A Possibilities of Time-Independent Entangled Photons

[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).

 

[DrChinese]

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."

a. 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.)

b. 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.

c. 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.)

d. 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.

I see your dilemma and acknowledge your points. Unfortunately I am not the person to bridge the published papers to your type of analysis. Without in any way taking away from your approach, I will simply point out your methodology/approach does not appear in the hundreds of papers I have read on the subject. Instead, the presentation is nearly identical in papers written in the 1999 to present time frame.

a. The answer is that QM is not only quantum nonlocal (our agreed upon terminology), it is also quantum non-spatiotemporal. Now before anyone objects to my usage of the term as speculative or overreaching, let me show you how and why this has actually been evident for many decades. Further, it really shouldn't be a surprise to anyone.

It should be clear from my post #91 that there is actually a pattern at play in the 6 experiments I referenced there. Each is a variant of separation in spacetime and order of measurement. Specifically, please look at papers 1. and 4, which I am reposting.

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. n/a
3. n/a
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.

Reference 1 includes a Delayed Choice version of Entanglement Swapping, as does the 4 reference. In these versions of swapping, the Bell State Measurement (BSM/BSA) occurs after both the other 2 photons have ceased to exist. In terms of what we've been discussing on photons that never co-exist: Photons 1 and 4 are entangled after photon 1 ceased to exist. In the Delayed Choice permutations: Photons 1 and 4 are entangled after both photons 1 and 4 cease to exist, but they did co-exist at some time. This has actually been confirmed in experiments going back to 2002, per the references.

So if your argument is that photon 1 cannot be placed in a Bell state (i.e. become an entangled component of a biphoton) after it ceases to exist, you're also gonna need to disavow all these referenced Delayed Choice experiments - as well as of all kinds of other Delayed Choice experiments as well. Whew, that's a lot of published material to deny! Check out this 28 page 2014 summary of "Delayed-choice gedanken experiments and their realizations" by top experimentalists Ma, Kofler and Zeilinger. The basic theme: The ordering of measurements (including delayed choice) has no affect on the quantum expectation value.

All of this goes back to Wheeler, circa 1978 and after, with speculations on whether choices made mid-experiment affect outcomes. The first application of delayed choice to entanglement swapping that I am aware of is "Delayed choice for entanglement swapping" (Peres, 1999). His analysis of Bell states looks almost identical to those I have presented in various posts. He shows how the initial Product state of 2 entangled pairs becomes of the 4 Bell states. He famously said: "In summary, there is nothing paradoxical in the experiments outlined above. However, one has to clearly understand quantum mechanics and to firmly believe in its correctness to see that there is no paradox."

b. I think post #98 in its entirety shows plenty of mathematical analysis, all saying the same thing. It is normal and natural for authors to reference other works, without an expectation that every individual formula presented need be justified. I scanned backwards in the literature and found a variety of interrelated papers in the 1990's by Gisin, Peres and stuff like this.

I have never seen you demand a fraction of what I have presented so far in any thread, much less a thread in Quantum Foundations. You have asked for citations, and received them - gold ones, by any standard. All I can say at this point is: Asked and answered. Again, if you or anyone have a counter-reference unrelated to a specific interpretation of QM, then that would be welcome.

c. Agreed.

d. I think any survey of the literature is going to show about the same as what I have shown already. Obviously, the type of explanation you seek is absent. You can deduce anything you like from that. I think it is odd that in all my searching, not a single author has expressed any concerns along the lines of yours (that some key assumption is "missing" from the literature). I checked many of the references in a variety of papers looking for something like that, but they pretty much all start from the 4 Bell states. I only went back to the 90's though.

---

Now obviously, almost every interpretation tackles experiments like those presented in a different manner. So I am not trying to make any statement about that. But the experimental results should be accepted without question. And the description (mathematical or otherwise) of the authors of these papers should be accepted as that of garden variety QM. This is stuff that has been out there for literally decades.

So I am hoping that we can return to the discussion of Measurement in QM/QFT. What is it? Is it physical? When does it occur? What triggers it? Where does it occur? And I certainly hope that the cited experiments co-authored by a Nobel prize winner (and others equally well-regarded) can be used for discussion purposes without further debate.

 

[PeterDonis]

The answer is that QM is not only quantum nonlocal (our agreed upon terminology), it is also quantum non-spatiotemporal.

