# BM/dBB and Entanglement, Non-contextuality

Gold Member
1. I have previously stated my opinion that Bohmian Mechanics (BM, or dBB), insofar as it is a hidden variable theory (HV), must be non-contextual. On the other hand, most Bohmians consider BM to be contextual in order to satisfy the requirements of Bell, Kochen-Specker (KS), etc.

It seems contradictory to me that: BM asserts there are well-defined values for all observables at all times, yet those values are both a) contextual (dependent on how measured); and b) match the usual QM predictions (such as the cos^2(theta) relationship for photon polarization).

2. Today, Ghose introduced a paper entitled: "On Entangled Multi-particle Systems in Bohmian Theory". From the abstract:

"Arguments are presented to show that in the case of entangled systems there are certain difficulties in implementing the usual Bohmian interpretation of the wave function in a straightforward manner. Specific examples are given."

From the paper:

"The analyses of the two constrained systems considered above in this paper show that the Quantum Equilibrium Hypothesis (QEH) cannot be invoked for wave functions of entangled multiparticle systems in general. Wave functions (3) and (13) are examples. In the first example (3) the initial conditions cannot be chosen to fit the quantum mechanical distribution at arbitrary times, and in the second example (13) the particle distribution is incompatible with QEH at all times, independent of initial conditions."

This conclusion seems reasonable to me. It would seem that there should be initial conditions which lead to entangled states compatible with QM; which Ghose says is not true for his example.

3. In other words, I think that BM will always have problems with physical explanations of entanglement scenarios. Which was, in fact, the best thing it had going for it after Bell. I guess my question is this: are there other ways to formulate non-local theories (which are compatible with the QM predictions) OTHER than the Bohmian approach? Perhaps there is a another kind of non-local interaction which comes into play when an observation is made? In the Bohmian approach, the particle positions are paramount; but perhaps there is something else which happens which is explicitly non-local. If you want non-local, is Bohmian the only choice?

Demystifier
Gold Member
I guess my question is this: are there other ways to formulate non-local theories (which are compatible with the QM predictions) OTHER than the Bohmian approach? Perhaps there is a another kind of non-local interaction which comes into play when an observation is made? In the Bohmian approach, the particle positions are paramount; but perhaps there is something else which happens which is explicitly non-local. If you want non-local, is Bohmian the only choice?
There certainly are other possibilities. For example, spontaneous objective collapse theories (like Ghirardi-Rimini-Weber theory) are a large class of such theories. In these theories, a collapse of the wave function is something that really happens, irrespective of measurements and observers. This collapse is a nonlocal process. In such theories, the Schrodinger equation is replaced by a modified equation that contains an additional non-linear term. In principle, such theories can be distinguished from standard QM experimentally.

Demystifier
Gold Member
It seems contradictory to me that: BM asserts there are well-defined values for all observables at all times, yet those values are both a) contextual (dependent on how measured); and b) match the usual QM predictions (such as the cos^2(theta) relationship for photon polarization).
First, BM does NOT assert that there are well-defined values for ALL observables at all times. Instead, it asserts this only for SOME observables - particle positions and velocities.
Second, I see no contradiction. The act of measurement is a physical process, so it would be much more strange if it did NOT affect the values of physical quantities.

Demystifier
Gold Member
By the way, what Ghose says is a nonsense.

There is a simple general theorem that says that Ghose cannot be right, but Ghose claims he found a (not so simple) counterexample, and yet he does not discuss why then the general theorem fails. This is like discovering a complicated perpetuum mobile without discussing why the simple general theorem of energy conservation fails in this case.

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vanesch
Staff Emeritus
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Actually, it is an interesting question as to whether BM is to be considered contextual or non-contextual. If I have it right, non-contextual means that the result of measurement (or its probability or the like) of a certain quantity is independent of the fact whether one extracts "other" information at the same time.

