Bohmian Prediction of Bell Inequality Violations

[Amadeo]

As you all know, one can predict violations of Bell's inequality and the CHSH inequality using standard quantum theory.

I am curious as to whether Bohm's hidden variables theory can also predict these violations. I have heard many times that Bohm's theory makes all of the same predictions as standard quantum theory. But, I would like to see for myself that this is indeed true with respect to violations of Bell's inequality and the CHSH inequality.

Can anyone provide a reference that includes a full demonstration that Bohm's HVT predicts violations of Bell's inequality and/or the CHSH inequality?

Thank you.

 

[PeterDonis]

I have heard many times that Bohm's theory makes all of the same predictions as standard quantum theory.

Yes. That's because it uses the same or equivalent math to "standard quantum theory". Same math, same predictions.

 

[Amadeo]

Yes. That's because it uses the same or equivalent math to "standard quantum theory". Same math, same predictions.

Very good. Can you provide a reference which demonstrates how Bohm's hidden variable theory involves the same math and predictions respecting violations of the Bell and CHSH inequalities?

 

[PeterDonis]

Can you provide a reference

Where have you looked?

 

[Amadeo]

Where have you looked?

Google scholar

 

[PeterDonis]

Google scholar

What did you find?

And have you tried arxiv.org?

 

[Amadeo]

Nothing.

Yes.

 

[PeterDonis]

Nothing.

Nothing at all on Bohmian mechanics, period? I strongly doubt that.

If you have found sources on Bohmian mechanics but they haven't answered your question, you should be able to find some specific parts of those sources where you think they should have answered it but didn't, or where they gave what they claim is an answer but you think it's incomplete. Just saying "I found nothing" is not credible. Bohmian mechanics is a well-known interpretation that has been around for decades, and has a lot of literature on it, plenty of which discusses how its predictions compare with those of "standard quantum theory".

Yes.

Same response as above.

 

[PeterDonis]

I have heard many times that Bohm's theory makes all of the same predictions as standard quantum theory.

Where have you "heard" this? Please give references.

 

[Amadeo]

Nothing at all on Bohmian mechanics, period? I strongly doubt that.

If you have found sources on Bohmian mechanics but they haven't answered your question, you should be able to find some specific parts of those sources where you think they should have answered it but didn't, or where they gave what they claim is an answer but you think it's incomplete. Just saying "I found nothing" is not credible. Bohmian mechanics is a well-known interpretation that has been around for decades, and has a lot of literature on it, plenty of which discusses how its predictions compare with those of "standard quantum theory".


Same response as above.

I didn't say I found nothing on Bohmian Mechanics in general. I found nothing about the the subject with which this thread is concerned-- which is, again, precisely how Bohm's theory can be used to predict Bell and CHSH inequality violations.

As for where I have heard that Bohm's theory is equivalent in its predictions to standard quantum theory, I do not remember specifically where I have heard it, besides, of course, in your first post a few minutes ago.

My inquiry is very simple. If you know where I might be able to find a derivation of Bell and CHSH inequality violations using Bohm's theory, please let me know. If not, I see no discernible reason for you to post in this thread.

 

[PeterDonis]

I didn't say I found nothing on Bohmian Mechanics in general. I found nothing about the the subject with which this thread is concerned-- which is, again, precisely how Bohm's theory can be used to predict Bell and CHSH inequality violations.

And that is not credible either, because, as I said, there is plenty of discussion in the literature about how Bohmian Mechanics uses equivalent math to "standard quantum theory" and therefore makes the same predictions. That applies to every prediction, which means it applies to the predictions of Bell and CHSH inequality violations.

As for where I have heard that Bohm's theory is equivalent in its predictions to standard quantum theory, I do not remember specifically where I have heard it

Really? You've read literature on Bohmian Mechanics and can't remember where you "heard" this? That's not credible either.

If you know where I might be able to find a derivation of Bell and CHSH inequality violations using Bohm's theory, please let me know.

My point is that, if you actually have searched the literature as you say you have, you have already found enough information to answer your question. So the fact that you're still asking it makes me think one of two things: either you haven't actually read any of the literature, in which case you need to go do that so you can ask a more focused question based on some specific reference (particularly in a thread that you marked as "A" level, which means you already are supposed to have graduate level knowledge of the subject matter), or you have some other agenda.

 

[PeterDonis]

I see no discernible reason for you to post in this thread.

I am posting because I am a moderator and I find it difficult to understand why you are asking the question in the first place in an "A" level thread, given what I said in my previous post.

 

[renormalize]

I am curious as to whether Bohm's hidden variables theory can also predict these violations. I have heard many times that Bohm's theory makes all of the same predictions as standard quantum theory. But, I would like to see for myself that this is indeed true with respect to violations of Bell's inequality and the CHSH inequality.

You might take a look at:
Gisin, Nicolas. "Why Bohmian mechanics? One-and two-time position measurements, Bell inequalities, philosophy, and physics." Entropy 20.2 (2018): 105.
and the references therein.

 

[Amadeo]

I am posting because I am a moderator and I find it difficult to understand why you are asking the question in the first place in an "A" level thread, given what I said in my previous post.

