biphoton components) at all times they existed. At what time(s), and by which measurement(s), did biphoton (1 & 4) come into existence? I am not sure that can be precisely pinned down except by assumption or definition. My goal would be to consider these swapping examples when defining quantum measurement, else they run the risk of becoming counterexamples to an otherwise consistent definition or interpretation.

 

[A. Neumaier]

1. And yet, the Bell state statistics appear. So there must be an associated wave function.

No. There are many examples in mathematics where the same statistis arises from very different situations. For example, the Born statistics of a qubit arises from unrelated classical models (Gleason's theorem does not hold for a qubit).

2. their Bell state presumably exists prior to the measurement

You and perhaps they presume it but based on no other evidence than the same statistics. To prove that a Bell state exists you must derive its wave function from a mathematical model of the experiment.

At what time(s), and by which measurement(s), did biphoton (1 & 4) come into existence?

Never, since there does not even exist a precise definition of being entangled at different times. Thus the term can only have a figurative meaning.

Only the Bell statistics comes into existence, but the latter is a classical object!

 

[PeterDonis]

the Bell state statistics appear. So there must be an associated wave function

This is simply wrong. I explicitly showed the math in a previous thread for the case where photon 1 is measured (and hence destroyed) before photon 4 is created. In that math there is never a wave function that includes both photons in any joint state, Bell state or otherwise. But the Bell state statistics appear.

 

[PeterDonis]

they did indicate the associated Bell states

But as far as I can tell, they do not do the kind of analysis I did in the previous thread, where they show explicit wave functions at different times. They just write down a single Bell state wave function and wave their hands and say that's how the statistics are obtained.

While we want to place events into a chronological order and describe evolution accordingly, there is a subjective element to this process.

I disagree. There is no subjectivity at all to an analysis that uses the Schrodinger equation to compute the wave function at different times. It's all just deterministic math.

 

[DrChinese]

I explicitly showed the math in a previous thread for the case where photon 1 is measured (and hence destroyed) before photon 4 is created. In that math there is never a wave function that includes both photons in any joint state, Bell state or otherwise. But the Bell state statistics appear.

That’s my point. Those statistics cannot appear without 1 & 4 entanglement, period. Those photons are not represented by an entangled wave function at any single point in time when they both exist. Ergo there is an assumption to be questioned.

In every single published paper on the subject I’ve read, such as the 2 I quoted, they agree the final 1&4 photon pair is in an entangled Bell State. We already established that such entanglement correlations could not occur without the swapping operation, as photon 1 cannot be maximally entangled to both 2 and 4. So either those authors and I have made an unwarranted assumption, or you have, or we both have.

You are certain of your proof of course, but a matching reference would always be nice. Because I say a Bell State is a Bell State, as real as real can be.

 

[PeterDonis]

Those statistics cannot appear without 1 & 4 entanglement, period.

I'm not going to keep arguing about this. I already showed the math in a previous thread. If you insist I can try to find it, but we discussed it at the time.

Those photons are not represented by an entangled wave function at any single point in time when they both exist.

Yes, exactly. That's why "1 & 4 entanglement" has to be interpreted very carefully for this case, so as not to require such a wave function. I am not claiming that such a wave function exists or that "entanglement" in the sense of the appropriate correlations being observed requires one.

In every single published paper on the subject I’ve read, such as the 2 I quoted, they agree the final 1&4 photon pair is in an entangled Bell State.

Then you should ask the authors how they account for this in the case under discussion, where there is no time at which both the 1 & 4 photons exist (and we can make sure that this is true in any frame by putting the photon 1 measurement in the past light cone of the photon 4 creation), so there is no time at which there is a photon 1 & 4 wave function at all, let alone one in the Bell state.

I am not going to try to speculate in any detail about how they would respond. However, generally speaking, I do not expect published papers, particularly advanced papers on complex topics where the paper is only treating one particular aspect and is relying on many other references (which is the case for the papers on this topic that have been referenced) to laboriously cover every detail or to maintain strict logical rigor and care in every statement. So my off the cuff guess would be that, if challenged, they would say that all they really meant was that photons 1 & 4 were measured to have the appropriate correlations for the Bell state, not that there was an actual Bell state wave function containing those photons at any time.

I say a Bell State is a Bell State, as real as real can be.

Then how do you account for the fact that, in the case under discussion, there is never any time where any such Bell state wave function exists? Wouldn't it be simpler just to adopt the solution I gave above, where "Bell state" refers to the measured correlations and does not imply any claim about a wave function at a particular time? And wouldn't you expect the authors of these papers, who certainly know more about the topic than you or I do, to do the same, rather than making a claim that is obviously false?

 

[A. Neumaier]

That’s my point. Those statistics cannot appear without 1 & 4 entanglement, period.

So you define entanglement in terms of the statistics, not as usual in terms of the decomposition of a wave function!?? This is very strange.

Or do you have a theorem that deduces your claim?

Those photons are not represented by an entangled wave function at any single point in time when they both exist. Ergo there is an assumption to be questioned.

The one to be questioned is your unproved assumption that statistics implies entanglement.

In every single published paper on the subject I’ve read, such as the 2 I quoted, they agree the final 1&4 photon pair is in an entangled Bell State.