For instance, if I do a measurement that answers the question: is a certain physical quantity (say, X), smaller or larger than 5.0 (a one-bit answer), then a theory that gives me a result (a probability of having "yes" or "no" to that question) would be non-contextual if, when ever I'd perform *another* measurement, say: is X in the interval (-inf,-5), (-5,0), (0,5) or (5,inf) (a two-bit answer), the prediction of its answer would be the same, whether I do this more refined experiment and then add together the probabilities for the first three results, and call that "the result that X is smaller than 5.0", and take the fourth, and call that "a measurement to find "the result that X is larger than 5.0".

In other words, the fact of measuring X with larger precision doesn't change (after lumping together) the result of measuring X with a coarser precision.

If that's the case, then we call the theory non-contextual.

If, on the other hand, the probability of having X smaller than 5.0 is depending on the precision by which we measure X, then the theory is contextual.

Now, instead of "increasing precision of measuring X", you can just as well add "measuring another variable we're a priori not interested in".

vanesh, but measuring value with a higher precision requires aother experiment setup: more energetic particles/shorter wavelength. So how we can compare apples to oranges?

vanesch
Staff Emeritus
Gold Member
vanesh, but measuring value with a higher precision requires aother experiment setup: more energetic particles/shorter wavelength. So how we can compare apples to oranges?

I see what you mean, as in HEP experiments. However, the idea is not on the level of the experiment (the phenomenon: a higher-energy collision is a different physical phenomenon from which it is not always evident how to derive lower-energy information) - but rather on the level of the detection. If you have a detection process which is coarser-grained than another one, do both give (after rebinning say) the same result ?
Standard quantum mechanics is supposed to be non-contextual (see Gleason's theorem and all that). If you measure the position of a particle's position with a slit + detector, then you should find the same distribution if the slit is relatively large than if you make the slit finer, and convolve the result with the aperture of the larger one.
Mind you, I'm talking about detecting the particle there - I'm not talking about what would happen afterwards. The particle will of course be in a different state if it interacted with a fine slit or a large one. So we're not talking about the influence a fine or a coarse experiment might have on the state of the system ; we're talking about the result we would obtain by doing a coarse or fine measurement.

Well, well, well, I have an impression that it has some interpretation-dependent flavor. Like questions 'If I registered something, but did not look at the indicator, did collapse occur?'

As I understand you are talking about making some measurement but then ignoring some information. If we use the Decoherence approach to the measurement, then in the basis of the device itself (device which got 2 bits, complete info) there is one result, but in the observer's basis (observer gets only 1 bit) the result is different (but the second bit of info can leak and decoherence him a little bit later)

In any case the definition of 'contextuality' uses measurement, so it must be defined what approach is used: CI collapse or Quantum Decoherence, and for QD what basis is used.

Demystifier
Gold Member
1. I have previously stated my opinion that Bohmian Mechanics (BM, or dBB), insofar as it is a hidden variable theory (HV), must be non-contextual.
To resolve this, I think it is crucial to distinguish two seemingly similar concepts: measurement and observation.
Hidden variables (including the Bohmian ones) are contextual in the sense that they depend on measurements, but they are non-contextual in the sense that they do not (necessarily) depend on observations. The difference is the following: Measurement is a physical process that may occur even without conscious observers. By contrast, observation is something that happens in the mind of a conscious observer.

By the way, what Ghose says is a nonsense.

There is a simple general theorem that says that Ghose cannot be right, but Ghose claims he found a (not so simple) counterexample, and yet he does not discuss why then the general theorem fails. This is like discovering a complicated perpetuum mobile without discussing why the simple general theorem of energy conservation fails in this case.
He makes it quite clear why the general theorem fails. In fact, that is what the entire paper is about. The general theorem requires three things. He then argues that these three things may not be compatible, then proceeds to prove this is the case with explicit examples.

If you think what he wrote is nonsense then please explain here where his math is incorrect.

I guess my question is this: are there other ways to formulate non-local theories (which are compatible with the QM predictions) OTHER than the Bohmian approach? Perhaps there is a another kind of non-local interaction which comes into play when an observation is made? In the Bohmian approach, the particle positions are paramount; but perhaps there is something else which happens which is explicitly non-local. If you want non-local, is Bohmian the only choice?
I would be interested in discussing this. The only paper I know of that tackles such questions is Leggett's inequality.

