Entanglement swapping and Bohmian mechanics

[Demystifier]

How important is it (in Bohmian terms) to include the measurement apparatus as a factor?

It is absolutely essential. Without that you cannot explain the effective collapse of the wave function, and even more importantly, you cannot explain how it is possible to measure any other observable except the position. Furthermore, without apparatus and environment you couldn't talk about the macroscopic description, without which there would be no time arrow, so you couldn't properly understand the notions of cause and effect. Finally, there would be no explanation of the macroscopic world we all see and love.

 

[Demystifier]

From @Demystifier:

Furthermore, at the macroscopic level we also have a time arrow, due to which the past causes the future, so the correlation can also be interpreted causally as
$$ (M_A,A)_2 \rightarrow (B,M_{BC},C)_3 \leftarrow (M_D,D)_2$$
Thus we see that, at the macroscopic level, we can say that the results of measurements in the past cause the result of measurement in the future.

---

OK, I follow this. And presumably if the experimental ordering were reversed, we'd have:

$(M_A,A)_2 \leftarrow (B,M_{BC},C)_3 \rightarrow (M_D,D)_2$

And the results must be the same, correct? (I know this is a simple question.)

Almost correct, my only correction is that you also have to change the labels 2 and 3 in that case, namely
$(M_A,A)_3 \leftarrow (B,M_{BC},C)_2 \rightarrow (M_D,D)_3$
because $t_2$ is before $t_3$.

 

[Morbert]

1. It's literally the definition of cherry picking. Read and quote Ma's entire paragraph. Or Megidish.

2. I am quite comfortable with my understanding of Ma, Megisdish, Hensen, Weihs, etc. I am not stuck anywhere. It is slow going getting answers to some of my questions. Here's one for you:

You keep trying to move off the point, which will just result in mistakes compounding mistakes. It is not cherry picking to point out Ma says (2) is a rewriting of (1). Do you accept what Ma says plainly? Do you accept that (2) is a rewriting of (1)?

 

[Morbert]

Re/ Bohmian mechanics and entanglement swapping: I'm finding it conceptually straightforward but computationally fiddly. Considering spin/electrons instead of polarization/photons since that's more readily NRQM-friendly: The state takes a form like $$|\Psi_0\rangle
=
|\Phi_{\mathrm{spatial}}\rangle
|\Psi^-\rangle_{12}
|\Psi^-\rangle_{34}
\otimes
\left(
\frac{1}{\sqrt{3}}
\sum_{a=x,y,z}
|a\rangle_{q_A}
\right)
\otimes
\left(
\frac{1}{\sqrt{3}}
\sum_{b=x,y,z}
|b\rangle_{q_B}
\right)
\otimes
|A_0\rangle
|B_0\rangle
|V_0\rangle
$$where $q_A$ and $q_B$ are QRNGs modeling Alice's and Bob's choice of measurement axes, and $|A_0\rangle, |B_0\rangle,|V_0\rangle$ are the apparatuses. The Bohmian configuration is $$
Q_0
=
\left(
\mathbf{Q}_1,
\mathbf{Q}_2,
\mathbf{Q}_3,
\mathbf{Q}_4,
Q_{q_A},
Q_{q_B},
Q_A,
Q_B,
Q_V
\right)$$The Bohmian guidance equation will then give us trajectories and hence the data read off from the apparatuses including all observed correlations reported in these experiments.

The work seems to be in actually constructing the toy configuration space to model the essentials.

 

[javisot]

I agree that as a mathematical exercise

At the end of the day, what do "post" and "pre" mean in a context where there is no inherent order?

 

[javisot]

It is absolutely essential. Without that you cannot explain the effective collapse of the wave function, and even more importantly, you cannot explain how it is possible to measure any other observable except the position. Furthermore, without apparatus and environment you couldn't talk about the macroscopic description, without which there would be no time arrow, so you could'n properly understand the notions of cause and effect. Finally, there would be no explanation of the macroscopic world we all see and love.

It is a profound and intriguing idea, it is like using the macroscopic description as an arrow of time for QM.