Which argument do they offer? Just claiming something without proof is not a good argument. If the only argument is the resulting statistics, this proves nothing about entanglement unless there is a proof that only entanglement can create Bell statistics.

 

[PeterDonis]

So you define entanglement in terms of the statistics, not as usual in terms of the decomposition of a wave function!?? This is very strange.

It seems to me to be the only way to interpret the use of terms like "entanglement" and "Bell state" in the papers @DrChinese has referenced. Otherwise we would have to conclude that the authors were making claims which they must have known were false.

 

[DrChinese]

1. I'm not going to keep arguing about this. I already showed the math in a previous thread. If you insist I can try to find it, but we discussed it at the time.

2. Yes, exactly. That's why "1 & 4 entanglement" has to be interpreted very carefully for this case, so as not to require such a wave function. I am not claiming that such a wave function exists or that "entanglement" in the sense of the appropriate correlations being observed requires one.

3. Then you should ask the authors how they account for this in the case under discussion, where there is no time at which both the 1 & 4 photons exist (and we can make sure that this is true in any frame by putting the photon 1 measurement in the past light cone of the photon 4 creation), so there is no time at which there is a photon 1 & 4 wave function at all, let alone one in the Bell state.

4. I am not going to try to speculate in any detail about how they would respond. However, generally speaking, I do not expect published papers, particularly advanced papers on complex topics where the paper is only treating one particular aspect and is relying on many other references (which is the case for the papers on this topic that have been referenced) to laboriously cover every detail or to maintain strict logical rigor and care in every statement. So my off the cuff guess would be that, if challenged, they would say that all they really meant was that photons 1 & 4 were measured to have the appropriate correlations for the Bell state, not that there was an actual Bell state wave function containing those photons at any time.

5. Then how do you account for the fact that, in the case under discussion, there is never any time where any such Bell state wave function exists? Wouldn't it be simpler just to adopt the solution I gave above, where "Bell state" refers to the measured correlations and does not imply any claim about a wave function at a particular time? And wouldn't you expect the authors of these papers, who certainly know more about the topic than you or I do, to do the same, rather than making a claim that is obviously false?

1. I don't want to argue about this or anything. If you feel something is more than a friendly academic discussion, it's not worth pursuing further.

2. Agreed.

3. Agreed that there is no one "time" at which 1 & 4 are in a Bell State. There is also no one time you can say the entanglement between 1 and 2 ended, leaving 1 to be absorbed by a detector and have no further opportunity to influence or be influenced by the context of a measurement on 4 (which will be perfectly correlated to 1). Keep in mind that 4 need not co-exist nor ever exist in a common backward light cone with 1. As to what the authors of all these papers account for this, I think their statements on this are perfectly clear and unambiguous.

4. I'm a little surprised you don't give more weight to their wording, which I believe is chosen quite carefully (given the many papers that say much the same thing). But perhaps you are correct and they agree with you.
5. I would simply say that quantum mechanics does not require a biphoton to either exist within a locally evolved space, nor at a single point in time. The thing we call "quantum nonlocality" includes something called "nonlocality in time" as well. Some versions of this include what are called "Temporal Bell Inequalities" (TBI). The general theme in these papers is that there is no particular theoretical difference between spatially-separated entangled systems as compared to temporally separated entangled systems (or combinations of both).

Admittedly the literature on this is a smaller set than for spatial non-locality, primarily because the research on this is somewhat newer and the experimental setups are quite novel and complex. I did not need to seek out any references on this, because I already had a few saved...

TBI (2011): Violation of a temporal Bell inequality for single spins in solid by over 50 standard deviations
"Quantum non-locality has been experimentally investigated by testing different forms of Bell's inequality... Much less explored are temporal Bell inequalities, which are not subject to the locality assumption, but impose a constrain on the system's time-correlations. In this paper, we report on the experimental violation of a temporal Bell's inequality..."

Two-photon interference of temporally separated photons (2016)
"We present experimental demonstrations of two-photon interference involving temporally separated photons within two types of interferometers: a Mach-Zehnder interferometer and a polarization-based Michelson interferometer. The two-photon states are probabilistically prepared in a symmetrically superposed state within the two interferometer arms by introducing a large time delay between two input photons; this state is composed of two temporally separated photons, which are in two different or the same spatial modes. We then observe two-photon interference fringes involving both the Hong-Ou-Mandel interference effect and the interference of path-entangled two-photon states simultaneously in a single interferometric setup."

Probing the Non-Classicality of Temporal Correlations (2017)
Here we aim to provide a fair comparison of classical and quantum models of temporal correlations on a single particle, as well as timelike-separated correlations on multiple particles. ... This provides a clearer picture of the role of quantum correlations in timelike separated scenarios..."

 

[PeterDonis]

Agreed that there is no one "time" at which 1 & 4 are in a Bell State.

Good. This is the primary point I have been making.

I'm a little surprised you don't give more weight to their wording

I'm a little surprised that, given your agreement on the point above, you don't draw the obvious conclusion that the authors must also know this, and therefore they would not intend to say anything that would contradict it. So any wording they choose would have to be interpreted to not imply the obviously false claim that there is a Bell state at any one time.

The thing we call "quantum nonlocality" includes something called "nonlocality in time" as well.

