Earlier thread: Entanglement swapping and Bohmian mechanics

The primary sources for experimental implementation are two important papers, plus a host of supporting work by top teams:

a) Ma et al, (2012): Experimental delayed-choice entanglement swapping
This paper demonstrates entanglement of Photons 1 and 4, produced from different PDC crystals, whereby the choice to entangle them via BSM on Photons 2 and 3 (i.e. a swap) or not is made randomly, and subsequent to their polarization measurements. Product State statistics are produced when no swap is executed.

b) Megidish et al, (2012): Entanglement Between Photons that have Never Coexisted
This paper demonstrates entanglement of Photons 1 and 4, produced from the same PDC crystal but at different times. Photon 1 is measured and ceases to exist before Photon 4 is created. Product State statistics are produced when Photon 3 is delayed so that it does not overlap with Photon 2 to enable a BSM (Bell State Measurement).

c) A number of other papers are available describing the underlying setups, here are a few well known ones (I may add some here from time to time):
High-fidelity entanglement swapping with fully independent sources
Experimental loophole-free violation of a Bell inequality using entangled electronspins separated by 1.3 km

We are fortunate to have some excellent resources in our PF members, many of whom are quite versed in Bohmian theory. We have also been referring to several papers helping us to refine some of the key tenets of BM as it relates to spin. Here are some additional papers we have been discussing:

d) Norsen, (2013): The Pilot-Wave Perspective on Spin
See his opening paragraph containing (1) and (2), and also his Fig. 6 and related text.

e) Huggett (2009): Entanglement Exchange and Bohmian Mechanics
His Fig. 1 is a schematic of what he calls a "Bell-ometer". This is the same thing as a Bell State Measurement (BSM) device as labeled in most papers. He labels his inputs to the BSM (Bell-ometer) as Particles 1 and 3, where in others they are marked Photons 2 and 3. Otherwise, the setup is essentially the same. This makes it convenient for comparison purposes with actual experiments. Note that this paper was written prior to the advent of the a) and b) papers above. In other words, the Delayed Choice option is not being presented. This paper analyzes the more traditional "swap-first" style.

I will use a follow-on post to "bring us up to speed".

-DrC

We discussed how Delayed Choice Entanglement Swapping is explained/described/modeled in Bohmian Mechanics. We managed to come to some common ground - and some disputed ground. I will try to state where we are and what it is I am asking as best as I can.

Common ground:
-If it is a specific axiom of Bohmian Mechanics - such as nonlocality or definite particle positions - I accept it as valid for purposes of the discussion.

Disputed ground:
i) For some, simply saying BM makes the same predictions as orthodox QM is sufficient and no further discussion is to be expected. I don't accept this as adequate, as there are specific issues around DCES that are not reasonably explained in the literature. In fact, comparatively speaking, there is almost nothing written as of now regarding details of how BM treats DCES. Huggett is one.
ii) What is orthodox QM (oQM)? I look at the works of experimentalists in the area of DCES, and assume their written presentation (theory and results) would be considered orthodox. Others look at oQM as an Interpretation of Quantum Mechanics. So typically I tend to compare oQM to BM to discover differences. Whereas some might not start where I do.

@Demystifier (and others) has generously taken some time to explain some of the points he sees as relevant. Specifically, he sees the Measurement apparatus for Photon 1 as becoming entangled with Photon 2 when Photon 1 is measured and ceases to exist. While this explanation is not exactly common to BM descriptions, I accept it for purposes of discussion.

I am going to attempt to walk through the logic of Huggett's paper, since it is something we can all look at and follow.

What I am asking:
a) I am trying to understand how BM explains both nonlocality in space (something that is built into the very premise of BM) and temporal nonlocality (which is not).
b) I am also trying to understand how BM explains deterministic outcomes in either swap direction - i.e. you get the same result regardless of whether the BSM occurs early or occurs later.
c) What is the nature of "collapse" in BM? Some say there is effective collapse (FAPP), some say no, some say yes. Somehow that seems relevant to the ordering of operations in a swap.