Experimentally, yes, I agree. It's the theoretical basis that I'm not clear on--but that probably means I need to do more digging into the literature. In my copious free time.

 

[PeterDonis]

I am hoping that we can return to the discussion of Measurement in QM/QFT. What is it? Is it physical? When does it occur? What triggers it? Where does it occur?

I believe the short answer to this is "decoherence", but while the ordinary QM picture of this seems to me to be fairly straightforward, I'm not sure how the picture changes in QFT.

I certainly hope that the cited experiments co-authored by a Nobel prize winner (and others equally well-regarded) can be used for discussion purposes without further debate.

Well, the theoretical background that I'm still not clear on is not irrelevant, since any interpretation (and this thread is in the interpretations subforum) has to rely on the theoretical background at least to some extent. To the extent that interpretations can ignore the details and simply rely on the experimentally observed correlations, yes, they would not need the theoretical details I have been asking for. The question is to what extent that is actually true.

 

[gentzen]

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.

I see your dilemma and acknowledge your points. Unfortunately I am not the person to bridge the published papers to your type of analysis. Without in any way taking away from your approach, I will simply point out your methodology/approach does not appear in the hundreds of papers I have read on the subject. Instead, the presentation is nearly identical in papers written in the 1999 to present time frame.

Do I guess correctly that Peter Donis initially did his analysis when you wanted to know how that stuff works in MWI? In that case, one explanation for the "disconnect" might be that the theoretical analysis in the papers implicitly relied on the Copenhagen interpretation, especially on the "update" of the wavefunction when new knowledge becomes available. From a MWI perspective that considers the wavefunction as something physically objective, this could feel like an invalid or incomplete analysis.

 

[PeterDonis]

To the extent that interpretations can ignore the details and simply rely on the experimentally observed correlations, yes, they would not need the theoretical details I have been asking for. The question is to what extent that is actually true.

To give an example: @gentzen has just commented that various papers that have been referenced appear to be using a more or less Copenhagen-type interpretation. With an interpretation like that, as @gentzen says, since the wave function is not claimed to be physically real, I think it would be perfectly fine to say, look, we know the correlations between 1 & 4 are the same regardless of the time ordering or whether those two photons ever even coexist. So it seems fine to use the same math to make the predictions in all these cases. We're not claiming that there is an actual physically real Bell state with photons 1 & 4 when they never coexist. We're just saying the Bell state is the math we are using to make predictions, and that math works.

But in a realist interpretation, which would include the MWI but is by no means limited to that, the above does not work at all. In a realist interpretation, there has to be some physically real thing that the math describes, that causes the correlations. In the ordinary NRQM version of a realist interpretation, that physically real thing is described by the wave function in the math, which evolves in time according to the Schrodinger equation (either literally all the time, in the MWI, or all the time except when "measurements" or something, occur). But if photons 1 & 4 never coexist, there is never any such wave function--which means now we are stuck looking for what physically real thing causes the correlations and what math we can use to represent it, since the ordinary NRQM math with a wave function and the Schrodinger equation can't be it.

@DrChinese has, I believe, stated that he is using a realist interpretation, which is why I have been making the point I made in the previous paragraph multiple times in this thread. But I don't know whether the authors of the papers that have been referenced are using a realist interpretation or not; it's not really clear to me from the papers (and, as I noted above, @gentzen, at least, seems to think they are using a Copenhagen-like interpretation, which is not realist). So one reason why the papers don't address the questions I have been asking is that the authors don't even see the issues since in the interpretation they are using those issues are not there.

 

[PeterDonis]

Do I guess correctly that Peter Donis initially did his analysis when you wanted to know how that stuff works in MWI?

You do. It was in the "Is the MWI local?" thread, which was fairly recent. I did the analysis to illustrate that the MWI does give an explanation for the correlations, but I agreed with @DrChinese that its explanation is not local, because of the nonlocality of the wave function.

 

[Morbert]

If we adopt a MWI interpretation of NRQM with some presentist account of time, then presumably the swap will not act on the "1,2" and "3,4" biphoton systems, but rather the the "3,4" biphoton system and the "detector environment, 2" system (in the sense that these are the subsystems at time $\tau$, just before the BSM).

 

[PeterDonis]

If we adopt a MWI interpretation of NRQM with some presentist account of time, then presumably the swap will not act on the "1,2" and "3,4" biphoton systems, but rather the the "3,4" biphoton system and the "detector environment, 2" system (in the sense that these are the subsystems at time $\tau$, just before the BSM).

