I Bohmian Prediction of Bell Inequality Violations

[PeterDonis]

I was unaware before today, but, according to Norsen, there is a “widespread belief…among physicists” that Bohmian theory cannot “account successfully for phenomena involving spin”.

Yes, because of a particular "spin" (pun intended) that was put on theorems like Kochen-Specker. The Norsen paper (now that I realize which one it is) discusses this.

From what I can gather, the hidden assumption behind the "widespread belief" Norsen describes was, basically, that because in wave function space the spin degrees of freedom are additional to the position degrees of freedom, any Bohmian-type model that could make predictions about spin would have to involve hidden, unobservable "positions" (or some type of hidden variables) in spin space as well as in position space. Norsen's paper makes clear that that is not the case: there are no hidden variables in spin space in the Bohmian model in addition to the hidden, unobservable positions in position space. The latter are entirely sufficient to make all the same predictions about spin measurements that standard QM makes.

Btw, the "widespread belief" Norsen describes is rather disappointing in view of the fact, which Norsen mentions, that Bohm's paper in 1955 on the pilot wave model included a treatment of how the model makes predictions about spin measurements, which is basically the same one Norsen gives (though Norsen adopts Bell's later formulation).

 

[Demystifier]

There are certain predictions made in standard quantum theory which make no use of the position space wavefunction whatsoever.

That is not really true. First, we usually assume that two entangled particles are far away from each other, and "far away" can only be defined if the wave functions are defined also in the position space. Second, we assume that the spin of each particle is measured, and this can only be true if there is a macroscopic measuring apparatus, with its own wave function in the position space. In practice we usually don't write these position dependences explicitly, but implicitly the position dependences are there, they are assumed tacitly.

So the right question is this. Consider standard (not Bohmian) QM, but take into account that the particles (with spin) and the measuring apparatuses all have wave functions that depend on positions. Does standard QM with all these positions taken into account make the same measurable predictions as "truncated" standard QM written down without positions? If you can understand how standard QM makes the same measurable predictions in the two cases, then you will trivially understand how Bohmian mechanics make the same predictions too. For that purpose, see also the paper in my signature below.

 

[Demystifier]

I can't judge whether this reference: On the Role of Density Matrices in Bohmian Mechanics addresses your questions, but the last line of its abstract is suggestive:
"In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system."

For a related paper see also my recent https://arxiv.org/abs/2308.10500 where spin is treated on an equal footing with all other degrees of freedom that are not measured directly.

 

[DrChinese]

Thank you.

Although the other references are better, Plato is not a bad place to look either:

https://plato.stanford.edu/entries/qm-bohm/#Cont

 

[stevendaryl]

There is probably no reference which explicitly does what you want. Not because it is hard, but because it is trivial. Once you know the general explanation why Bohm's theory always makes the same predictions as standard quantum theory, applying this to the special case of CHSH inequality is trivial. If you want to understand how Bohm's theory explains violation of CHSH, I recommend you to first study Bohm's theory in general. You cannot understand a special case if you don't understand the general theory.

In my opinion the equivalence between Bohmian mechanics and standard QM is not as trivial as that.

Let me take as standard QM the following recipe (I think due to Von Neumann):

1. We describe the system we are interested in as a wave function (or more generally, a density matrix, but I'm going to assume a pure state here).
2. We let the wave function evolve under Schrodinger's equation until the time of measurement.
3. We perform a measurement, which gives an eigenvalue of the operator corresponding to the observable being measured.
4. The probability for each possible value is given by the Born rule (the square of the amplitude corresponding to that value)
5. Afterwards, we use a collapsed wave function for future measurements.

The Bohmian model was created to give exactly the same results as this recipe except for two differences:

Point A: at step 3, the Bohmian model was only constructed to be equivalent to the standard recipe in the special case in which the experimenters measure particle positions.

Point B: the Bohmian model doesn't have step 5.

I can certainly believe that it's true that the Bohmian model makes the same predictions as the standard recipe, but because of points A and B, demonstrating this seems far from trivial. I know the hand-wavy argument that all measurements ultimately boil down to position measurements (or we can make it so, by arranging the experiment so that systems go one direction if they are in one state and a different direction if they are in another state).