 

[Roberto Pavani]

It is a profound and intriguing idea, it is like using the macroscopic description as an arrow of time for QM.

Still, Maxwell's equations do not require an arrow of time

 

[javisot]

Still, Maxwell's equations do not require an arrow of time

Or, as in GR, we can impose conditions that lead to a certain arrow of time, but in essence, the field equations are symmetric under time reversal.

 

[Lord Jestocost]

I am interested in an answer in terms of your own preferred interpretation. You don't need to tell me what interpretation it is (or you can - your call). I just want to know your opinion.

I guess you will get no satisfactory answer.

All interpretations of quantum theory (QT), Bohmian mechanics is no exemption, use Born’s rule for the calculation of probabilities for observable outcomes of physical experiments and use all other standard links between the mathematical formalism of QT and the physical phenomena. One can find the essential rules of QT in https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/.

All interpretations of QT agree by construction with QT as far as its experimentally testable physical aspects are concerned and can thus deliver nothing new regarding physics or physical experiments. Therefore, using the title of a paper by Christopher A. Fuchs and Asher Peres: Quantum Theory Needs No ‘Interpretation’ (Christopher A. Fuchs and Asher Peres, Physics Today 53 (3), 70–71 (2000)).

To my mind, the relevant question regarding physics would thus merely be: Does one accept the fundamentally probabilistic world view of QT or not?

 

[Morbert]

I guess you will get no satisfactory answer.

A basic level of understanding of quantum theory is needed for any kind of advancement of the discussion. Fundamentally misunderstanding basic quantum theory has gummed up discussions before. There, @DrChinese denies an even more basic expansion |HH⟩ = (|Φ+⟩ + |Φ-⟩)/√2

 

[Demystifier]

It is a profound and intriguing idea, it is like using the macroscopic description as an arrow of time for QM.

The existence of the arrow of time is a matter of statistical physics, either classical or quantum. To see how that works in QM from a Bohmian point of view, see my https://arxiv.org/abs/2308.10500
especially Sec. 5.3.

 

[Lord Jestocost]

A basic level of understanding of quantum theory is needed for any kind of advancement of the discussion. Fundamentally misunderstanding basic quantum theory has gummed up discussions before.

From a former post by @FactChecker:

"I think I can safely say that nobody understands quantum mechanics."
"in mathematics you don't understand things. You just get used to them.”
-- Richard Feynman

Post in thread 'How do entangled particles know what the other is doing?'

Jun 5, 2026

"I think I can safely say that nobody understands quantum mechanics."
"in mathematics you don't understand things. You just get used to them.”
-- Richard Feynman

Mathematically, they are one entity. If you are given that $x^2 = 1$, then you know that there are two solutions, $1$ and $-1$. When you are told that one solution is $x_1 = -1$, you know instantly that the other solution, $x_2$, is $x_2=-1$. And the value of either $x_1$ or $x_2$ are not determined until the other is.
That is not one causing the other, and it is not a case of a "hidden variable". Simply...

• FactChecker

• 

Probably, a new paper by Časlav Brukner, entitled “Schrödinger’s Equation at 100: The Wave Picture That Helped and Possibly Hurt” (https://arxiv.org/abs/2604.26325), gives some hints to understand Feynman. One reads from the abstract:

"Schrödinger’s equation gave early quantum theory a visual language that looked like
physics again: a wave evolving by a linear differential equation. This essay argues that the
same success also seeded a recurring impulse to keep quantum theory “classical-looking”
by treating the wave function as a physical wave. Schrödinger quickly realized that, for
many-particle systems, the wave function is naturally defined on configuration space rather
than ordinary physical space, blocking any straightforward reading of it as a literal classical
wave. Read through Mach and Boltzmann, who shaped his intellectual outlook most deeply,
his achievement appears double-edged: it provided an extraordinarily powerful picture
for calculation and discovery, while also warning against taking that picture too literally.
I argue that this tension never fully disappeared. It still reappears in modern physics
whenever the wave function, or in quantum field theory the field itself, is treated as ontology
rather than as part of a representation tied to measurement and observational context, a
point sharpened by Bell-type no-go theorems. The centenary moral is: use pictures boldly,
but demote them ontologically."

