Entaglement and hidden variables

[andreab1987]

Hi everybody!

I would like to understand how entaglement is calculated in Bohm's theory.
I know that in Bohm theory, the particle's spin is not an intrinsic property of the particle, but it depends on the global wave function (particle + device used to mesasur the spin).
So, in the case of two electrons in a singlet state, when we change the direction of the device, also the global function is changed, which affects the results of measurements on the second electron.
I need understand if and how this explanation can be given quantitatively in the hideen variables framework, in order to obtain the same results of standard quantum theory.

 

[Demystifier]

I would like to understand how entaglement is calculated in Bohm's theory.
I know that in Bohm theory, the particle's spin is not an intrinsic property of the particle, but it depends on the global wave function (particle + device used to mesasur the spin).
So, in the case of two electrons in a singlet state, when we change the direction of the device, also the global function is changed, which affects the results of measurements on the second electron.

That's all correct.

I need understand if and how this explanation can be given quantitatively in the hideen variables framework, in order to obtain the same results of standard quantum theory.

The global wave function in Bohm's theory described above is calculated in an exactly the same way as in standard quantum theory. Does it help?

 

[andreab1987]

The global wave function in Bohm's theory described above is calculated in an exactly the same way as in standard quantum theory. Does it help?

The problem is that in standard quantum theory it is not necessary to calculate the global function (i.e. the wave function of particles+ device) because it is sufficent to use the collapse of the wave function of the electrons only. I don't understand how the standard quantum results can be obtained in Bohm's approach, without the collapse, since it is certainly not possible to calculate the global function.

 

[Demystifier]

The problem is that in standard quantum theory it is not necessary to calculate the global function (i.e. the wave function of particles+ device) because it is sufficent to use the collapse of the wave function of the electrons only.

In most cases of practical interest, this is true. Yet, in some cases it is necessary to calculate the effects of quantum environment (which can be thought of as a device), because otherwise you cannot know the basis in which the collapse postulate should be applied.

I don't understand how the standard quantum results can be obtained in Bohm's approach, without the collapse, since it is certainly not possible to calculate the global function.

It is certainly not possible to calculate it exactly, but is possible to do it approximately. Anyway, irrespective of the particular result of such a calculation, there is a general theorem: Whatever the result of such a calculation will be, the standard and Bohm approaches will give the same measurable predictions!

 

[andreab1987]

It is certainly not possible to calculate it exactly, but is possible to do it approximately. Anyway, irrespective of the particular result of such a calculation, there is a general theorem: Whatever the result of such a calculation will be, the standard and Bohm approaches will give the same measurable predictions!

Thank you very much for the answer, but of course I cannot accept such a theorem without a convincing proof. Can you give me some link where I can find either the proof of such theorem or a quantitative solution of the entaglement problem with the hidden variables?

 

[Demystifier]

Thank you very much for the answer, but of course I cannot accept such a theorem without a convincing proof. Can you give me some link where I can find either the proof of such theorem or a quantitative solution of the entaglement problem with the hidden variables?

See e.g. the Appendix of
http://xxx.lanl.gov/abs/quant-ph/0208185 [Found.Phys.Lett. 17 (2004) 363]
and references therein.

 

[andreab1987]

See e.g. the Appendix of
http://xxx.lanl.gov/abs/quant-ph/0208185 [Found.Phys.Lett. 17 (2004) 363]
and references therein.

Thank you very much for the information.

I have found also these information:

No entangled pair of electrons has ever been produced. There is a race to do so, and it may have just been accomplished this year using a Cooper Bardeen pair. Everyone is awaiting the first Aspect-like experiment with such a pair. Entaglement has beed experimentally tested only with photons.

Bohm introduced electron pairs into the discussion of entanglement because there is as yet no way to define photon photon paths (or even photon wavefunctions!). This is also discussed in Holland. The natural way would be to use the Poynting vector of the electromagnetic field as the `current', but the photon `density' E^2-B^2 is not necessarily positive. Other formulae based on the e.m. stress tensor for the field are not Lorentz invariant.