In terms of Bell inequality violations, this is obviously true, and the experiments we have been discussing are excellent illustrations of the point.

However, given the point we agree on above, this "quantum nonlocality" obviously cannot depend on a Bell state of photons 1 & 4 existing at any time. So either it doesn't require such a Bell state at all, or somebody needs to develop a different mathematical treatment of these experiments that doesn't use the Schrodinger equation and its time evolution, but instead makes use of the Bell states in a different way that shows how they come into play when the photons never coexist.

 

[DrChinese]

1. So you define entanglement in terms of the statistics, not as usual in terms of the decomposition of a wave function!?? This is very strange.

Or do you have a theorem that deduces your claim?

The one to be questioned is your unproved assumption that statistics implies entanglement.

2. Which argument do they offer? Just claiming something without proof is not a good argument. If the only argument is the resulting statistics, this proves nothing about entanglement unless there is a proof that only entanglement can create Bell statistics.

1. It is generally agreed that violation of Bell/CHSH inequalities or similar is sufficient to demonstrate entanglement. In a N=2 photon state, that would be one of the 4 Bell main Bell states. It is also standard to provide such statistical evidence whenever a paper claims to have created entanglement. Normally, such papers do not start with any kind of Schrödinger equation, although I am sure there are exceptions.

2. There are all kinds of papers that demonstrate that classical (local realistic) systems cannot violate those inequalities, and that violation is itself evidence of entanglement. First of all, that is actually included somewhere in most experimental demonstrations of entanglement. Although I can't find a general purpose proof at the tips of my fingers, I would be glad to provide references of same.

 

[DrChinese]

I'm a little surprised that, given your agreement on the point above, you don't draw the obvious conclusion that the authors must also know this, and therefore they would not intend to say anything that would contradict it. So any wording they choose would have to be interpreted to not imply the obviously false claim that there is a Bell state at any one time. ... So either it doesn't require such a Bell state at all, or somebody needs to develop a different mathematical treatment of these experiments that doesn't use the Schrodinger equation and its time evolution, but instead makes use of the Bell states in a different way that shows how they come into play when the photons never coexist.

The ψ+ Bell polarization state HV>_ab + VH>_ab (for example) for a system of 2 quantum particles (a & b) says it all. Entangled photons a and b (composing a biphoton, Fock N=2) can be anywhere, at anytime. I.e. the biphoton can have both spatial and temporal extent, and there is no particular restriction on one versus the other in standard QM. The pair can also be entangled into a non-separable state before or after they are created, no restriction there either. They don't need to be from a common source, co-exist in a common backward light cone, or in fact even need to be of the same particle type.

I don't think there is anything in the above paragraph that is not generally accepted at this time. That there are unusual time ordering structures in quantum entanglement was deduced - from theory alone - at least by 2000 (Peres). Not sure that can be explained by the Schrödinger equation any better than the earlier citation*.

But how do biphotons in time get to such a state? I think that's the question where we see things differently. You are asking for the Schrödinger equation that leads to this Bell state. I can't answer that properly, I guess because the standard form of the Schrödinger equation only contains a single time parameter (but can accommodate 3N space parameters). This places time on a different footing than space. But this issue has been recognized by a number of authors. Here are some recent papers that touch on exactly that point, not that any of this will necessary satisfy you regarding Bell States across time.

Temporal teleportation with pseudo-density operators: how dynamics emerges from temporal entanglement
Marletto, Vedral, Virzì, Avella, Piacentini, Gramegna, Degiovanni, Genovese (2021)
"Pseudo-density operators (PDOs) were introduced in [1] in order to express quantum spatial and temporal correlations on an equal footing. In usual quantum theory, quantum states, represented as density matrices, are given at a fixed time and then evolved in time through some completely positive (CP) map. This is at odds with relativity, where the line of simultaneity is observer dependent... The PDO formulation seeks to rectify this by representing statistics from events with a unique mathematical object, the pseudo density operator, irrespective of whether the events are space-like, time-like or light-like."*BTW: All 4 of the photons (for each trial) in this experiment are created in the same PDC crystal. That places them in a common light cone. But I could imagine a similar experiment in which photons 1 & 2 are created spacelike separated from 3 & 4, while also insuring 1 & 4 never co-existed. Of course, that would not change the expected outcome.

 

[PeterDonis]

The ψ+ Bell polarization state HV>_ab + VH>_ab (for example) for a system of 2 quantum particles (a & b) says it all.

Is there an explicit mathematical analysis in any of the papers you have referenced that shows how this works for the case where the two photons never coexist?

 

[DrChinese]

Is there an explicit mathematical analysis in any of the papers you have referenced that shows how this works for the case where the two photons never coexist?

See formula (2) in the original reference, which shows the explicit 4 photon state prior to the Bell State Measurement. They describe: "In order to generate consecutive photon pairs at well defined times, a pulsed laser is used to pump a single PDC polarization entangled photon source [4]. It is a probabilistic source, and thus there is a probability that two pairs will be created, each pair from one of two consecutive pulses, separated by the laser period time τ. The four-photon state is (2)"

Then, they present the resulting state as formula (3), which contains all 4 possible Bell states (one of which I used as an example in post #72). They describe: "In order to project the second photon of the first pair and the first photon of the second pair onto a Bell state, the former is delayed by τ in a delay line. The same delay is also applied to the second photon of the second pair and the resulting state can be reordered and written as (3)"

I guess I just don't get what's ambiguous or not sufficiently spelled out here. They said:

"In this work we demonstrate how the time at which quantum measurements are taken and their order, has no effect on the outcome of a quantum mechanical experiment, by entangling two photons that exist at separate times..."

