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

[Sambuco]

Mjelva: "Assuming the projection postulate [can be applied here], Alice and Bob’s measurements have the effect of projecting the state |Ψ(0)〉into one of the following four product states, each with probability p=1/4: ... On any given run of the experiment, the total system will be in one of the these possible states."

If it is in one of the product states, as he says here, then according to your 1. it does not lead to entanglement swapping. You are taking both sides of the same question.

I'll try to show that we (Mjelva, @Morbert and me) are not wrong. I really hope we all can agree. When I shared Mjelva's paper on the OP, I wished to discuss the implications the forward-in-time analysis has on how the different interpretations of QM deal with DCES. I understand we spend some time with the nuances of the forward-in-time analysis, but it's a bit frustating that we'are stuck still discussing basic linear algebra.

I believe that the problem lies in:
1. You confuse the state assigned to any given run of the experiment, which is a product state as you correctly showed Mjelva said in his eq. pre-(4), with the state of the whole system composed of all the runs, which is a mixture of product states, as Mjelva showed through the densitiy operator in his eq. (4).
2. You (wrongly) believe that a mixture of product states cannot reproduces the statistical data in Ma's paper.

To be clearer (one more time), let's do the math. Please, @DrChinese, follow me on this calculation. I'll consider the case in Ma's experiment where Alice and Bob always measure in the $L/R$ basis. After preparation at $t_0$, the state of the system is:

$\ket{\Psi(t_0)} = \ket{\psi^-}_{1,2} \otimes \ket{\psi^-}_{3,4}$

At time $t_1$, Alice and Bob measure photons 1 and 4, respectively. They could obtain four different combinations: RR, RL, LR, LL. According to the projection postulate as in Zweibach's textbook, Alice and Bob project the initial state onto four different product states with 25% chance:

$\ket{\Psi(t_1)}_A = \ket{R}_1 \otimes \ket{R}_4 \otimes \ket{L}_2 \otimes \ket{L}_3$
$\ket{\Psi(t_1)}_B = \ket{R}_1 \otimes \ket{L}_4 \otimes \ket{L}_2 \otimes \ket{R}_3$
$\ket{\Psi(t_1)}_C = \ket{L}_1 \otimes \ket{R}_4 \otimes \ket{R}_2 \otimes \ket{L}_3$
$\ket{\Psi(t_1)}_D = \ket{L}_1 \otimes \ket{L}_4 \otimes \ket{R}_2 \otimes \ket{R}_3$

So, if we want to define the state of the whole system including all the runs, it will be a mixture:

$\rho(t_1) = \frac{1}{4} (\ket{\Psi(t_1)}_A\bra{\Psi(t_1)}_A + \ket{\Psi(t_1)}_B\bra{\Psi(t_1)}_B + \ket{\Psi(t_1)}_C\bra{\Psi(t_1)}_C + \ket{\Psi(t_1)}_D\bra{\Psi(t_1)}_D)$

Something very important: the state $\rho(t_1)$ does not represent one of the four subsets $A,B,C,D$. The state $\rho(t_1)$ is assigned to the whole system formed by all the runs taken together, so it is a mixture of product states. Now, the question is: This state is able to reproduce the statistical data in Ma's paper? Let's see.

First, we can rewrite the states $\ket{\Psi(t_1)}_{A,B,C,D}$ with photons 2&3 in the Bell-states basis:

$\ket{\Psi(t_1)}_A = \ket{R}_1 \otimes \ket{R}_4 \otimes \frac{1}{2} (\ket{\phi^-}_{2,3}-i\ket{\psi^+}_{2,3})$
$\ket{\Psi(t_1)}_B = \ket{R}_1 \otimes \ket{L}_4 \otimes \frac{1}{2} (\ket{\phi^+}_{2,3}+i\ket{\psi^-}_{2,3})$
$\ket{\Psi(t_1)}_C = \ket{L}_1 \otimes \ket{R}_4 \otimes \frac{1}{2} (\ket{\phi^+}_{2,3}-i\ket{\psi^-}_{2,3})$
$\ket{\Psi(t_1)}_D = \ket{L}_1 \otimes \ket{L}_4 \otimes \frac{1}{2} (\ket{\phi^-}_{2,3}+i\ket{\psi^+}_{2,3})$