Point B is, I think, complicated to prove rigorously. Suppose you have a multipart measurement. For example, we have two entangled particles, and we measure one property of one particle and then at a later time, we measure a different property of the other particle. The recipe above would say that we collapse the wave function at the first measurement, and then use the collapsed wave function to compute probabilities for the second measurement. An alternative approach is to consider the two measurements as a single compound measurement. Then we only need to apply the Born rule to the compound measurement, and we don't need the collapse rule. So the compound measurement approach would (I assume) give the same result as the Bohmian model (if point A is taken care of). But it's nontrivial (at least, I don't know of a trivial proof) to show that the one at a time measurements with a collapse in the middle gives the same result as the single compound measurement.

 

[stevendaryl]

In my opinion the equivalence between Bohmian mechanics and standard QM is not as trivial as that.

Let me take as standard QM the following recipe (I think due to Von Neumann):

1. We describe the system we are interested in as a wave function (or more generally, a density matrix, but I'm going to assume a pure state here).
2. We let the wave function evolve under Schrodinger's equation until the time of measurement.
3. We perform a measurement, which gives an eigenvalue of the operator corresponding to the observable being measured.
4. The probability for each possible value is given by the Born rule (the square of the amplitude corresponding to that value)
5. Afterwards, we use a collapsed wave function for future measurements.

The Bohmian model was created to give exactly the same results as this recipe except for two differences:

Point A: at step 3, the Bohmian model was only constructed to be equivalent to the standard recipe in the special case in which the experimenters measure particle positions.

Point B: the Bohmian model doesn't have step 5.

I can certainly believe that it's true that the Bohmian model makes the same predictions as the standard recipe, but because of points A and B, demonstrating this seems far from trivial. I know the hand-wavy argument that all measurements ultimately boil down to position measurements (or we can make it so, by arranging the experiment so that systems go one direction if they are in one state and a different direction if they are in another state).

Point B is, I think, complicated to prove rigorously. Suppose you have a multipart measurement. For example, we have two entangled particles, and we measure one property of one particle and then at a later time, we measure a different property of the other particle. The recipe above would say that we collapse the wave function at the first measurement, and then use the collapsed wave function to compute probabilities for the second measurement. An alternative approach is to consider the two measurements as a single compound measurement. Then we only need to apply the Born rule to the compound measurement, and we don't need the collapse rule. So the compound measurement approach would (I assume) give the same result as the Bohmian model (if point A is taken care of). But it's nontrivial (at least, I don't know of a trivial proof) to show that the one at a time measurements with a collapse in the middle gives the same result as the single compound measurement.

The business of collapse seems to me to be an ambiguity in the application of Bohmian mechanics. The wave function in Bohmian mechanics serves double duty: (1) It provides a "quantum potential" term to the equations of motion for a particle, and (2) Its square gives the initial probability distribution of the particle.

If you do a two-part measurement on a particle, then after the first measurement, you now have more information about where the particle is than you did at the beginning. That means that the probability distribution has changed. Does this additional information change the wave function? If so, then there is a collapse-like effect in Bohmian mechanics, as well. If not, then the correspondence between wave function and probability distributions would be broken.

 

[DrChinese]

If so, then there is a collapse-like effect in Bohmian mechanics, as well.

I have discussed something like this point with @Demystifier previously. He agreed that Bohmian Mechanics is contextual in addition to its usual description as nonlocal. I would say that contextuality implies a "collapse-like effect", which would then occur nonlocally in BM.

Of course, he'll probably prefer to weigh in on that himself.

 

[Demystifier]

Let me take as standard QM the following recipe (I think due to Von Neumann):

1. We describe the system we are interested in as a wave function (or more generally, a density matrix, but I'm going to assume a pure state here).
2. We let the wave function evolve under Schrodinger's equation until the time of measurement.
3. We perform a measurement, which gives an eigenvalue of the operator corresponding to the observable being measured.
4. The probability for each possible value is given by the Born rule (the square of the amplitude corresponding to that value)
5. Afterwards, we use a collapsed wave function for future measurements.

You are missing the crucial part here, the entanglement with wave function of the measuring apparatus (or the environment). This is absolutely essential if you want to understand the von Neumann theory of measurement, the theory of decoherence, the many-world interpretation, and/or the Bohmian interpretation. Without that, in particular, you cannot understand where the effective collapse in Bohmian mechanics comes from, even though there is no true collapse. See e.g. my "Bohmian mechanics for instrumentalists", or my lecture http://thphys.irb.hr/wiki/main/images/e/e6/QFound4.pdf

 

[PeterDonis]

at step 3, the Bohmian model was only constructed to be equivalent to the standard recipe in the special case in which the experimenters measure particle positions.