(Bold by LJ)

 

[Morbert]

"I think I can safely say that nobody understands quantum mechanics."
"in mathematics you don't understand things. You just get used to them.”
-- Richard Feynman

"For instance, our fundamental law that any state can be made up from a linear combination of base states is written as $\ket{\psi} = \sum_i C_i \ket{i}$" -- Richard Feynman

https://www.feynmanlectures.caltech.edu/III_20.html

 

[Lord Jestocost]

The notion “state function” for $\ket{\psi}$ in QT might indeed confuse lay-persons who are interested in physics. They might mix up this quantum mechanical notion with the notion “state of a system” as it is used in classical physics. This confusion might give rise to the “interpretational nonsense” that the notion “superposition” (as used in the mathematical formalism of QT) would imply that two or more different physical states might actually exist at the same time.

In his book “The Structure of Physics” (the book is a newly arranged and revised English version of "Aufbau der Physik" by Carl Friedrich von Weizsäcker) Carl Friedrich von Weizsäcker remarks:

“But we can now see that the name "state" for the wavefunction $\ket{\psi}$ is misleading. $\ket{\psi}$ is nothing but a catalog of knowledge that follows from one observed fact and which determines the probabilities for possible future events.”

 

[PeterDonis]

It is absolutely essential.

It's worth noting that this viewpoint is not limited to the Bohmian interpretation. Chapter 9 of Ballentine's book, in which of course he uses the ensemble interpretation, discusses the reasons why you need to include the macroscopic measuring apparatus in the analysis in order to get a proper account of measurement.

 

[Demystifier]

It's worth noting that this viewpoint is not limited to the Bohmian interpretation. Chapter 9 of Ballentine's book, in which of course he uses the ensemble interpretation, discusses the reasons why you need to include the macroscopic measuring apparatus in the analysis in order to get a proper account of measurement.

Yes. I would like to add that the first person who understood the importance of the quantum description of the measuring apparatus was von Neumann in 1932, in his book on mathematics of QM. This is now called von Neumann measurement scheme.

 

[PeterDonis]

Why write a formula for these that doesn't include other potential future partners?

Um, because we're not talking about any experiment that includes them?

By your logic, every wave function we write down for any scenario whatever must include the entire universe. After all, there could be "potential future partners" anywhere, right?

 

[PeterDonis]

You might be interested to know that Riedmatten uses the same phrase you and several others are using: "rewritten".

Yep. He also says that when photons 2 and 3 are measured in the Bell basis via a BSM, they are projected onto one Bell state. Not four.

Which, to me, and I suspect to everyone in this thread except you, means a description of the state after the projection--i.e., after the BSM--i.e., after the swap--would be described by an expression that had one Bell state in it. Not four.

But Riedmatten's "rewritten" (2), like Ma's (2) and Megidish's (3), has four Bell states in it. Not one.

So, to me, and I suspect to everyone in this thread except you, Riedmatten's (2), Ma's (2), and Megidish's (3) are describing the pre-swap state. Because they have four Bell states in them. Not one.

 

[DrChinese]

At the end of the day, what do "post" and "pre" mean in a context where there is no inherent order?

There is still an initial context and a final context. In between elements of a swap are not significant as to order. So pre- means initial; post- means final.

 

[DrChinese]

Um, because we're not talking about any experiment that includes them?

By your logic, every wave function we write down for any scenario whatever must include the entire universe. After all, there could be "potential future partners" anywhere, right?

That's exactly my point. That there can be 3 pairs included in a swap - i.e. 3 photons converging at a BS - is not hypothetical. Or any number N. That is a physically realizable scenario. What's special and 2 and 3 that places them in a Bell state before interaction?

So saying there is a mathematical relationship prior to there being a physical relationship is not useful on its own.