What the states above predict if Victor decides to perform a BSM? To be short, take only the case when Victor measures $\phi^-$. According to the Born's rule, Victor will measure $\phi^-$ 50% of the times Alice and Bob measured $RR$ and 50% of the times Alice and Bob measured $LL$. So, if you took all the runs, i.e. the system whose state is a mixture of product states described by the density operator $\rho(t_1)$, and construct a subset formed only by the runs where Victor measured $\phi^-$, you could see perfect correlation between Alice's and Bob's measurements in the $L/R$ basis. This is the experimental result observed in Ma's paper! We could do the same calculations considering any other basis for Alice's and Bob's measurements.

So what did we proved above? Well, we can see that if we solve the time-dependent Schrödinger equation evolving the initial state of the system from time $t_0$ until time $t_2$, unitarily between measurements and "collapsing" the state upon measurements, we can reproduce statistical data in Ma's paper with photons 1&4 never being in an entangled state even when Victor performed a BSM. If you like, you could say that these forward-in-time analysis does not involve entangled states for 1&4. This is the entire point of the forward-in-time treatment, you could reproduces all the data without 1&4 entangled states!

On the other hand, the experiment can be interpreted in a very different way by giving priority to Victor's measurements. In that case, when he decides to perform a BSM, he projects the state of the system and post-selects the runs where he obtained only one of the Bell states (let's say $\phi^-$), the state of the remaining photons 1&4 will be in an entangled state $\phi^-$, which correcly predicts 1&4 results even if they were already measured before the BSM (swap) was performed. This is how authors present the results in Ma's paper. I use the word "present", instead of "interpret" because they (Zeilinger's group) does not take the state of the system as something real. They don't favor a $\Psi$-ontic interpretation, but a $\Psi$-epistemic one.

What is really interesting about all these analysis is that both descriptions correctly predict the same data. If you argue against retrocausality, the forward-in-time analysis à la Mjelva (or Cohen, by the way) is perfect for you, because all the data can be explained by evolving the state of the system from past to future with 1&4 photons never being in an entangled state. In that sense, this allows to (re)-interpret the well-known conclusion by Peres "each subset behaves as if it consisted of entangled pairs of distant particles". Some authors (Bacciagaluppi and others) interpret that as a "proof" against the reality of entanglement. In my view, it could be even saw as a hint against $\Psi$-ontic interpretations or even against causation. Of course, these last thoughts are interpretation-dependent.

As I said at the beginning of this post, I really hope we can all agree on the math, so we can move on to the physical analysis of these results.

Lucas.

 

[javisot20]

I do not understand the conclusions, even if it is true that "entanglement=selection criterion", that would lead to the conclusion that both the entanglement and the selection criterion are physical descriptions of the same physical phenomenon (this tells us nothing about the nature of the wave function)

 

[PeterDonis]

You confuse the state assigned to any given run of the experiment, which is a product state as you correctly showed Mjelva said in his eq. pre-(4), with the state of the whole system composed of all the runs

In an interpretation like the one @DrChinese is using, there is no such thing as "the state of the whole system composed of all the runs". Each run has its own individual state.

Conversely, as I've already commented, in a statistical interpretation such as the one used in Ballentine's textbook, there is no such thing as "the state assigned to any given run of the experiment"; quantum states never describe anything except abstract ensembles of runs.

So it's not a matter of "confusion" here; it's a matter of different interpretations not even having any common ground between them about what the quantum state represents.

I'm not aware of any interpretation of QM that gives both meanings to the quantum state. If you know of one, you need to give a reference for it. Otherwise your statement quoted above is simply irrelevant, since it does not apply to any known interpretation of QM.

I think many of the comments in this thread have failed to realize just how fundamentally different different interpretations of QM can be even on such a basic item as this.

 

[jbergman]

A. This is contradictory to this discussion on many levels.

1. We've been talking statistical results of a sufficiently large set of specific outcomes (say RR) of Alice and Bob's measurements on chosen basis (say R/L), with a BSM performed on an unbiased basis (say H/V).
2. No one is saying anything about what would have (counterfactually) happened if a different basis for the BSM was selected (say +/- instead of H/V).
3. The Ma experiment clearly shows the expected correlations on all 3 bases for measurements on Alice and Bob.