I have also found some interesting information here

http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/deBB_10/conference.html

In particular Valentini's talk "In search of a breakdown in quantum theory", explains why Bohm's theory does not reproduce standard quantum theory results.

From what I can understand, hidden variable theory is still very far from a complete and consistent explanation of quantum physics.

 

[Demystifier]

In particular Valentini's talk "In search of a breakdown in quantum theory", explains why Bohm's theory does not reproduce standard quantum theory results.

No, you misunderstood Valentini. Bohm's theory reproduces standard quantum theory, but at the same time Bohm's theory is reacher than standard quantum theory, in the sense that it also contains nonequilibrium solutions which differ from standard quantum theory. Such nonequilibrium solutions have a small probability of actual realization in nature, and it is questionable if we would ever discover them even if Bohm's theory was correct. To say that Bohm's theory does not reproduce standard quantum theory is like saying that kinetic theory of gases
http://en.wikipedia.org/wiki/Kinetic_theory_of_gases
does not reproduce thermodynamics. It does, but it is also reacher than that.

 

[andreab1987]

No, you misunderstood Valentini. Bohm's theory reproduces standard quantum theory, but at the same time Bohm's theory is reacher than standard quantum theory, in the sense that it also contains nonequilibrium solutions which differ from standard quantum theory. Such nonequilibrium solutions have a small probability of actual realization in nature, and it is questionable if we would ever discover them even if Bohm's theory was correct. To say that Bohm's theory does not reproduce standard quantum theory is like saying that kinetic theory of gases
http://en.wikipedia.org/wiki/Kinetic_theory_of_gases
does not reproduce thermodynamics. It does, but it is also reacher than that.

Actally Valentii writes:

(1)
The quantum equilibrium state is unstable in Bohm’sdynamics.
(2) There is no tendency to relax to quantum
equilibrium in Bohm’sdynamics.
(3) If the universe started in a non-equilibrium state, we would (almost certainly) not see equilibrium today, and in particular there would be no bound states (atoms etc)

Conclude: Bohm’sdynamics is untenable
Agrees with QT only if assume very special initial conditions.
The dynamics is unstable, and small deviations from initial equilibrium do not relax.
Small departures from equilibrium (e.g. In the remote past) would in fact grow with time.
If you believe in Bohm’sdynamics, it would be unreasonable to expect to see effective QT today, in contradiction with observation

Then he adds that the situation is different with de broglie original theory.

 

[Demystifier]

Actally Valentii writes:

(1)
The quantum equilibrium state is unstable in Bohm’sdynamics.
(2) There is no tendency to relax to quantum
equilibrium in Bohm’sdynamics.
(3) If the universe started in a non-equilibrium state, we would (almost certainly) not see equilibrium today, and in particular there would be no bound states (atoms etc)

Ah, now I see what you mean. Well, the point is that here, by "Bohm" dynamics, Valentini means something ELSE. More precisely, something that almost nobody really takes seriously, and almost nobody calls "Bohm dynamics". But it is a subtle matter of terminology, so I don't want to spend much time to explain it. In short, what other people call "Bohm dynamics" or "de Broglie-Bohm dynamics", Valentini calls "de Broglie dynamics". In fact, even Bohm himself was not an adherent of what Valentini calls "Bohm dynamics".

 

[andreab1987]

Ah, now I see what you mean. Well, the point is that here, by "Bohm" dynamics, Valentini means something ELSE. More precisely, something that almost nobody really takes seriously, and almost nobody calls "Bohm dynamics". But it is a subtle matter of terminology, so I don't want to spend much time to explain it. In short, what other people call "Bohm dynamics" or "de Broglie-Bohm dynamics", Valentini calls "de Broglie dynamics". In fact, even Bohm himself was not an adherent of what Valentini calls "Bohm dynamics".

Thank you very much for the clarification.

I am expert in standard quantum mechanics, but not in hidden variables.
Since you seem to know well HV, can you confirm my previous statement about photons, i.e.