Thanks in advance,

-DrC

t=0:
Ψ = (α1,2) (α3,4) [Huggett] or
|Ψ〉1234 = |Ψ−〉12⨂|Ψ−〉34 [Ma et al - same thing but different labeling]

Question: What is a |Ψ−〉 Bell state in BM? (and please don't say the same as in oQM )

a) How does that spin state connect to a particle's definite position and velocity? A spin outcome is supposed to be "predetermined/deterministic", but velocity is a vector with a single direction. How does that vector map to an infinite number of spin bases with specific outcomes lacking "true" randomness (since that doesn't exist in BM, there is only the appearance of randomness)?

b) If each particle has its own deterministic evolution, how do a pair of particles end up synced as to spin? Presumably through spin-related action at a distance from particle 1 to particle 2 (as Norsen specified). But that means that particle 2's deterministic spin evolution was solely dependent on particle 1. Which goes against the premise.

c) Note that those same particles can be entangled but NOT be spin entangled. For instance, they can be both spin entangled and momentum entangled, but independently so. If there is one position variable controlling both spin and momentum outcomes, why do they commute? I would think a measurement on one basis dependent on position would affect the other basis, also dependent on position.

d) If every particle position in the universe contributes to action of the "pilot wave" or "guiding equation", why doesn't those positions affect particle 2 after particle 1 is measured? I recognize this has something to do with configuration space, as @Demystifier has pointed out on a number of occasions. But apparently, potential nonlocal effects are "walled off" in some cases, but not in others. How does this happen?

Thanks.-DrC

Please, I can't be the only person who has wondered about these points.

Now, entanglement exchange, as presented in §1, involves ‘internal’ degrees of freedom, such as spin, so we need to explain how they are treated in BM. ... Therefore the (spin) result must depend on the initial position, q⁡(t=0), of the particle.

It’s easy to see why this result should hold: the velocity field is a local functional of the wavefunction in configuration space, since its value at (q,Q) depends only on Ψ and its derivative at (q,Q).
The effective wavefunction is central to the Bohmian interpretation of wavefunction ‘collapse’. ... That packet is thus the effective one, and, for the reasons discussed, the system will behave as if that were the entire wavefunction – as if the wavefunction had collapsed into the wavepacket corresponding to the observed outcome!

The crucial feature of the Bohm theory is that it assigns definite particle positions at all times. Therefore, the particles are in some sense “pushed” around by the wave function on the configuration space – hence BM is sometimes referred to as the ‘pilot wave’ theory. ... In Bohm’s perspective probability is an epistemic tool. The particles always have definite positions – from which they evolve in an entirely deterministic way – and the use of probability is a measure of our ignorance of how they really behave.

... since the dynamics is of the standard, unitary, Schrödinger kind, it is linear. Thus if the initial state of the particles is a superposition of Bell states, then the outcome will involve a superposition of pointer wavefunctions... However, the pointer can only end up in one position, which must be inside the support of its wavefunction. The Born Rule gives zero probability for being elsewhere: it must yield a definite outcome even in this case. As we discussed in the previous section, that location will pick out a term from the RHS of (3.3) as the effective wavefunction, and there will thus be an effective collapse.

Taking Huggett's paper and formalism as a base, but reversing measurement ordering, we have Huggett's initial state $$\alpha_{12}\alpha_{34}\phi_1\phi_2\phi_3\phi_4\,\psi_0$$Instead of the BSM on 1 and 3, we'll look at the scenario where 2 and 4 are measured, correlating the spatial pointer states with the internal states of 2 and 4, yielding the state$$\begin{aligned}\frac12\Big(&
a_1b_2a_3b_4\,\phi_2^b\phi_4^b\\
&+a_1b_2b_3a_4\,\phi_2^b\phi_4^a\\
&+b_1a_2a_3b_4\,\phi_2^a\phi_4^b\\
&+b_1a_2b_3a_4\,\phi_2^a\phi_4^a
\Big)\phi_1\phi_3\psi_0.
\end{aligned}$$Each of these terms is itself an effective wavefunction. Let's say the pointer states ##\phi_2^b,\phi_4^b## are recorded. Then the effective wavefunction is $$a_1b_2a_3b_4\,\phi_2^b\phi_4^b\phi_1\phi_3\psi_0$$A BSM on 1 and 3 can only yield pointer outcomes ##\psi_\gamma,\psi_\delta## and so this effective wavefunction evolves to $$b_2b_4\,\phi_2^b\phi_4^b\frac{1}{\sqrt{2}}(\gamma_{1,3}\psi_\gamma + \delta_{1,3}\psi_\delta)\phi'_1\phi'_3$$My understanding of the Bohmian implications (and @Demystifier please critique): Say the initial configuration of 1 and 3 was one that was deterministically destined for a Bell outcome ##\psi_\alpha##. Now, the choice to measure 2 and 4 with outcomes ##\phi_2^b \phi_4^b## has nonlocally forbidden that outcome, and changed that destination to either ##\psi_\gamma## or ##\psi_\delta##.