"Nonlocal hidden-variable theories and quantum mechanics: An incompatibility theorem"
Found. Phys. 2003, vol 33, pg 1469

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vanesch
Staff Emeritus
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As I understand you are talking about making some measurement but then ignoring some information.

No, not really, because you cannot "ignore" information. You can decide not to take it into account, but it is (as you say, in decoherence) "registered somewhere".

The point is that your measurement did not really "distinguish", for fundamental reasons, between "different outcomes". For instance, look at a mirror, and measure the recoil of a photon. If you measure the recoil, it means that you "measured" the position of the photon to be "on the mirror". But you don't know (and there's no way to find out - even in principle) where exactly on the mirror. Now if you cut your mirror in 4 independent mirrors, you now have a different setup, that does allow you to know the position of the photon better. And then the question is: is the probability of seeing your photon "on one of the 4 sub-mirrors" (so the sum of the probabilities of mirror 1, mirror 2, mirror 3 and mirror 4) in this second experiment going to be different from the probability of seeing the photon on the large mirror in the first experiment.

If the answer to the question is "yes" then the theory is contextual, otherwise (if the probabilities are the same) the theory is non-contextual.

As far as I know, quantum mechanics will make this the same outcome, and hence it will be non-contextual.

Of course, the *state* of the world is different afterwards.

But if you calculate that the probability to detect a photon on the large (uncut) mirror is 28%, then it is hardly thinkable that you would find, in quantum theory, that if you cut the mirror in 4 pieces, that the probability to detect the photon on the first one, to be 10%, on the second, to be 15%, on the third, to be 10% and on the fourth, to be 5% (with a total of 10+15+10+5 = 40%).

It is not because you cut your mirror into 4 pieces that the probability of seeing the photon on one of them is 40%, while if you keep it as one single mirror, that it is going to be 28%.

Demystifier
Gold Member
He makes it quite clear why the general theorem fails. In fact, that is what the entire paper is about. The general theorem requires three things. He then argues that these three things may not be compatible, then proceeds to prove this is the case with explicit examples.

If you think what he wrote is nonsense then please explain here where his math is incorrect.
He says that velocity conditions imply constraints on particle positions. However, the truth is that
1. Velocity conditions do NOT imply constraints on INITIAL particle positions.
2. Consequently, the initial particle positions can be chosen as required by his (iii).
3. Therefore, if these initial positions are chosen as in 2., from (iii) it follows that particle positions have the quantum distribution for ANY t.
Q.E.D.

1. Velocity conditions do NOT imply constraints on INITIAL particle positions.
Given the positions and wavefunctions, the velocities are known. For the velocity conditions to be held, therefore only the positions which satisfy that condition are allowed.

Demystifier
Gold Member
Given the positions and wavefunctions, the velocities are known. For the velocity conditions to be held, therefore only the positions which satisfy that condition are allowed.
To understand why this reasoning is wrong, let me make an analogous (wrong) statement in classical Newtonian mechanics:
Given the positions and forces, the accelerations are known. For the acceleration conditions to be held, therefore only the positions which satisfy that condition are allowed.

Do you understand the mistake now?

To understand why this reasoning is wrong, let me make an analogous (wrong) statement in classical Newtonian mechanics:
Given the positions and forces, the accelerations are known. For the acceleration conditions to be held, therefore only the positions which satisfy that condition are allowed.

Do you understand the mistake now?
Your analogy doesn't make sense. In Newtonian mechanics, there is no entanglement and therefore, for lack of a better phrase, the accelerations are separable. Thus you cannot obtain an acceleration constraint. If you however restrict yourself to systems where particles meet some acceleration condition, then yes indeed only the positions which satisfy that condition are allowed.

You haven't shown anything. Your "proof" was to, in the first statement, disagree with Ghose's work. You wrote:
"He says that velocity conditions imply constraints on particle positions. However, the truth is that
1. Velocity conditions do NOT imply constraints on INITIAL particle positions."

Ending your statements with QED does not make it a proof. All you did was state that you disagree with him. You did not prove any of your statements.