B. Mjelva: "Assuming the projection postulate [can be applied here], Alice and Bob’s measurements have the effect of projecting the state |Ψ(0)〉into one of the following four product states, each with probability p=1/4: ... On any given run of the experiment, the total system will be in one of the these possible states."

And yet: No ONE (his words) of these 4 product states will experimentally lead to DCES statistics, in contradiction to Mjelva's basic premise. There cannot be such intermediate Product states. It doesn't matter if you try to call them some special name ("proper mixtures of product states"*), which is not what Mjelva says above.


*Which is simply a classical statistical mixture of Product states, and bearing no relationship whatsoever to a quantum superposition of states.

I hate to jump in as this point may have already been discussed, but Mjelva states in the intro, "I will argue that this foliation-dependence is not so much a problem for Egg’s view as it is a problem for a projection-based interpretation of quantum mechanics."

In other words,

In an interpretation like the one @DrChinese is using, there is no such thing as "the state of the whole system composed of all the runs". Each run has its own individual state.

Conversely, as I've already commented, in a statistical interpretation such as the one used in Ballentine's textbook, there is no such thing as "the state assigned to any given run of the experiment"; quantum states never describe anything except abstract ensembles of runs.

So it's not a matter of "confusion" here; it's a matter of different interpretations not even having any common ground between them about what the quantum state represents.

I'm not aware of any interpretation of QM that gives both meanings to the quantum state. If you know of one, you need to give a reference for it. Otherwise your statement quoted above is simply irrelevant, since it does not apply to any known interpretation of QM.

I think many of the comments in this thread have failed to realize just how fundamentally different different interpretations of QM can be even on such a basic item as this.

Is this what Mjelva is trying to highlight? From the intro's description of section 4 of the paper,

This is demonstrated both on an account which assumes that following a measurement, the state of the system is projected into an eigenstate corresponding to the measurement outcome, and on an Everettian account, where the quantum state remains in a superposition of states corresponding to different outcomes following the measurements. I show that while a projection-based account leads to different explanations of the correlations in the non-delayed and delayed-choice-cases, there is no such asymmetry on an Everettian account.

 

[Sambuco]

So it's not a matter of "confusion" here; it's a matter of different interpretations not even having any common ground between them about what the quantum state represents.

Well, we didn't start to discuss interpretations yet, we're still discussing about the math. The disagreement was because in a forward-in-time analysis of DCES, the quantum state of each run after Alice's and Bob's measurements is a product state and @DrChinese thought that these states cannot be consistent with the statistical data in Ma's paper. This discrepancy is not interpretation-dependent.

Anyway, I fully agree with what you said about how the different interpretations view the quantum state in very different ways. In that sense, I like your response in another thread where you said that answers about which physical properties are described by the wave function range from "none" to "all of them".

Otherwise your statement quoted above is simply irrelevant, since it does not apply to any known interpretation of QM.

As I said above, we were discussing the math in Mjelva's paper, not the interpretation (yet), so in the statement you selected, I don't adhere to any interpretation, I use "quantum state" as a synonym of "state vector", i.e. the thing that we evolves in time from the initial preparation up to the last measurement to calculate the probabilities of certain measurement outcomes.

I'm not aware of any interpretation of QM that gives both meanings to the quantum state. If you know of one, you need to give a reference for it.

Thanks for your question because this is the kind of things I wished to discuss when I shared the paper in the OP. Information-based interpretations have no problem considering both quantum states (the state of a single run and the state of the whole system formed by all the runs) on an equal footing, as $\Psi$-epistemic. For example, Rovelli's relational interpretation (RQM). For those who are not well versed on this interpretation, I recommend some papers: the seminal work on RQM (https://arxiv.org/abs/quant-ph/9609002), two recent shorter introductions (https://arxiv.org/abs/2109.09170, https://arxiv.org/abs/1712.02894), and a nice discussion about relative and stable facts (https://arxiv.org/abs/2006.15543).