They then show the theory, the math, the experiment and results. What's in question? I already said, maybe they agree with you. I guess it's just me reading something into their paper that they didn't intend when they presented their (3). I am not trying to be argumentative, but you have simply rejected every quote and reference I have supplied, and not provided a single quote or reference to the contrary (other than of course your own words).

I may as well bow out, if what I referenced in this post doesn't do the trick. This thread is about quantum measurements, and I believe I have posted the suitable quotes/references on the topic.

 

[PeterDonis]

See formula (2) in the original reference,

I'm not looking for a formula that has a Bell State in it; of course I already know that if you assume that that formula is the correct one, it predicts the correlations. All of the papers you have referenced appear to do the same thing: they assume that that Bell State is the correct one, with no supporting argument. In an experimental paper, that might be fine, if there is a theoretical argument already existing in the literature that explains why the assumption that that Bell State is the correct one is justified.

What I'm looking for is that theoretical argument: a theoretical analysis that explains, starting from first principles (i.e., either using the Schrodinger Equation, or using some other fundamental equation and explaining why you're using that one instead of the Schrodinger Equation, in a regime where, as you yourself have noted, relativistic effects are negligible and so NRQM should be sufficient), why that Bell State is relevant, even though the photons never coexist and so there is never a wave function at any time that includes that Bell State. That is what I have not found in any references that have so far been given.

 

[PeterDonis]

They then show the theory

No, they don't. They assume the theory. They assume that Bell States are the correct ones even though the photons never coexist and there is never a wave function at any time that includes them. They never justify that assumption with any theoretical argument.

 

[Morbert]

How do you define entanglement of two photons not coexisting in time?
The Schrödinger equation assumes that complete states are given at fixed times. Without this, the terminology becomes ambiguous.

The non-coexistence is accommodated for by the use of temporal modes (in this case, $0$, $\tau$, and $2\tau$).

A skim of literature shows temporal modes are often used in photonics. They are presumably uncontroversial given the linear nature of the Schroedinger equation. This paper gives a general form (equation 1). In this paper temporal modes are even used to respresent a single-photon state as a photon-number Bell state.

 

[PeterDonis]

They are presumably uncontroversial given the linear nature of the Schroedinger equation.

I'm not sure what you mean by this. You appear to be saying that if I have two solutions of the Schrodinger equation, $\psi(t)$ and $\phi(t + T)$, where $T$ is some constant time offset, I can superpose them to get another solution. But how would that work? What time is the resulting solution a function of?

In the papers you referenced I don't see any sign of a Schrodinger equation or any analysis that would answer that question. I'll take another look to see if I've missed something.

 

[A. Neumaier]

1. It is generally agreed that violation of Bell/CHSH inequalities or similar is sufficient to demonstrate entanglement.

Not with the standard definition of entanglement. It is only sufficient to demonstrate the violation of local hidden variable assumptions.

In a N=2 photon state, that would be one of the 4 Bell main Bell states.

yes, but your claim of the converse is mathematically meaningless.

2. There are all kinds of papers that demonstrate that classical (local realistic) systems cannot violate those inequalities, and that violation is itself evidence of entanglement.

But quantum systems are well-known to be able to violate it, and the experiments were done with quantum systems, not with classical systems!

See formula (2) in the original reference, which shows the explicit 4 photon state prior to the Bell State Measurement.

So we have an entangled 4-photon state in the standard meaning, a tetraphoton, but not an entangled 2-photon state, no biphoton.

The paper just shows that certain tetraphotons measured at different times produce the same correlations as a biphoton in a Bell state, and tries to sell it as ''temporal entanglement''.

by entangling two photons that exist at separate times..."

But they don't substantiate this claim, and indeed, the claim is meaningless.

The non-coexistence is accommodated for by the use of temporal modes (in this case, $0$, $\tau$, and $2\tau$).

A skim of literature shows temporal modes are often used in photonics. They are presumably uncontroversial given the linear nature of the Schroedinger equation. This paper gives a general form (equation 1).

But these temporal modes were not used in the experiments cited by Dr. Chinese.

 

[PeterDonis]

temporal modes are often used in photonics

As described in the papers you referenced, these modes are created by carefully chosen operations on a single source (such as a two-level atom in an excited state). But in the experiments we have been discussing, photons 1 & 4, the ones that never coexist, are created from different sources, independently. So the "temporal mode" treatment as described in the papers is not relevant for those experiments.

 

[Morbert]

But these temporal modes were not used in the experiments cited by Dr. Chinese.