That would be one way for the MWI to handle this, yes. But note that in this case there is no Bell state for photons 1 & 4. The correlations would be accounted for by an entanglement between photon 4 and the "detector environment" degrees of freedom associated with the photon 1 measurement. (In fact this is more or less what I did in the analysis in the previous thread that I have been referring to.)

 

[A. Neumaier]

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.

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.

This produces the entangled 4-photon GHZ state (2) of arXiv:1204.1997 that I was referring to as a tetrastate.

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.

Yes, indeed. They showed that a particular measurement scheme applied to the entangled 4-photon state (whather a GHZ state or a product of two Bell states does not matter) with measurement at two different times produced the same statistics as a Bell state would have done.

But it didn't produce a Bell-state! Instead the first measurement reduced the number of particles, and the second reduced it again. A Bell state never appeared, and the authors of arXiv:1204.1997 never claimed that.

The authors claimed (and achieved) to show how to entangle photons to a 4-photon, 6-photon, 8-photon etc state with nonclassical measurement statistics.

They did not claim that they produced temporally entangled photons.

It was published in Physical Review Letters 110, 210403; 22 May 2013. Obviously it passed peer review.

There is no dispute about that. But publication of a paper in a peer reviewed journal is no guarantee for the correctness of every statement of its content.

Perhaps you are aware of published work that contradicts this result.

The very definition of entanglement contradicts the existence of temporally entangled photons. Entanglement requires a state in the Schrödinger picture belonging to the Hilbert space in question, and such a state always means a state at a fixed time.

So I question your evaluation of their results as "meaningless".

I only evaluate as meaningless the interpretation of their results as having created temporally entangled photons.

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 pattern is clear: They measured multiphoton states at different times and obtained a statistics identical with that of measuring a Bell state. In no case, they produced a temporally entangled state, since the latter is a thing that cannot be consistently defined.

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.

This has nothing to do with relativistic or not. The Hilbert spaces in question are finite-dimensional, so no field theory is involved.

 

[A. Neumaier]

The first application of delayed choice to entanglement swapping that I am aware of is "Delayed choice for entanglement swapping" (Peres, 1999). His analysis of Bell states looks almost identical to those I have presented in various posts.

With the exception that he is precise about the implications. The abstract says:

each subset behaves as if it consisted of entangled pairs of distant particles, that have never communicated in the past, even indirectly via other particles.

Note the subjunctive: The claim is only identical behavior of the statistics, not the creation of a meaningless temporally entangled state!

From HERE (my primary reference

It seems that this is the paper that coined the notion of ''entangled pair of photons that have never coexisted'', since they state that

Previous demonstrations [...] entangled photons that were separated spatially, but not temporally, i.e., all the photons that were entangled, existed and were measured at the same time.

But they do not give any definition of the meaning of the new concept. Thus it must be inferred by reading between the lines. Clearly it means no more than pairs of photons that reproduce the statistics of entangled photons, in the above as-if sense of Peres.

Figure 1 shows that we have between I and II a 2-photon state, between II and III a 1-photon state, between III and IV a 3-photon state, between IV and V a 1-photon state, and after V a 0-photon state. These are the physical states actually present in the experiments - well-defined states at each time.

By the experimental set-up, the statistical behavior of the chain is identical to that we'd have obtained by starting with the entangled 4-photon state given by equation (2), measured in several stages to produce a temporally stretched Bell statistics for the photon pair 1, 4. Thus, in the careful formulation of Peres, the whole set-up ''behaves as if it consisted of entangled pairs of distant particles, that have never communicated in the past, even indirectly via other particles.''

Talking of temporal modes replaces the physical states by imagined (physically unrealized) states that would behave the same as the physical states, if they would have been created in place of the actual set-up and subjected to the same projective measurements. This works for simple combination schemes where such a would-be equivalent behavior can be justified, and may justify talking informally about ''temporally entangled photons'', using loose language as used elsewhere, e.g., when talking about virtual particles. It does, not, however, justify talking of temporally entangled 2-photon states or biphotons, since quantum-mechanical states are mathematically well-defined objects with a clear single time meaning.

the cited experiments co-authored by a Nobel prize winner

Who of the authors of your primary source is a Nobel prize winner? Which year?