This is all so tiresome. I have no secret agenda. There is no conspiracy. I have looked around and not found what I am looking for. I do have graduate training in the subject. If you think that I miscategorizeed the thread, then by all means change it-- I couldn't care less. There is no legitimate reason for you to be so curt and impolite. The question is as simple and focused as can be: if anyone knows a place where I can find a Bohmian derivation of Bell and CHSH inequality violations, please let me know. What kind of forum is this when such a straightforward and legitimate question elicits such a rude response from the moderator?

 

[Amadeo]

You might take a look at:
Gisin, Nicolas. "Why Bohmian mechanics? One-and two-time position measurements, Bell inequalities, philosophy, and physics." Entropy 20.2 (2018): 105.
and the references therein.

Thank you.

 

[PeterDonis]

I have looked around and not found what I am looking for.

And that, by itself, is not enough information. What have you found? What sources have you read? Hasn't any source you've read discussed anything about how the predictions of Bohmian mechanics compare with those of "standard quantum theory"?

It would be very helpful if you would provide more information about these things than just "I found nothing".

 

[PeterDonis]

Moderator's note: Thread level changed to "I".

 

[Demystifier]

I have heard many times that Bohm's theory makes all of the same predictions as standard quantum theory. But, I would like to see for myself that this is indeed true with respect to violations of Bell's inequality and the CHSH inequality.

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.

 

[Demystifier]

You might take a look at:
Gisin, Nicolas. "Why Bohmian mechanics? One-and two-time position measurements, Bell inequalities, philosophy, and physics." Entropy 20.2 (2018): 105.
and the references therein.

Interesting paper, but probably not what OP wants.

 

[Demystifier]

Also an analogy could help. Suppose that someone said: "I have heard that statistical mechanics always makes the same predictions as thermodynamics. But I want to see how statistical mechanics explains the working of refrigerator. Is there a paper that explains the working of refrigerator by statistical mechanics?" It is likely that there is no such paper, and those who understand both thermodynamics and statistical mechanics will understand why.

 

[PeroK]

It all hinges on the probability of $\cos^2 \frac \theta 2$ for a measurement of spin at an angle $\theta$ from some reference direction. If you derive that using Bohmian Mechanics, either directly or indirectly, then all else follows.

 

[Demystifier]

It all hinges on the probability of $\cos^2 \frac \theta 2$ for a measurement of spin at an angle $\theta$ from some reference direction. If you derive that using Bohmian Mechanics, either directly or indirectly, then all else follows.

For spin measurements in Bohmian mechanics see https://arxiv.org/abs/1305.1280. It is quite general, so there is no derivation of the formula you mention above. But with results of this paper, derivation of this formula, or of violation of Bell/CHSH inequalities, should be straightforward.

 

[Amadeo]

Also an analogy could help. Suppose that someone said: "I have heard that statistical mechanics always makes the same predictions as thermodynamics. But I want to see how statistical mechanics explains the working of refrigerator. Is there a paper that explains the working of refrigerator by statistical mechanics?" It is likely that there is no such paper, and those who understand both thermodynamics and statistical mechanics will understand why.

Thank you for your response. I think the analogy is useful, but I do not think it fully applies.

Bohmian mechanics involves the assumption of a guiding equation, which is derived from the position space wavefunction of a system, which, of course, is derived via the Schrodinger equation. Assuming certain initial distributions of the initial conditions of the particles in a system governed by such a guiding equation, the probability distribution of particle trajectories will match the shape of the position space wavefunction. Therefore, all of the predictions of Bohmian mechanics will match those of standard quantum theory, (in analogy to the way in which all of the predictions of statistical mechanics will match all of those of thermodynamics). However, we can only be assured that this is the case as long as the predictions of standard quantum theory make use only of a position space wavefunciton. There are certain predictions made in standard quantum theory which make no use of the position space wavefunction whatsoever.

Take, for instance, the standard QM prediction of the expectation value of the product of the outcome of two spin measurements on a singlet state in a Bell test experiment, (which turns out to be proportional to the cosine of the angle between the orientations of the detectors-- in violation of Bell's inequality). This prediction makes no use of position wavefunctions whatsoever. It makes use only of spin operators and the singlet state. Unlike the eigenstates of regular angular momentum, the spin eigenstates are not expressible as position space wavefuncitons. Therefore, in this example, there is no position space wavefunction that we can use to derive the Bohmian guiding equation which is necessary to make predictions in accordance with Bohm's hidden variable theory in the first place. Therefore, it is not at all clear (to me) how Bohmian mechanics is to predict Bell inequality violations.

That said, if there were another way to predict Bell inequality violations using only position space wavefunctions in standard quantum mechanics, then, of course, it would be trivial to see how Bohm's theory would make the same prediction.

It seems that one might predict Bell inequality violations using only position wavefuncitons in standard QM in the following way: Properly define the initial conditions and Hamiltonian for a system of a pair of spin 1/2 particles each moving through differently oriented Stern-Gerlatch magnetic fields. From this derive a position space wavefunction which turns out to give you the right probabilities that the particles will move up or down in their respective analyzers in a way that reproduces the result that the expectation values of the product of the measurements violates Bell inequalities (by being proportional to the cosine of the angle between the orientations of the SG analyzer fields).