And as you have correctly pointed out: When you apply his logic to the Swapping scenario, everything seems to work out and come back to the usual quantum mechanical result. Superficially, QED.

So it will be natural that I might have questions in the application of his logic. It's the missing pieces, such as the questions I posed above in #7, that I will be asking. I will be very specific in my next post.

Huggett: "The crucial feature of the Bohm theory is that it assigns definite particle positions at all times. Therefore, the particles are in some sense “pushed” around by the wave function on the configuration space – hence BM is sometimes referred to as the ‘pilot wave’ theory.".. The particles always have definite positions – from which they evolve in an entirely deterministic way – and the use of probability is a measure of our ignorance of how they really behave."

I think this matches normal Bohmian concepts. Now, what is a particle position for a photon? Is it a discrete point, some x/y/z at time t? Is it a wavelength from some point a to b? Because we just agreed that a BSM is required for a swap. And a BSM requires physical overlap of photons in a Beam Splitter. That overlap must occur within about 5+/- picoseconds*. Outside of some distance, the photons don't have an opportunity to interact (or do whatever it is they do during a swap). 5 ps at c translates into about 15 million nanometers (if I did my arithmetic correctly). Regardless, much much larger than the wavelength of a PDC photon. They have a characteristic wavelength of around 800 nm.

So if Photons 2 and 3 are to overlap in a Beam Splitter, each with a definite position and presumably a definite size: how do they ever get close enough to interact? If a Bohmian photon** were the size of a wavelength (I'm making this up as a starting point), and its position can be considered to be confined to some very very small region as it moves, and there is no opportunity for normal EM field effects to somehow intervene: the likelihood of overlap between the two photons would seem to near zero. But that doesn't happen. Instead, they do - nearly every time.

What gives? Can anyone help me on this?

*High-fidelity entanglement swapping with fully independent sources See Fig. 2.
** This isn't an issue in oQM, which does not assert a photon has a definite position or a definite size.

So if Photons 2 and 3 are to overlap in a Beam Splitter, each with a definite position and presumably a definite size: how do they ever get close enough to interact? If a Bohmian photon** were the size of a wavelength (I'm making this up as a starting point), and its position can be considered to be confined to some very very small region as it moves, and there is no opportunity for normal EM field effects to somehow intervene: the likelihood of overlap between the two photons would seem to near zero. But that doesn't happen. Instead, they do - nearly every time.

What gives? Can anyone help me on this?

The image of photons as tiny little balls does not allow for an understanding of certain aspects.

Question: What is a |Ψ−〉 Bell state in BM? (and please don't say the same as in oQM )

Of course, I told you all this several times, so I don't hope too much that this time you will understand what I am telling you. But since you asked, I felt a duty to answer.

1. Imagine two photons entering a beam splitter through the two input ports. If their wavepackets are sufficiently matched in arrival time, frequency, polarization, spatial mode, then the quantum states associated with the two paths are indistinguishable. In Bohmian mechanics the particle positions are not the whole physical state. The guiding wavefunction has nonlocal configuration-space structure, and it is that structure that produces the interference.

2. The image of photons as tiny little balls does not allow for an understanding of certain aspects.

2. Photons as "tiny little balls" was not my idea. Huggett and other Bohmians say photons have a definite position. Well, exactly what is "definite"? That's what I'm asking.

Either they do have definite positions - a unique feature of Bohmian theory. Or they have the kind of physical extent that orthodox QM claims. They can't be the same.

Nonlocal interaction - instantaneous action at a distance - is also consistent. What else would action at a distance mean? It seems to mean that anything anywhere could have a nonlocal impact under BM.

But... what in the heck are "wave functions/pilot waves" that sometimes influence our photons, and sometimes have no influence on our photons? Exactly what is the point of saying "photons have definite positions of vanishingly near zero size" and "action at a distance" when you simply pick and choose when and where such statements are to be applied? The rule seems to be: We pick and choose at our convenience.