 

[PeterDonis]

That there can be 3 pairs included in a swap - i.e. 3 photons converging at a BS - is not hypothetical. Or any number N. That is a physically realizable scenario.

Sure, but none of these are the scenario that's presented in the papers you yourself referenced. So what's the point of talking about them? Nobody but you has brought up these things. Why are you doing it? To me, and I suspect to everyone in the thread except you, statements like this, while true, are just noise, irrelevant to the thread.

What's special and 2 and 3 that places them in a Bell state before interaction?

They aren't in a Bell state before interaction. I've already said this more times than I can count, but I'll say it once more: before the BSM, photons 2 and 3 are in a superposition of 4 Bell states, which is separable because the entangled parts of the different Bell state terms in the superposition destructively interfere and cancel each other out. That is what Ma's (2) and the other equivalent expressions in the other threads are describing.

After the BSM, there is no such interference. (If you ask why not, well, then we get into the importance of the measuring device, in this case the BSM apparatus itself, in understanding what happens during a measurement. The papers you reference don't bother going into this because they're written for experts and they assume they don't need to belabor the obvious. But evidently it's not obvious to you, so apparently in this thread we need to discuss it. There have already been a number of posts in the thread about it--indeed, @Morbert mentioned it in #40 to explain how he got the state he wrote down there. But if we need to discuss it more, I guess we'll have to.)

saying there is a mathematical relationship prior to there being a physical relationship

If "mathematical relationship" means "being in an entangled state in the math", nobody except you is saying any such thing. See above.

 

[Morbert]

The notion “state function” for $\ket{\psi}$ in QT might indeed confuse lay-persons who are interested in physics. They might mix up this quantum mechanical notion with the notion “state of a system” as it is used in classical physics. This confusion might give rise to the “interpretational nonsense” that the notion “superposition” (as used in the mathematical formalism of QT) would imply that two or more different physical states might actually exist at the same time.

In his book “The Structure of Physics” (the book is a newly arranged and revised English version of "Aufbau der Physik" by Carl Friedrich von Weizsäcker) Carl Friedrich von Weizsäcker remarks:

“But we can now see that the name "state" for the wavefunction $\ket{\psi}$ is misleading. $\ket{\psi}$ is nothing but a catalog of knowledge that follows from one observed fact and which determines the probabilities for possible future events.”

Bohmian mechanics treats a wavefunction as nomological or ontological, and supplies a primitive ontology. It's quite distinct from the interpretation of QM as a theory of tests and responses.

But we have a pre-interpretational problem. Before a wavefunction can be interpreted, it must be correctly written down.

 

[DrChinese]

Yep. He also says that when photons 2 and 3 are measured in the Bell basis via a BSM, they are projected onto one Bell state. Not four.

Oh really? And exactly when does that happen? After the projecting Beam Splitter? Or upon passing polarizers and detection (i.e. after the one Bell state is identified) ?

Again, by analogy (and you might need to review the concept of analogy after my previous one): When an entangled photon emerges from a PDC setup, is it |V> or |H>? Oh, it's in a superposition! Just as 2 & 3 are when they emerge from the BS. A superposition of 4 Bell states.

 

[PeterDonis]

And exactly when does that happen?

The Riedmatten paper doesn't say. It says "When photons B and C are measured in the Bell basis (Eq. 3), i.e. projected onto one of the four Bell states via a so-called Bell state measurement, photons A and D are projected onto the corresponding entangled state" (my emphasis). The paper doesn't specify when that "when" is. Does it matter?