"there is as yet no way to define photon photon paths (or even photon wavefunctions!). The natural way would be to use the Poynting vector of the electromagnetic field as the `current', but the photon `density' E^2-B^2 is not necessarily positive. Other formulae based on the e.m. stress tensor for the field are not Lorentz invariant."

 

[Demystifier]

Since you seem to know well HV, can you confirm my previous statement about photons, i.e.

"there is as yet no way to define photon photon paths (or even photon wavefunctions!). The natural way would be to use the Poynting vector of the electromagnetic field as the `current', but the photon `density' E^2-B^2 is not necessarily positive. Other formulae based on the e.m. stress tensor for the field are not Lorentz invariant."

I disagree with that statement. It is quite easy to write a Lorentz invariant equation for photon trajectories. However, in some cases, such an equation leads to velocities larger than the velocity of light. Some people find it unacceptable (which is why they think that there is a problem with photon trajectories), but they are wrong. As shown in the paper I mentioned a few posts above, faster than light trajectories are neither inconsistent nor in conflict with observations. As shown there, the theory with faster than light trajectories also explains why the MEASURED velocity cannot be faster than light.

 

[andreab1987]

I disagree with that statement. It is quite easy to write a Lorentz invariant equation for photon trajectories. However, in some cases, such an equation leads to velocities larger than the velocity of light. Some people find it unacceptable (which is why they think that there is a problem with photon trajectories), but they are wrong. As shown in the paper I mentioned a few posts above, faster than light trajectories are neither inconsistent nor in conflict with observations. As shown there, the theory with faster than light trajectories also explains why the MEASURED velocity cannot be faster than light.

Thank you again for your help.

I still have some problems to understand how entaglement is described in the hidden variables framework.
If I consider the simple case of spin singlet |u>|d> - |d>|u> in standard QM (where u means spin up and d spin down) everything is clear. If Observer 1 chooses to measure spin along the z-axis and found up, the wave function collapses so Observer 2 will find spin d if he chooses to measure spin along z while he finds 50% probability d and 50% probability u if he chooses to measure spin along x .
Even if this case is so simple, I do not understand how this situation is described with hidden variables. Could you do that?

 

[Demystifier]

Thank you again for your help.

I still have some problems to understand how entaglement is described in the hidden variables framework.
If I consider the simple case of spin singlet |u>|d> - |d>|u> in standard QM (where u means spin up and d spin down) everything is clear. If Observer 1 chooses to measure spin along the z-axis and found up, the wave function collapses so Observer 2 will find spin d if he chooses to measure spin along z while he finds 50% probability d and 50% probability u if he chooses to measure spin along x .
Even if this case is so simple, I do not understand how this situation is described with hidden variables. Could you do that?

Read the Appendix on the general theory of quantum measurements I have already mentioned. If you understand that, you will also know the answer to your question. If you don't understand that Appendix, ask a question about that Appendix.

 

[andreab1987]

Read the Appendix on the general theory of quantum measurements I have already mentioned. If you understand that, you will also know the answer to your question. If you don't understand that Appendix, ask a question about that Appendix.

I have read the paper at the link you indicated, but there was no appendix.
The title of the paper is : Bohmian particle trajectories in relativistic fermionic quantum field theory

Are you sure you gave the correct link?

 

[Doc Al]

I have read the paper at the link you indicated, but there was no appendix.

The appendix begins on page 11.

 

[andreab1987]

The appendix begins on page 11.

At page 11 there are only the conclusions and bibliography.
We are certainly reading two different papers; the paper I have found at the link you gave

http://xxx.lanl.gov/abs/quant-ph/0208185

is: "Bohmian particle trajectories in relativistic fermionic quantum field theory" by H. Nikolic

The pdf file of this paper can be found at
http://xxx.lanl.gov/PS_cache/quant-ph/pdf/0302/0302152v3.pdf

but there is no appendix.

 

[Doc Al]

At page 11 there are only the conclusions and bibliography.
We are certainly reading two different papers; the paper I have found at the link you gave

http://xxx.lanl.gov/abs/quant-ph/0208185

Go to that link and click PDF. On page 11 begins the appendix, titled "APPENDIX:
THE GENERAL THEORY OF QUANTUM MEASUREMENTS".