2. Photons as "tiny little balls" was not my idea. Huggett and other Bohmians say photons have a definite position. Well, exactly what is "definite"? That's what I'm asking.

Either they do have definite positions - a unique feature of Bohmian theory. Or they have the kind of physical extent that orthodox QM claims. They can't be the same.

The situation of "the location of a photon" is quite complicated, even in "standard" QM/QFT. And I noted this to you a long time ago (maybe you remember?). I would assume that the rigorous way to treat photons in Bohmian Mechanics is also via a field-theoretic approach. However I am entirely unfamiliar with Bohmian Field Theory. Above my paygrade.

1. The answers to all your questions are implicitly written down in the math of the Bohmian equations of motion. But I am aware that your math skills are limited, so I will try to answer your questions qualitatively, without math.

2. The infinite number of spin bases is encoded in the infinite number of possible states of the apparatuses that measure the spin. As I told you a million times, there is no spin in BM without a measuring apparatus. The position (and velocity) of the particle alone does not predetermine spin. The spin is predetermined by positions of all particles that interact with it, which is a huge number of particles because the apparatus is made of a huge number of particles.

3. It is not true that each particle has its own deterministic evolution. ... As I said above, it is not true that one position variable controls spin and momentum outcomes. It is controlled by many position variables, corresponding to many particles constituting the measuring apparatus.

4. It is not true that every particle in the universe contributes to it. Many do, but not all. Moreover, among those that do, they do not all contribute equally. Some contribute more, some less.

5. ... since you asked, I felt a duty to answer.

6. In BM temporal nonlocality is illusion. The photon B in the future is not nonlocally connected to the photon A in the past. Instead, B in the future is nonlocally connected to the successor of A, because the successor exists in the future. But the successor is not one particle. The successor is quantum information related to A spread among many particles of the apparatus, of the computer memory, and of their environment. (The quantum information is carried by the wave function that guides the Bohmian particles, not by Bohmian particles themselves.)

7. Essentially, that's because the successor carries the same quantum information as does the original photon A.

8. All Bohmians agree that there is effective (FAPP) collapse, and also that there is no fundamental collapse. This collapse takes place whenever a measurement is performed, so, since the swap operations are measurements, it depends on the ordering of operations.

2. I'll hold you to this.

3. I'll hold you to this.

4. I'll hold you to this.

5. Thanks as always!

6. I'll hold you to this.

7. I'll hold you to this.

8. I'll hold you to this.

But: if anyone makes assertions about BM, hopefully I am free to point out what I see are inconsistencies. To quote an old old song by Sly Stone: "Sometimes I'm right and I can be wrong".

My entire discussion on Bohmian theory centers around trying to fit puzzle pieces together. You are very sure of yourself, and there's no good reason you shouldn't be. You are an expert, and certainly well more expert than myself. But if I'm the first person you have encountered that is questioning some of these points, I'd be surprised. Otherwise, to quote Passon (2004): Why isn't every physicist a Bohmian? The answer is: Obviously, all of the pieces don't fit together quite as well as you might hope. Ma and Megidish are not a Bohmians, apparently neither are their co-authors. Maybe Bell was; but if he was, he was not consistently emphatic about it.

So to get to the point: you have said - as I have always understood - that particles have definite positions and their movement (I assume you mean velocity as well) are affected by particles nonlocal to them. All good, reasonable hypotheses and ones that neatly - as I have repeatedly expressed - accounts for some details of what is generally called "quantum nonlocality". Hey, it's explicit! That's a good thing!!

But the devil is in the details. You can provide formula after formula to "prove" any single point, I will assume that you can for discussion purposes. But is it a fair application of the underlying rule? You say above: Some particles contribute more, some particles contribute less. A "reasonable" statement, for sure, easily accepted by me. But why? Apparently - and correct me if I'm wrong - it's not a matter of separation distance. Do you really need a formula to show me that? It's obvious from experiment*! Because we both know there is no exact formula to demonstrate the precise Bohmian effect individual particle A has on each individual particle B, and vice versa.

Or, if I'm off base, just tell me I am. Again, I'll accept whatever you tell me. But... I really don't think my math skills are the issue here.

*If such influence did depend on proximity: perfect correlations wouldn't exist when particles travel a hundred kilometers past mountains, the sea, buildings, etc.

Could you cite a quote from the Huggett paper or other Bohmians saying photons (specifically) have a definite position? I believe in the Huggett paper, he is just using "some 2-state system" to develop the formalism (same as Peres actually, who specialized to spin-1/2 in his original DCES paper) but did not specialize to photons.