Ghose mathematically derived the constraint on the positions. Please don't argue with analogies, for it will only derail the conversation. Argue mathematically by showing where his math is incorrect. I find his well laid our argument far more convincing that your blunt statement he is wrong.

Demystifier
Gold Member
Now I have also explicitly identified the mistake in the Ghose paper.
The mistake is in the interpretation of Eq. (12). By construction, this equation is valid for a specific pair of particles that satisfy x1-x2=0 at time t0. So far so good. But after writing this equation he assumes that this equation is generally valid, i.e., that it is valid for EVERY PAIR of particles. But this is wrong. Another pair of particles may satisfy x1-x2=0 at some other time t'0 not equal to t0.

Hmm... that does sound convincing.
Thank you for taking the time to explain it.

What about the one with the time independent constraint? Your counter-argument does not work there.

Demystifier
Gold Member
In Newtonian mechanics, there is no entanglement and therefore, for lack of a better phrase, the accelerations are separable.
This is not true. For example, Newtonian gravity is based on a potential of the form
V(x1,x2)=const/|x1-x2|
This provides an instantaneous nonlocal force between the particles, so accelerations of the two particles are not independent. This is very much analogous to the entanglement.

Demystifier
Gold Member
Hmm... that does sound convincing.
Thank you for taking the time to explain it.

What about the one with the time independent constraint? Your counter-argument does not work there.
The mistake in his second example is even more trivial.
Eq. (28) is the condition needed for separability. That's fine. However, separability is not necessary for integrability. Therefore, contrary to the Ghose's claim, (28) is NOT the integrability condition. The equations are integrable even when (28) is not valid.

This is not true. For example, Newtonian gravity is based on a potential of the form
V(x1,x2)=const/|x1-x2|
This provides an instantaneous nonlocal force between the particles, so accelerations of the two particles are not independent. This is very much analogous to the entanglement.
No, it is not analogous. You are constraining yourself to a system in which the accelerations are related a certain way. So, again, yes indeed only the positions which validate those acceleration constraints are allowed. But because the positions are not entangled, ALL positions validate those conditions.

The mistake in his second example is even more trivial.
Eq. (28) is the condition needed for separability. That's fine. However, separability is not necessary for integrability. Therefore, contrary to the Ghose's claim, (28) is NOT the integrability condition. The equations are integrable even when (28) is not valid.
This one is not as clear to me. Rereading, I too do not follow his argument there. I will think about this some more as I may be missing something.

Thank you for pointing that out.

Demystifier
Gold Member
Now I have also sent an e-mail to Ghose. I am curious about his response.

Gold Member
There certainly are other possibilities. For example, spontaneous objective collapse theories (like Ghirardi-Rimini-Weber theory) are a large class of such theories. In these theories, a collapse of the wave function is something that really happens, irrespective of measurements and observers. This collapse is a nonlocal process. In such theories, the Schrodinger equation is replaced by a modified equation that contains an additional non-linear term. In principle, such theories can be distinguished from standard QM experimentally.

Thanks for this. For anyone who is interested, I found a couple of useful links on this subject:

Collapse Theories (by Ghirardi)

On the Common Structure of Bohmian Mechanics and the Ghirardi–Rimini–Weber Theory (by well known experts on both)

One possibilty is that the entangled particles correlated states do not percieve a separation, alternatively said, a spacetime separation is not the one that correlated states are using to remain correlated.

i.e. not a Lorentz covariant type of 'space' - SR, GR, relativistic spacetime rather an underlying quantum state space (which is not actually a 3D space at all!).

We can see this when enatngled particles of different ages (due to one taking a high speed trip) still have exactly correlated states. i.e. there is a disconnect between the particles 'age' and its entangled correlation of quantum statetes.

Some recent research in QFT (somewhere in this forum too) is leading in this direction although it does not specifically say what I said, it appears (IMO) to point in that direction:
“Reconciling Spacetime and the Quantum: Relational Blockworld and the Quantum Liar Paradox,” W.M. Stuckey, Michael Silberstein & Michael Cifone, Foundations of Physics 38, No. 4, 348 – 383 (2008), quant-ph/0510090.