Lucas.

 

[Morbert]

Is this what Mjelva is trying to highlight? From the intro's description of section 4 of the paper,

What Mjelva is highlighting is that you get the same statistics whether or not Victor carries out a BSM before or after Alice and Bob carry out their measurements, but under a projection based account, the explanation differs.

If Victor carries out a BSM before Alice or Bob carry out their measurements, then 1 & 4 are projected onto a Bell state upon Victor's measurement of 2 & 3. I.e. 1 & 4 become entangled. This entanglement explains the Bell correlations Alice and Bob will observe.

If Victor carries out a BSM after Alice or Bob carry out their measurements, then Victor's BSM has no effect on 1 & 4. Instead, runs post-selected by each of Victor's BSM results will exhibit Bell correlations in Alice's and Bob's data.

For experiments with spacelike separated measurements, there will be a relativity of entanglement and hence of explanation.

No such divisions are needed in some alternative accounts like an Everettian account.

I gave some thoughts on this in my first post in this thread:

As an aside: His distinction between entangled particles (pre-selection) and particles that exhibited Bell-inequality-violating correlations (post-selection) is interesting though I suspect anti-realist interpretations would dissolve any physical significance of that distinction, and he shows that for spacelike separated configurations of W experiments, they become relative.

Note that @DrChinese's objection is more fundamental. He maintains that a projection based account will yield probabilities and correlations that contradict experiment.

 

[PeterDonis]

This discrepancy is not interpretation-dependent.

Yes, it is. See my post #203. In order to get the correct prediction from the product state, you have to use the mixture of all four product states; just one product state for a given run won't do it. But in the interpretation @DrChinese is using, the mixture of all four product states has no meaning since it does not apply to any individual run, it only applies to the abstract ensemble of runs, and @DrChinese is not using an ensemble interpretation a la Ballentine. In the interpretation @DrChinese is using, each individual run has to be analyzed using the entire measurement context, and the BSM operation has to be performed on the entire entangled state that is initially prepared; otherwise you get the wrong answer.

 

[Morbert]

each individual run has to be analyzed using the entire measurement context, and the BSM operation has to be performed on the entire entangled state that is initially prepared; otherwise you get the wrong answer.

As this is a forward-in-time analysis and the experiment is delayed-choice, the BSM operation is performed on whichever product state Alice and Bob projected onto earlier. The computed probabilities will all be consistent with experiment, as the product state from Alice's and Bob's measurement will restrict what results Victor can get from a BSM. I showed this for the particular case of Alice and Bob measuring in the +/- basis and getting the result ++ in an earlier post

The Ma experiment: The tetraphoton is prepared in the state $$\ket{\psi^-}_{12}\ket{\psi^-}_{34}$$Alice and Bob measure in the +/- basis and get the result ++, projecting the state (rule 7) onto $$\ket{++}_{14}\ket{--}_{23}$$The probability that Victor's BSM gets the result $\phi^-$ for [this run] is (rule 6)$$p(\phi^-) = |\bra{\phi^-}--\rangle|^2 = 0$$This can be shown by expanding $\ket{--}$ in the Bell basis: $$\ket{--} = (\ket{\phi^+} - \ket{\psi^+})/\sqrt{2}$$Consistent with the Ma experiment, Victor can't get the result $\phi^-$ when Alice's and Bob's measurements are correlated in the +/- basis.

This exercise can be carried out for any combination of outcomes.

 

[PeterDonis]

As this is a forward-in-time analysis and the experiment is delayed-choice, the BSM operation is performed on whichever product state Alice and Bob projected onto earlier.

If you are using an interpretation in which such an analysis is allowed, yes. But not all interpretations allow such an analysis, since it involves applying the projection postulate without taking into account the entire measurement context.

 

[DrChinese]

What Mjelva is highlighting is that you get the same statistics whether or not Victor carries out a BSM before or after Alice and Bob carry out their measurements, but under a projection based account, the explanation differs.

If Victor carries out a BSM before Alice or Bob carry out their measurements, then 1 & 4 are projected onto a Bell state upon Victor's measurement of 2 & 3. I.e. 1 & 4 become entangled. This entanglement explains the Bell correlations Alice and Bob will observe.