(The link to the pdf is: http://xxx.lanl.gov/PS_cache/quant-ph/pdf/0208/0208185v2.pdf)

is: "Bohmian particle trajectories in relativistic fermionic quantum field theory" by H. Nikolic

Somehow you got the wrong paper. Follow the links above and get: "Bohmian particle trajectories in relativistic bosonic quantum field theory" by H. Nikolic

 

[Demystifier]

Thank you Doc Al for indicating the correct paper.

 

[andreab1987]

Go to that link and click PDF. On page 11 begins the appendix, titled "APPENDIX:
THE GENERAL THEORY OF QUANTUM MEASUREMENTS".

(The link to the pdf is: http://xxx.lanl.gov/PS_cache/quant-ph/pdf/0208/0208185v2.pdf)


Somehow you got the wrong paper. Follow the links above and get: "Bohmian particle trajectories in relativistic bosonic quantum field theory" by H. Nikolic

Thank you very much for providing the correct link.

However eq. 42 in the appedix does not seem correct to me.
For example, consider the momentum p as the operator A.
So we are measuring the momentum of an electron through a detector.
Before the electron interacts with the detector, the needle in the detector indicates the value 0 (since no momentum has yet beed detected), and this for every value of the electron momentum.
After the interaction with the detector, the electron has lost its momentum, while the detector has a given value of its variable y.
This situation cannot clearly be described by equation 42.
Besides, the value of the observable A is not a constant of the motion(as it is claimed) because the interaction with the detector changes the electron momentum.

What do you think?

 

[Demystifier]

Thank you very much for providing the correct link.

However eq. 42 in the appedix does not seem correct to me.
For example, consider the momentum p as the operator A.
So we are measuring the momentum of an electron through a detector.
Before the electron interacts with the detector, the needle in the detector indicates the value 0 (since no momentum has yet beed detected), and this for every value of the electron momentum.
After the interaction with the detector, the electron has lost its momentum, while the detector has a given value of its variable y.
This situation cannot clearly be described by equation 42.
Besides, the value of the observable A is not a constant of the motion(as it is claimed) because the interaction with the detector changes the electron momentum.

What do you think?

Eq. (42) describes the so-called ideal measurements, in which the value of the measured observable does not change by repeated measurement. For example, an ideal measurement of momentum would involve carefully chosen interaction such that momentum is not changed by measurement (which, in practice, may be difficult to achieve). You are right that realistic measurements are usually not ideal. (Although, in the case of spin measurements which you were initially interested in, I think they are not so far from ideal.) Yet, the analysis in the paper can easily be adjusted to non-ideal measurements. All you need to do is to replace psi_a in (42) by some psi'_a. The same modification propagates also to (43), while (44) and (45) remain the same.

 

[andreab1987]

Eq. (42) describes the so-called ideal measurements, in which the value of the measured observable does not change by repeated measurement. For example, an ideal measurement of momentum would involve carefully chosen interaction such that momentum is not changed by measurement (which, in practice, may be difficult to achieve). You are right that realistic measurements are usually not ideal. (Although, in the case of spin measurements which you were initially interested in, I think they are not so far from ideal.) Yet, the analysis in the paper can easily be adjusted to non-ideal measurements. All you need to do is to replace psi_a in (42) by some psi'_a. The same modification propagates also to (43), while (44) and (45) remain the same.

I disagree. Eq. 42 is wrong because it is not possible to use a simple decomposition of the wave function as a sum of terms such as Psi_a X_a with X_a non-overlapping, but the corret sum such be given for two independent index a and b with terms Psi_a X_b.
In other words, the author is right only when he says that the de-broglie-bohn theory does not reproduce the standard quantum mechanics results for momentum distribution (and this is sufficient to cast away de broglie-bohn theory given that all experiments confirm standard quantum mechanics predictions). Then the author tries to devolope a proof based on a strong postulate about the form of the wave function; this postulate is not only arbitrary, but it is clearly wrong because it does not describe correctly the process of measurement.