 

[romsofia]

Re/ Bohmian mechanics and entanglement swapping: I'm finding it conceptually straightforward but computationally fiddly. Considering spin/electrons instead of polarization/photons since that's more readily NRQM-friendly: The state takes a form like $$|\Psi_0\rangle
=
|\Phi_{\mathrm{spatial}}\rangle
|\Psi^-\rangle_{12}
|\Psi^-\rangle_{34}
\otimes
\left(
\frac{1}{\sqrt{3}}
\sum_{a=x,y,z}
|a\rangle_{q_A}
\right)
\otimes
\left(
\frac{1}{\sqrt{3}}
\sum_{b=x,y,z}
|b\rangle_{q_B}
\right)
\otimes
|A_0\rangle
|B_0\rangle
|V_0\rangle
$$where $q_A$ and $q_B$ are QRNGs modeling Alice's and Bob's choice of measurement axes, and $|A_0\rangle, |B_0\rangle,|V_0\rangle$ are the apparatuses. The Bohmian configuration is $$
Q_0
=
\left(
\mathbf{Q}_1,
\mathbf{Q}_2,
\mathbf{Q}_3,
\mathbf{Q}_4,
Q_{q_A},
Q_{q_B},
Q_A,
Q_B,
Q_V
\right)$$The Bohmian guidance equation will then give us trajectories and hence the data read off from the apparatuses including all observed correlations reported in these experiments.

The work seems to be in actually constructing the toy configuration space to model the essentials.

Not sure if it's been brought up in this thread, but section 3/4 of this paper seems to work out the details in a toy model (I think stern-gerlach) using a configuration space: https://arxiv.org/pdf/0905.4036

I'm still slowly reading through it, but maybe it will be of help to you.

 

[PeterDonis]

section 3/4 of this paper seems to work out the details in a toy model (I think stern-gerlach) using a configuration space: https://arxiv.org/pdf/0905.4036

This looks like the kind of Bohmian analysis of entanglement swapping that @Demystifier was looking for earlier in the thread.

 

[Demystifier]

This looks like the kind of Bohmian analysis of entanglement swapping that @Demystifier was looking for earlier in the thread.

Yes, exactly. I see it as complementary to my own pictorial description with links between particles and without explicit wave functions. The analysis in this paper is more rigorous than mine, while mine is intended to be more intuitive. What I have done is something like "Feynman diagrams" for Bohmian mechanics.

 

[Lord Jestocost]

But we have a pre-interpretational problem.

Maybe, I don’t get the point.
I recommend that you read Richard Healey’s paper “Quantum Theory: A Pragmatist Approach”!

 

[DrChinese]

The Riedmatten paper doesn't say. It says "When photons B and C are measured in the Bell basis (Eq. 3), i.e. projected onto one of the four Bell states via a so-called Bell state measurement, photons A and D are projected onto the corresponding entangled state" (my emphasis). The paper doesn't specify when that "when" is. Does it matter?

It matters in the sense that you are the one quoting Riedmatten. I say the photons emerging from the BS (post-swap) are in a superposition (similar to the analogy I presented). You are the one saying that the pre-swap state matches (2), while I say the post-swap state matches (2). Seems like it might matter.

Sure, but none of these [other entangled photon pairs, 5/6 or 7/8 etc] are the scenario that's presented in the papers you yourself referenced. So what's the point of talking about them? Nobody but you has brought up these things. Why are you doing it? To me, and I suspect to everyone in the thread except you, statements like this, while true, are just noise, irrelevant to the thread.

Hmm, not relevant?

|Ψ〉1234 = 12(|Ψ+〉14⨂|Ψ+〉23 − |Ψ−〉14⨂|Ψ−〉23 − |Φ+〉14⨂|Φ+〉23 + |Φ−〉14⨂|Φ−〉23)
|Ψ〉1256 = 12(|Ψ+〉16⨂|Ψ+〉25 − |Ψ−〉16⨂|Ψ−〉25 − |Φ+〉16⨂|Φ+〉25 + |Φ−〉16⨂|Φ−〉25)

These cannot both be simultaneously true in a physical sense. Your line of thinking implies they are. One of these - and only one - can be true IFF two photons overlap physically* in a Beam Splitter.

*And of course indistinguishably as well. So obviously the full BSM is a physical mechanism, and not just an "informational" one. We know this because: In experiments (Megidish, Ma) where the Bell state signatures match informationally - but there is not indistinguishability - there is no swap.

 

[PeterDonis]

not relevant?

Yes. Where in any of the experiments we've been discussing are photons 5 and 6? Who cares about them?