The paper @DrChinese cites, cites this one in turn. From the paper:

The four spatial modes of previous schemes are replaced by two spatial modes (1 and 2 after the projecting PBS, 1’ and 2’ before it) and three temporal modes (0, τ , and 2τ)
$$|\Psi^{(4)}_{GHZ}\rangle = \frac{1}{\sqrt{2}}(|h_{1'}^0h_2^\tau h_1^\tau h_{2'}^{2\tau}\rangle + |v_{1'}^0v_2^\tau v_1^\tau v_{2'}^{2\tau}\rangle)$$

 

[PeterDonis]

The paper @DrChinese cites, cites this one in turn. From the paper:

This paper is still creating all the photons from a single source (a single pump pulse is used for all three of the BBO crystals). As far as I can tell, that is a necessary requirement for creating entanglement between different temporal modes in the manner described. But, again, in the experiments we have been discussing, photons 1 & 4, which never coexist, are created by different, independent sources.

 

[Morbert]

This paper is still creating all the photons from a single source (a single pump pulse is used for all three of the BBO crystals). As far as I can tell, that is a necessary requirement for creating entanglement between different temporal modes in the manner described. But, again, in the experiments we have been discussing, photons 1 & 4, which never coexist, are created by different, independent sources.

The single pump pulse is shown in fig 1a and is the "resource intensive" scheme. The resource efficient scheme involving the temporal modes is fig 1b which iiuc corresponds to the apparatus in @DrChinese's references

 

[PeterDonis]

The resource efficient scheme involving the temporal modes is fig 1b which iiuc corresponds to the apparatus in @DrChinese's references

No, it doesn't. If you look at the states that the paper says are produced, they are all single entangled states, just containing more and more photons (depending either on how many BBOs you use, in the fig 1a setup, or how many pump pulses you put through the single BBO, in the fig 1b setup). None of them are a product state of two independently entangled photon pairs, which is the relevant type of state for the experiments we have been discussing.

 

[Morbert]

No, it doesn't. If you look at the states that the paper says are produced, they are all single entangled states, just containing more and more photons (depending either on how many BBOs you use, in the fig 1a setup, or how many pump pulses you put through the single BBO, in the fig 1b setup). None of them are a product state of two independently entangled photon pairs, which is the relevant type of state for the experiments we have been discussing.

Fig 1b from that paper

Fig 2 from @DrChinese 's paper

Also, @DrChinese 's paper cites that paper

We realized this scenario with the experimental setup presented in Fig. 2 [23]

 

[PeterDonis]

Fig 2 from @DrChinese 's paper

@DrChinese has referenced quite a few papers. Which one?

 

[Morbert]

@DrChinese has referenced quite a few papers. Which one?

https://arxiv.org/pdf/1209.4191.pdf

 

[PeterDonis]

https://arxiv.org/pdf/1209.4191.pdf

Ok. In that paper the state that is produced is, as I described before, a product state of two entangled photon pairs. (In equation 3 of the paper the state before the projection operation on photons 2 & 3 is given, but the following text makes it clear that after the projection operation, you simply have one of the four Bell states given in equation 3, and each of those is as I described.)

But in the other paper, the one you referenced, the state after the projection operation on the photons at time $\tau$ (which correspond to photons 2 & 3) is a four-photon entangled state (the GHZ state). And the process that paper describes, as I said before, is set up to produce a single entangled state containing any even number of photons (just do the BBO operation more times).

I don't see any explanation in either paper of how to produce a product state of two separate entangled photon pairs using the given setup.

 

[DrChinese]

I. So we have an entangled 4-photon state in the standard meaning, a tetraphoton, but not an entangled 2-photon state, no biphoton. The paper just shows that certain tetraphotons measured at different times produce the same correlations as a biphoton in a Bell state, and tries to sell it as ''temporal entanglement''.

II. But they don't substantiate this claim, and indeed, the claim is meaningless.

I. Sorry, perhaps either I (most likely) or the authors of the paper were not clear. There is an N=4 photon state, but there is no tetraphoton as you mention. The 4 photon state(s) in the generalized entanglement swapping regime starts as a Product state of 2 biphotons*, and end also as a Product state of 2 biphotons (although your method of photon counting produces a different result in some specific cases):

a. Initial Product state: |ψ-₁₂> |ψ-₃₄>
b. Execute Swap: Photons 2 & 3 are allowed to overlap such that their identities become indistinguishable. You could also say that biphotons (1 & 2) and (3 & 4) are allowed to overlap.
c. Final Product state: |φ+/-₁₄> |φ+/-₂₃>II. The paper follows standard QM predictive methods, using quantum theory (that measurement order in entanglement swaps is irrelevant to the observed statistics) that was published at least 25 years ago. Their specific hypothesis was that measuring Photon 1 before the swap (BSM) would lead to predicted violations of Bell-type inequalities just the same as if they had measured Photon 1 after the swap. That hypothesis was substantiated, yet another confirmation of quantum mechanics - and to nobody's surprise.

It was published in Physical Review Letters 110, 210403; 22 May 2013. Obviously it passed peer review. There have been no published works that contradict or otherwise criticize their work (that I am aware of). Perhaps you are aware of published work that contradicts this result. So I question your evaluation of their results as "meaningless". I would say this is an important paper that deserves proper respect and consideration, even if you disagree with elements of it - which you are certainly free to do. I recognize your expertise in the math around the Schrödinger equation from your extensive published works. But I really think you should take a second look here.