However, it seems to me that this is not a viable approach, since there seems to be no difference between the Hamiltonian for such a system in which the two particles are in the entangled singlet state (in which case Bell inequality violations are observed), and the Hamiltonian for such a system in which the two particles are just in a mixed state of spin up and spin down (in which case Bell inequality violations are not observed).

Thank you for the reference. I will see if it answers some of these questions.

 

[renormalize]

Unlike the eigenstates of regular angular momentum, the spin eigenstates are not expressible as position space wavefuncitons. Therefore, in this example, there is no position space wavefunction that we can use to derive the Bohmian guiding equation which is necessary to make predictions in accordance with Bohm's hidden variable theory in the first place. Therefore, it is not at all clear (to me) how Bohmian mechanics is to predict Bell inequality violations.

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."

 

[PeterDonis]

we can only be assured that this is the case as long as the predictions of standard quantum theory make use only of a position space wavefunciton

No, that's not correct. Bohmian mechanics does make predictions about observables like spin which are not operators on the position space degrees of freedom. Its underlying explanation of those predictions is very different from standard QM (for example, it accounts for the results of a Stern-Gerlach experiment by appealing to random variation in which part of the incoming wave packet the unobservable position of the particle being measured resides--heuristically, the upper half of the wave packet gets directed to the spin up output channel, and the lower half of the wave packet gets directed to the spin down output channel), but the predictions themselves are the same.

Basically Bohmian mechanics relies on the fact that all measurements ultimately come down to particular events taking place in space and time, i.e., within the degrees of freedom that are covered by the position space wave function; it's impossible to have a measurement of other degrees of freedom that has no effects at all in position space. This argument is made, for example, in Figure 1 of the Gisin paper referenced earlier and the discussion accompanying it (in that figure a beam splitter is shown instead of a Stern-Gerlach apparatus, but the argument is the same).

 

[Amadeo]

No, that's not correct. Bohmian mechanics does make predictions about observables like spin which are not operators on the position space degrees of freedom. Its underlying explanation of those predictions is very different from standard QM (for example, it accounts for the results of a Stern-Gerlach experiment by appealing to random variation in which part of the incoming wave packet the unobservable position of the particle being measured resides--heuristically, the upper half of the wave packet gets directed to the spin up output channel, and the lower half of the wave packet gets directed to the spin down output channel), but the predictions themselves are the same.

Basically Bohmian mechanics relies on the fact that all measurements ultimately come down to particular events taking place in space and time, i.e., within the degrees of freedom that are covered by the position space wave function; it's impossible to have a measurement of other degrees of freedom that has no effects at all in position space. This argument is made, for example, in Figure 1 of the Gisin paper referenced earlier and the discussion accompanying it (in that figure a beam splitter is shown instead of a Stern-Gerlach apparatus, but the argument is the same).

I see that I misunderstood the guiding equation. It is not solely derived from the position space wave function of a system-- in which case the only way that spins involved in a system could affect the behavior of the particles would be by modifying the position space wavefunction through interaction terms in the Hamiltonian. The spin state part of the wavefunction is also a factor in the guiding equation. (Though, you can’t have a guiding equation if the wavefunction for the system has no position space component, while it is possible to have a guiding equation if the wavefunction for the system has no spin component). Even so, the derivation of concrete predictions about the outcome of spin measurements seems to be much more complicated in Bohmian theory than it is in standard QM. In contrast to predictions respecting systems with only position space wavefunctions, is not at all obvious (to me) that the predictions of Bohmian theory will always necessarily match those of standard QM when spin is involved. 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”.

So, I suppose my original question boils down to this: Can the Bohmian guiding equation (in conjunction with suitably chosen initial conditions of the particles) produce predictions of Bell and CHSH inequality violations granting that the spin state part of the wavefunction involved in the guiding equation is a singlet, and the position space part of the wavefunction involved in the guiding equation is defined in a way appropriate to the experimental configuration? If so, I would very much like to see it. As Bell inequality violations are one of the most significant effects predicted by standard QM, it would seem that proponents of the Bohmian theory would, by this time, have rigorously and precisely shown how this effect is also predicted by their theory.

Perhaps my question will be answered once I read through the Norsen paper.

 

[PeterDonis]

is not at all obvious (to me) that the predictions of Bohmian theory will always necessarily match those of standard QM when spin is involved

Again, I recommend looking at Figure 1 of the Gisin paper referenced earlier and the discussion surrounding it.

 

[PeterDonis]

according to Norsen

the Norsen paper

Which paper?

 

[Doc Al]

the Norsen paper.

Are you referring to Norsen's "The Pilot-wave Perspective on Spin”? Here: https://arxiv.org/abs/1305.1280

 

[PeterDonis]

Are you referring to Norsen's "The Pilot-wave Perspective on Spin”? Here: https://arxiv.org/abs/1305.1280

I think so since I now see that @Demystifier referenced that paper in post #22. I hadn't spotted that before.

 

[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.