If Victor carries out a BSM after Alice or Bob carry out their measurements, then Victor's BSM has no effect on 1 & 4. Instead, runs post-selected by each of Victor's BSM results will exhibit Bell correlations in Alice's and Bob's data.

For experiments with spacelike separated measurements, there will be a relativity of entanglement and hence of explanation.

In the forward in time only view: As @Morbert says, there must be differing explanations/descriptions precisely because the events occur in different sequence when there is delayed choice. So the concept of "relatively of entanglement" has been introduced around that.

Note that QM itself is silent about the ordering. You could say "the 2&3 BSM leads to 1&4 entanglement statistics in one, while the 1&4 outcomes lead to 2&3 BSM outcomes in the other." Of course, that same logic applies when there is ordinary 2 particle entanglement, when one particle is measured before the other. You could as easily claim a) Alice's earlier outcome is affected by Bob's later choice of measurement, as claim b) the reverse. Results are consistent with either explanation.

So basically: "Relativity of entanglement" simply means that the forward in time only premise is kept by assuming the same outcomes of experiments as if that same forward in time only premise didn't apply.

 

[DrChinese]

1. Alice and Bob measure in the +/- basis and get the result ++, projecting the state (rule 7) onto $$\ket{++}_{14}\ket{--}_{23}$$The probability that Victor's BSM gets the result $\phi^-$ for these runs is (rule 6)$$p(\phi^-) = |\bra{\phi^-}--\rangle|^2 = 0$$2. This can be shown by expanding $\ket{--}$ in the Bell basis: $$\ket{--} = (\ket{\phi^+} - \ket{\psi^+})/\sqrt{2}$$Consistent with the Ma experiment, Victor can't get the result $\phi^-$ when Alice's and Bob's measurements are correlated in the +/- basis.

1. Your first state - exactly as you have it - is one which can be prepared for each and every run*. That does not lead to DCES statistics on unbiased bases for a BSM. So either you should be presenting this in some different form*, or it is excluded experimentally. Which?

2. There is no such thing as what you describe here. A Product state of 2 particles is not equivalent to a expression of entangled states of those same particles. But the reverse can be true: An entangled state can lead to a product state of 2 particles.


BTW: your simplistic rule references are useless for these situations. We're discussing entangled systems of 2 and 4 particles, which are not discussed in the "7 rules".


*Presumably not a 4-fold Product state.

 

[PeterDonis]

or equivalently, physicsforum's prescription will get you the right predictions.

@DrChinese is correct that the "7 Rules" are a very simplified presentation and do not cover all possible experiments. In particular, they do not cover cases like the one under discussion, where you are trying to apply the projection postulate without taking into account the full measurement context.

In particular:

Applying it to these experiments In particular, a system prepared in state is evolved until Alice's and Bob's measurements with the time-dependent Schrödinger equation (TDSE) (rule 3). Measurement outcome probabilities are given by the Born rule (rule 6) and upon measurement, the state is updated (rule 7) and is evolved with the TDSE until Victor's measurement.

You are ignoring crucial caveats to the rules you cite:

Rule 3 applies to an isolated quantum system. But the only isolated quantum system in the experiment under discussion is the system of all four photons. You can't pick out just photons 1 & 4 and treat them as an isolated system, because they're entangled with photons 2 & 3. So you can't apply rule 3 the way you are doing it here.

Rule 6 applies to the probabilities of possible outcomes, but that is irrelevant to the analysis you're doing, because you're assuming particular outcomes for the photon 1 & 4 measurements, and then trying to analyze what that means for the photon 2 & 3 measurements. Nothing in your analysis considers probabilities at all. So rule 6 is irrelevant.

Rule 7 says that a measurement with a given outcome can be treated as a state preparation for future measurements on the same quantum system. Which, as above, needs to be an isolated quantum system. Rule 7 does not say you can treat a measurement with a given outcome on one system as a projection of some other system. So you can't apply rule 7 the way you are doing it here.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

After moderator review, the thread will remain closed as all relevant arguments have been thoroughly made. Thanks to all who participated.