 

[andreab1987]

All you need to do is to replace psi_a in (42) by some psi'_a. The same modification propagates also to (43), while (44) and (45) remain the same.

I would like to add that changing psi_a with psi'_a means nothing at all. You seem not to understand that you have to describe a measurement; so the wave function must evolve in time form an initial situation when the electron has a given momentum and the needle of the detector has value zero, from a situation when the needle has a given value and the electron has lost its momentum. This evolution is not given by the form chosen of wave funtion in eq. 42, and all the rest of the proof is then simply wrong.

Two final considerations:

1)measurements of spin do change completely the value of the spin because the different components of spin do not commute.

2) measurements of momentum without changing it are not only difficult to achieve in practice, but they are simply impossible.

 

[Demystifier]

I disagree. Eq. 42 is wrong because it is not possible to use a simple decomposition of the wave function as a sum of terms such as Psi_a X_a with X_a non-overlapping, but the corret sum such be given for two independent index a and b with terms Psi_a X_b.

You are right that a general decomposition has two indices. However, a general decomposition does not describe a measurement. Eq. (42) is a special case, and this case corresponds to a measurement. Besides, Eq. (42) and other related results in this Appendix are not new. The Appendix presents the standard results in QM that can even be found in some textbooks. (I can give you a few textbook references of that kind if you want.)

 

[Demystifier]

1)measurements of spin do change completely the value of the spin because the different components of spin do not commute.

When you measure spin many times in the SAME direction (say, z-direction), you will always get the same value (say, +1/2), provided that in the meantime you do not perform other measurements.

2) measurements of momentum without changing it are not only difficult to achieve in practice, but they are simply impossible.

No, they are not impossible. In fact, for ANY hermitian operator A it is possible to write down an interaction Hamiltonian that will measure A without changing it. The "only" problem is to realize such a Hamiltonian in a laboratory.

 

[Demystifier]

... but it is clearly wrong because it does not describe correctly the process of measurement.

I think you have never been reading any serious literature on the theory of quantum measurements. For some basics see e.g.
http://en.wikipedia.org/wiki/Quantum_measurement
especially sections "von Neumann measurement scheme" and "Measurements of the second kind".

(Of course, wikipedia should not be counted as "serious", but you can also find more serious references there.)

You should learn more before criticizing.

 

[andreab1987]

You are right that a general decomposition has two indices. However, a general decomposition does not describe a measurement. Eq. (42) is a special case, and this case corresponds to a measurement. Besides, Eq. (42) and other related results in this Appendix are not new. The Appendix presents the standard results in QM that can even be found in some textbooks. (I can give you a few textbook references of that kind if you want.)

You are wrong: Eq. 42 does not describe a measurement, as I have already explain to you; in fact the time evolution of the wave function in eq. 2 does not allow for the change in the position of the needle before and after the measurement.
I am very expert in quantum mechanics, while I think you are not.
Eq. 2 is simply a flaw in hidden-variable theories.

 

[Demystifier]

You are wrong: Eq. 42 does not describe a measurement, as I have already explain to you; in fact the time evolution of the wave function in eq. 2 does not allow for the change in the position of the needle before and after the measurement.
I am very expert in quantum mechanics, while I think you are not.
Eq. 2 is simply a flaw in hidden-variable theories.

Eq. (42) is a well-known and accepted result which has nothing to do with hidden variables. If you are convinced that (42) is wrong, you should publish a paper. It would be a very important result with wide consequences. Since you are an expert in QM, you must know very well how often Eq. (42) is used in the literature.

 

[andreab1987]

Eq. (42) is a well-known and accepted result which has nothing to do with hidden variables. If you are convinced that (42) is wrong, you should publish a paper. It would be a very important result with wide consequences. Since you are an expert in QM, you must know very well how often Eq. (42) is used in the literature.