Kinda sorta. The Bohmian way of describing this is that all measurements end up coming down to measurements of particle positions. For example, if we measure the spin of a spin-1/2 particle using a Stern-Gerlach apparatus, our measurement result is ultimately where the particle is detected on the detector screen downstream of the S-G magnet. So we're really measuring a position, and then inferring a spin from it.

 

[Demystifier]

I would say that contextuality implies a "collapse-like effect", which would then occur nonlocally in BM.

Contextuality arises due to entanglement with wave function of the measuring apparatus. The wave function of the measuring apparatus is also the most important thing to understand the "collapse-like effect". In fact, I claim, the entanglement with wave function of the measuring apparatus is the most underappreciated thing in all of quantum theory. It is so essential for proper understanding of quantum theory, and yet so few physicists are aware of that.

 

[stevendaryl]

You are missing the crucial part here, the entanglement with wave function of the measuring apparatus (or the environment). This is absolutely essential if you want to understand the von Neumann theory of measurement, the theory of decoherence, the many-world interpretation, and/or the Bohmian interpretation. Without that, in particular, you cannot understand where the effective collapse in Bohmian mechanics comes from, even though there is no true collapse. See e.g. my "Bohmian mechanics for instrumentalists", or my lecture http://thphys.irb.hr/wiki/main/images/e/e6/QFound4.pdf

My point is that it isn't trivial to show that the Bohmian model reproduces the same results as standard QM. It isn't an immediate consequence of the model's construction.

The operative word here is "trivial". I'm not denying that it's true, but I am denying that it's trivially true.

 

[stevendaryl]

My point is that it isn't trivial to show that the Bohmian model reproduces the same results as standard QM. It isn't an immediate consequence of the model's construction.

Presumably, the work of showing that Bohmian mechanics is equivalent to standard QM, including the apparent collapse, is probably the same work in showing that MWI makes the same predictions as standard QM. That isn't obvious, either, because MWI doesn't have collapse, either.

 

[stevendaryl]

This actually relates to @PeterDonis claim (in another thread) that by definition, different interpretations of QM must have the same observable consequences. I think that for what's called different interpretations of QM, there actually can be different predictions, but for all practical purposes, it's impossible to observe the differences. For example, collapse versus no-collapse interpretations can in principle make different predictions, because no-collapse interpretations predict the possibility of interference effects involving macroscopically different alternatives. In practices, decoherence makes it impossible to measure such interference.

 

[PeterDonis]

That isn't obvious, either, because MWI doesn't have collapse, either.

Collapse in standard QM, i.e., independent of any interpretation, is just a mathematical step that is taken after the result of a measurement is known, in order to make further predictions. This step is taken when using the MWI or the Bohmian interpretation. Where those interpretations differ from others is in how they explain why that mathematical step works, i.e., allows us to make accurate further predictions.

Some examples:

Copenhagen: Asking what happens "in reality" is a meaningless question. All we have is the mathematical machinery we use to make predictions. In that machinery, we apply the collapse postulate once we know the result of a measurement because that is what we have observed, empirically, to work.

"Objective collapse" interpretations: The mathematical step of applying the collapse postulate once we know the result of a measurement works because an actual, physical collapse process happens when a measurement result is determined.

MWI: The mathematical step of applying the collapse postulate once we know the result of a measurement works because decoherence separates the different "worlds" so they can't interfere with each other, and any observation of a particular result for a measurement limits all further predictions based on that observation to the particular "world" in which that result occurred. So the predictions obtained by applying the collapse postulate only apply to one particular branch of the wave function, not all of it.

Bohmian: Observing a measurement result doesn't change the wave function or the particle positions, but it does reveal information about them that was previously unknown. The mathematical step of applying the collapse postulate once we know the result of a measurement is the way we capture the information revealed by observing the measurement result in order to constrain our further predictions.

 

[Demystifier]

My point is that it isn't trivial to show that the Bohmian model reproduces the same results as standard QM. It isn't an immediate consequence of the model's construction.

The operative word here is "trivial". I'm not denying that it's true, but I am denying that it's trivially true.

OK, but do you think it's true? And if you do, why don't you explain, in your own words, why it is true?

 

[PeterDonis]

It isn't an immediate consequence of the model's construction.

It is if you include the required initial statistical distribution of particle positions in "the model's construction".