---

To summarize the line of research on the nature of measurements in entanglement swapping, which was the reason I posted in this thread to begin with:

At this time, there has been experimental verification (references below from 2002 to 2012) of the following:

1. Basic Entanglement Swap with Bell State Measurement (photons 2 & 3) performed prior to observation of entanglement between (photons 1 & 4). See Zeilinger et al, etc.
2. Entanglement Swap with fully independent sources with Bell State Measurement (photons 2 & 3) performed prior to observation of entanglement between (photons 1 & 4). See Zeilinger et al, etc.
3. Remote (photons 1 & 4 outside each others' light cone) Entanglement Swap with Bell State Measurement (photons 2 & 3) performed prior to observation of entanglement between (photons 1 & 4). See Gisin et al, etc.
4. Delayed Choice Entanglement Swap with Bell State Measurement (photons 2 & 3) performed subsequent to observation of entanglement between (photons 1 & 4). See Ma et al, etc.
5. Temporal Entanglement Swap with Bell State Measurement (photons 2 & 3) performed prior to observation of photon 4 but subsequent to observation of photon 4. See Eisenberg et al.
6. Failure of Temporal Entanglement Swap with Bell State Measurement (photons 2 & 3) performed without indistinguishability** of photons 2 & 3, prior to observation of photon 4 but subsequent to observation of photon 4. See Eisenberg et al.

Hopefully, the pattern is clear: the same results are obtained, regardless of measurement order. The only requirement that must be met is that the Bell State Measurement of photons (2 & 3) must result in indistinguishability between 2 & 3. When that is not accomplished, there is no swap and there is no Bell State consisting of 1 & 4. If your viewpoint (or interpretation or mental picture or whatever) does not accommodate the 6 experimental facts listed, you might want to review them to understand why not.

I am not aware of any aspects of relativistic QFT that would add to this discussion or lead to any predictions contrary to garden variety QM. But I would certainly be interested in viewpoints from anyone following this.*Each biphoton contains photons that are maximally entangled, but there is no entanglement (or correlation) whatsoever with the other biphoton.
** This outcome is predicted by theory.

 

[mattt]

The paper just shows that certain tetraphotons measured at different times produce the same correlations as a biphoton in a Bell state, and tries to sell it as ''temporal entanglement''.

This. In a nutshell.

 

[DrChinese]

This. In a nutshell.

Sorry, no. See my post #91, which explains why not. Arnold misread (or misinterpreted) something amongst the many posts. It would be easy for anyone to do.

Entanglement swapping protocols don't use 4-photon GHZ entanglement (what was referred to as "tetraphotons" in some posts). Obviously, with 4-photon GHZ entanglement there is no entanglement to swap within its components, as they are all already entangled with each other.

Instead, the Entanglement Swapping regime uses a Product state of biphotons (entangled photon pairs). These Product states can be even be daisy-chained together: 2 pairs=2 biphotons, 3 pairs=3 biphotons, etc. And any of the related measurements can be performed in any order the experimenter is able to achieve.

 

[mattt]

What I mean is that in the case where biphoton1,2 is created and photon1 is measured (and destroyed) before (for all possible reference frames) biphoton3,4 is created, there can be no biphoton1,4 Bell State (or any biphoton1,4 state).

I must conclude that you (and possibly the authors of that paper) are using these English words with a different meaning than the usual mathematical meaning.

 

[PeterDonis]

Entanglement swapping protocols don't use 4-photon GHZ entanglement

Yes, I agree. What I'm still not sure about is how the state they do use, which is, as you say, a product state of two biphotons (the entanglement swap just changes which photons are in each biphoton), is produced by the apparatus that is described. The paper that describes the theoretical background, at least in a little bit of detail, only talks about how the GHZ state is produced. The paper that you referenced that describes the entanglement swapping, and uses the two-biphoton states (with similar temporal markings to the 4-photon GHZ states in the other paper), does not describe how those two-biphoton states are produced by the apparatus, and it's not clear to me from the description in the first paper (the GHZ state paper) how pair-of-biphoton states could be produced. Possibly there are other papers (which would hopefully be somewhere in the references in those two papers) that go into more detail about the latter; but I haven't seen any yet.

 

[DrChinese]

Yes, I agree. What I'm still not sure about is how the state they do use, which is, as you say, a product state of two biphotons (the entanglement swap just changes which photons are in each biphoton), is produced by the apparatus that is described. The paper that describes the theoretical background, at least in a little bit of detail, only talks about how the GHZ state is produced. The paper that you referenced that describes the entanglement swapping, and uses the two-biphoton states (with similar temporal markings to the 4-photon GHZ states in the other paper), does not describe how those two-biphoton states are produced by the apparatus, and it's not clear to me from the description in the first paper (the GHZ state paper) how pair-of-biphoton states could be produced. Possibly there are other papers (which would hopefully be somewhere in the references in those two papers) that go into more detail about the latter; but I haven't seen any yet.