The situation of "the location of a photon" is quite complicated, even in "standard" QM/QFT. And I noted this to you a long time ago (maybe you remember?). I would assume that the rigorous way to treat photons in Bohmian Mechanics is also via a field-theoretic approach. However I am entirely unfamiliar with Bohmian Field Theory. Above my paygrade.

Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km

To be fair: would we really want to say "photons don't have definite positions, but electrons do" as an out? Surely if we perform an entanglement experiment on electrons moving sub-lightspeed, we'll get the same kind of Bohmian prediction as with photons. Or perhaps, is your assertion is that there is a substantive difference?

And... exactly what would a "field-theoretic" approach look like in BM anyway? Presumably, the pilot wave equation - a la Norsen's (2) - does it all. Also above my pay grade... by many orders of magnitude.

Whether the Bohmian particles have more roles to play than this is unclear. They cannot be observed directly, because … have you ever seen a Bohmian particle? What you can see are results of interactions with a subsystem. Those interactions are encoded in the Hamiltonian of the subsystem, and this Hamiltonian only acts on the wavefunction for the few particles of the subsystem that interests you.

So for me, a definite position means a tuple of three real numbers that I can plug into the wavefunction. If I should want more (i.e. some connection to observation), I would have to blur the position. For example by some variance determined by the wavefunction.

1. Bohmian particles have positions which can be plugged into the 3n-dimensional n-particle wavefunction. If you do this for all particles which currently don‘t interest you much, you end up with a wavefunction for the one or two (or four) particles which do interest you.

Whether the Bohmian particles have more roles to play than this is unclear. They cannot be observed directly, because … have you ever seen a Bohmian particle? What you can see are results of interactions with a subsystem. Those interactions are encoded in the Hamiltonian of the subsystem, and this Hamiltonian only acts on the wavefunction for the few particles of the subsystem that interests you.

2. So for me, a definite position means a tuple of three real numbers that I can plug into the wavefunction. If I should want more (i.e. some connection to observation), I would have to blur the position. For example by some variance determined by the wavefunction.

a) All particles in the universe contribute. (Goldstein)
b) Only the entangled particles and their measuring apparati contribute. (@Demystifier)
c) Only the first measured entangled partner contributes. (Norsen)

Just saying.

2. I don't follow. Why blur? The outcome isn't blurred. |H> or |V> is pretty definite. Perfect correlations are pretty definite.

Umm, exactly which measuring apparati have particle positions that interact with the photon (or any particle type) and predetermine spin? Photons 1 and 4 go through a polarizer (PBS) and then a detector. Photons 2 and 3 go into a 50:50 beam splitter* (BS), and then through a polarizer (PBS), and then a detector. 4 detectors must click, after the 4 photons go through 5 total beam splitters.

Care to explain how all these apparati ("made of a huge number of particles") influence the outcomes? Because I don't think their component particles do** - only their overall orientation is a variable. And absolutely nothing else. I can send them through fiber, through wave plates, mirrors, plain glass, etc. None of which has any effect on statistics.

* A "minor" detail: The PBSs for photons 1 and 4 are oriented on an unbiased basis relative to those PBSs of photons 2 and 3.
** Individual component particles of a PBS don't really carry any information about its overall orientation.

I would be interested to see how Bohmian Mechanics deals with photons in particular since the "particles are real" ontology, at least naively, seems in conflict with our difficulty in localizing a photon. I assume we have to invoke some kind of Bohmian Field Theory. But does that mean we no longer take particles as the ontologically real thing? I have no idea. In fact, in writing my previous post, I was hoping an expert in BM could enlighten both of us. :)

This question might not have any bearing on the DCES discussion itself, but it is an interesting one to me.

1. Well, I am interested in all the particles in the universe which contribute to entangled photon polarization outcomes. I am hearing 3 different answers (all previously quoted):

a) All particles in the universe contribute. (Goldstein)
b) Only the entangled particles and their measuring apparati contribute. (@Demystifier)
c) Only the first measured entangled partner contributes. (Norsen)

Just saying.

And... exactly what would a "field-theoretic" approach look like in BM anyway? Presumably, the pilot wave equation - a la Norsen's (2) - does it all. Also above my pay grade... by many orders of magnitude.