The 2 biphoton (2 photon entangled system, i.e. an entangled photon pair) Product state (N=2x2) used for the initial portion of the entanglement swapping can be produced through a number of interesting and innovative methods. Here are a few, including reference links:

a. Here they pass a laser pulse forward and backward (from a mirror) through a PDC crystal. Some pulses yield a biphoton in the forward direction or the backward direction, but usually not both. About 1 in 15 of those yield biphotons coming out both directions of the PDC crystal simultaneously. The biphoton pairs - 4 total photons labeled (0 & 1) and (2 & 3) - are in a Product State, as they are actually produced independently.

b. Here they synchronize a slave laser to a master laser.

c. Here they sync two source lasers to a common signal. Each laser pumps a PDC crystal to create a biphoton. The sources are 12+ km apart. Note that this setup allows the order of measurement to be selected at will, as well as the spatial separation. The authors so noted: "...the configuration of our experiment allows the space-like separation between any two measurements of those performed in the three nodes, and various of time-space relation can be achieved by combining both coiled optical fiber and deployed optical fiber."

d. Here they use the same pump laser in line serially to fire 2 PDC crystals.

e. Here they pump successive pulses into a single PDC crystal to create 2 biphotons, one of which is routed a little differently than the other.
x

Note: In all of these configurations, the creation of 2 entangled pairs suitable for the experiment occurs randomly, perhaps on the order of 10-100 per second. Only 4 fold coincidences are considered for data collection. Further, there is the requirement that the photon overlap in the Bell State Measurement (BSM) portion of the apparatus be narrow enough so that the outputs of the beam splitter are indistinguishable in all degrees of freedom. I.e. the source should not be able to be determined. Also, only 1 or 2 of the 4 possible Bell states can be determined using current optical technology. This does not affect the fidelity or validity of the results in any way.

 

[PeterDonis]

The 2 biphoton (2 photon entangled system, i.e. an entangled photon pair) Product state (N=2x2) used for the initial portion of the entanglement swapping can be produced through a number of interesting and innovative methods.

I'm not asking about how biphoton states are produced experimentally. I already know there are multiple ways of doing that.

I am asking about how products of two two-photon entangled Bell states are produced mathematically using a theory that, as far as I can tell, only explains how the apparatus described in both the entanglement swapping paper you referenced and the "temporal mode" paper that @Morbert referenced (which describes the theory, though not in very great detail) can produce 4-photon GHZ states.

 

[DrChinese]

1. I'm not asking about how biphoton states are produced experimentally. I already know there are multiple ways of doing that.

2I am asking about how products of two two-photon entangled Bell states are produced mathematically using a theory that, as far as I can tell, only explains how the apparatus described in both the entanglement swapping paper you referenced and the "temporal mode" paper that @Morbert referenced (which describes the theory, though not in very great detail) can produce 4-photon GHZ states.

1. Well, you said (so I answered): "The paper that you referenced that describes the entanglement swapping, and uses the two-biphoton states (with similar temporal markings to the 4-photon GHZ states in the other paper), does not describe how those two-biphoton states are produced by the apparatus..." So I provided 5 examples with references. Note that none of these produce 4-photon GHZ states. I think that you and several others like the theory presented in a reference @Morbert provided in post #83. But that is a different state than the 2 biphotons in a Product State - as I think you acknowledge.2. Here is what I believe is the mathematical representation you seek, which is presented verbatim in italics and in context without interruption from several papers:

a. From HERE, circa 2002 (provided to show that my primary reference is using well-established theory and experimental confirmation):

Initially, the system is composed of two independent entangled states and can be written in the following way:

|Ψ>_total = |Ψ−>₀₁ ⊗ |Ψ−>₂₃

Including equation (1) in (2) and rearranging the resulting terms by expressing photon 1 and photon 2 in the basis of Bell states leads to:

|Ψ>_Total = 1/2 [
|Ψ+>₀₃ |Ψ+>₁₂
-|Ψ−>₀₃ |Ψ−>₁₂
-|φ+>₀₃ |φ+>₁₂
+|φ−>₀₃ |φ−>₁₂ ].

Alice subjects photons 1 and 2 to a measurement in a Bell-state analyzer (BSA) [also called BSM in other papers], and if she finds them in the state |Ψ−>₁₂, then photons 0 and 3 measured by Bob, will be in the entangled state |Ψ−>₀₃. If Alice observes any of the other Bell-states for photons 1 and 2, photons 0 and 3 will also be perfectly entangled correspondingly.b. From HERE (my primary reference, and note that the starting and ending points are exactly the same as the one above, although labeled differently):

In order to generate consecutive photon pairs at well defined times, a pulsed laser is used to pump a single PDC polarization entangled photon source [4]. It is a probabilistic source, and thus there is a probability that two pairs will be created, each pair from one of two consecutive pulses, separated by the laser period time τ. The four-photon state is

[|Ψ>_total] = |Ψ−>^{0, 0}_ab ⊗ |Ψ−>^{τ, τ}_ab

where the subscripts are the spatial mode labels and the superscripts are the time labels of the photons. In order to project the second photon of the first pair and the first photon of the second pair onto a Bell state, the former is delayed by τ in a delay line. The same delay is also applied to the second photon of the second pair and the resulting state can be reordered and written as

|Ψ−>^{0, τ}_ab ⊗ |Ψ−>^{τ, 2τ}_ab = 1/2(
|Ψ+>^{0, 2τ}_ab |Ψ+>^{τ, τ}_ab
-|Ψ−>^{0, 2τ}_ab |Ψ−>^{τ, τ}_ab
-|φ+>^{0, 2τ}_ab |φ+>^{τ, τ}_ab
+|φ−>^{0, 2τ}_ab |φ−>^{τ, τ}_ab )

When the two photons of time τ (photons 2 and 3) are projected onto any Bell state, the first and last photons (1 and 4) collapse also into the same state and entanglement is swapped. The first and last photons, that did not share between them any correlations, become entangled.

---

Please note the nearly identical language in both of these experiments, which I would refer to as so common in experimental papers as to be unquestioned at this point. The point of difference - which is of course why the b. paper was published in PRL in the first place - is that time superscripts are added to the label to show that there is "Entanglement between photons that never coexisted" (the title of the paper).

If these unambiguous and well presented theoretical and experimental research developments, published in PRL and culminating in a Nobel for one of the authors, is not sufficient to convince readers: I'm just not sure what the point of this thread is. Here, we investigate some of the most detail elements of measurements in QFT (title of the thread) and change of state via entanglement swapping. When does measurement occur? Where does measurement occur? What causes change of state? When and where does that occur?

 

[PeterDonis]

I provided 5 examples with references. Note that none of these produce 4-photon GHZ states.

You provided 5 references that describe experiments. That's not what I'm asking for.

Here is what I believe is the mathematical representation you seek

I'll take a look.

 

[DrChinese]

1. What I mean is that in the case where biphoton1,2 is created and photon1 is measured (and destroyed) before (for all possible reference frames) biphoton3,4 is created, there can be no biphoton1,4 Bell State (or any biphoton1,4 state).

2. I must conclude that you (and possibly the authors of that paper) are using these English words with a different meaning than the usual mathematical meaning.

1. They say exactly the opposite, using nearly identical wording as you: "When the two photons of time τ (photons 2 and 3) are projected onto any Bell state, the first and last photons (1 and 4) collapse also into the same state and entanglement is swapped. The first and last photons, that did not share between them any correlations, become entangled."

2. If anyone is mangling the English, it can't be me - since I am quoting. So it must be the authors. This is a peer-reviewed paper from an impeccable source.

As to the usual mathematical meaning, I present that in detail in my post #98 above. If you see anything ambiguous or out of the ordinary in terms of presentation, please present your preferred alternative.

 

[PeterDonis]

I'll take a look.

Ok, I took a look. Here's the problem. From the 2002 paper, p. 5:

"A seemingly paradoxical situation arises — as suggested by Peres [4] — when Alice’s Bell-state analysis is delayed long after Bob’s measurements. This seems paradoxical, because Alice’s measurement projects photons 0 and 3 into an entangled state after they have been measured. Nevertheless, quantum mechanics predicts the same correlations."

My question is: how does QM predict the same correlations when the photons never coexist? The way I did it when I did the math using the Schrodinger equation in a previous thread, there are no Bell states with photons 1 & 4. Ever. Anywhere. So that analysis, while it certainly supports the claim that QM predicts the same correlations, does not support the claim that it does so by means of Bell states with photons 1 & 4--because there are no such states anywhere in the analysis. (And you have already agreed that, if photons 1 & 4 never coexist, there is no time at which such a Bell state exists.)

The 2002 paper does not give any mathematical analysis to back up the claim I quoted above. So I have no way of knowing why they think that claim is true. Is it just because the experiments show the same correlations? Or is it because someone has actually done a mathematical analysis, not the same as the one I did, that does involve a Bell state with photons 1 & 4 even though they never coexist? I don't mean just writing down such a Bell state; I mean showing how such a Bell state can arise from the dynamics even though photons 1 & 4 never coexist.

The other paper, from 2012, does a very short mathematical operation to obtain such a Bell state: it takes the state in equation (2) and rearranges it, applying a time delay to photons 2 & 4 and algebraically refactoring, to obtain equation (3), which is an entangled superposition of the 4 possible "double biphoton" states for photons 1 & 4, and photons 2 & 3. Each photon 1 & 4 state in that superposition is a Bell State. (In actual experiments, as you have said, only 1 or at most 2 of these can actually be distinguished after all measurements are made. But that's not important for what we're discussing here.)

However, this still doesn't help, because in equation (2), the paper is already assuming that it makes sense to write down a tensor product state between biphotons at different times. But this assumption is never justified by any first principles analysis. This paper appears to be depending on the other "temporal mode" paper that @Morbert referenced, which at least tries to construct a Hilbert space for such states. But there is still no dynamics; there is nothing corresponding to the Schrodinger equation or anything like it.

Perhaps the underlying assumption here is that standard NRQM, where you use the Schrodinger equation and you have a state of the system that evolves in time, is simply inapplicable to these types of experiments. But if that is the case, I would certainly like to see somebody justify that assumption and explain what should be put in its place.

 

[DrChinese]

The paper @DrChinese cites, cites this one in turn. From the paper...

Yes @Morbert, thanks for bringing up this reference. It is something of a companion to the reference I supplied, being from the same team. And is has additional detail as well.

So it turns out that the same basic apparatus can be used to prepare a variety of 4 photon states with temporal separation. Either GHZ or biphoton pairs. And various entangled states of 6 or more photons as well!

@PeterDonis: I had mentioned in post #96.e. that actually, the experiment I cite uses a single PDC crystal to produce the time separated biphotons. That is different from what might be expected, and it is possible something I said previously led you astray on this point (if you were astray).