Eq. 42 is never used in serious scientific papers on quantum mechanics. I have published some papers on Physical Review B, and read a lot of papers on Physical Review Letters and Physical Review B and I have never found eq. 42.
This is due to the fact that in standard quantum mechanics eq. 42 does not exist at all; nobody needs it to interpret experimental data; it is sufficent to evaluate the probability distribution directly from the wave function of the system, without taking the wave function of the detector into consideration.
I think that Eq. 42 is used only in pseudoscientific or fantascientific literature, and no serious scientific journal would take a paper about eq. 42 into consideration.

By the way, wikipedia is certainly not a serious source since everybody can write on it.
Besides, also von neumann scheme and the other sources are debatable (actually also wikipedia introduce the first equation in von Neumann measurement scheme with

" During the interaction of object and measuring instrument the unitary evolution is supposed to realize the following transition from the initial to the final total wave"

this is only a supposition. Such a supposition is wrong, as I have already explained.

 

[Demystifier]

I have published some papers on Physical Review B, and read a lot of papers on Physical Review Letters and Physical Review B and I have never found eq. 42.

That's because your speciality is practical QM, not foundations of QM. (Otherwise you would publish in PRA, not PRB).

I think that Eq. 42 is used only in pseudoscientific or fantascientific literature, and no serious scientific journal would take a paper about eq. 42 into consideration.

Well, PRL and PRA also have papers with this equation, but obviously you have not been reading THESE papers.

For example, have you ever read any paper on the theory of decoherence?

I am sure you are an expert in your field (whatever it is), but it does not mean that you know everything about QM.

 

[comote]

Maybe I am missing something, but isn't equation (42) just the Schmidt decomposition?

http://en.wikipedia.org/wiki/Schmidt_decomposition

 

[andreab1987]

I am sure you are an expert in your field (whatever it is), but it does not mean that you know everything about QM.

I have explained the reason why eq.42 is wrong and you have raised no counter arguments; you have only said that it is correct because someone else think it is correct, and this is not a valid argument.

I give you another argument proving that the so called "theory of quantum measurement" of the appendix quoted before, is wrong .

In standard quantum mechanics the probability of measuring a specific value q of momentum is given by the square modulus of the coefficient c_q in the decomosition of the electron wave function. This value is calculated for the wave function of the electron BEFORE the measurement (say at t=0), i.e. for the wave funtion of the electron only, without any interaction with the detector.
On the contrary, in the hidden variable theory, this probability is calculated after the measument has occurred, since the needle is explicitly considered; so the value of c_q is calculated at an instant t successive AFTER the measuremet.
Since in real measurements the eigenfunctions of the electron momentum are not eigenfunctions of the interaction hamiltonian, the value of c_q(t=0) is different from c_q(t), which means that the two probabilities are different.
In other words, the de-broglie bohm theory does not reproduce the standard quantum mechanics results (as it is said also in eq. 39). Since experiments confirm standard quantum mechanics, this is again sufficent to prove that both "quantum theory of measurement" and hidden variable theories are wrong.

 

[andreab1987]

Maybe I am missing something, but isn't equation (42) just the Schmidt decomposition?

http://en.wikipedia.org/wiki/Schmidt_decomposition

No it isn't.

In fact Schmidt theorem says that:

For any vector v in the tensor product , there exist orthonormal sets u and v ...

this means that the two orthonormal sets are in general different for any different vector v; in other words they depend on the choice of v, but there are no fixed orthonormal sets which can be used for every vector v.

In the case of eq. 42 the orthonormal sets is chosen with eigenvectors of the operator A corresponding to the quantity to be measured; so this orhonormal sets does not depend on the vector v, i.e. the wave function of the electron.

 

[Demystifier]

I have explained the reason why eq.42 is wrong and you have raised no counter arguments; you have only said that it is correct because someone else think it is correct, and this is not a valid argument.

I gave you some arguments, but you ignored them.

Let me also note that Eq. (42) plays an important role in decoherence theory, which, by the way, is an observational fact. If you don't know about decoherence, see e.g.
http://xxx.lanl.gov/pdf/quant-ph/0312059 [Rev.Mod.Phys.76:1267-1305,2004]
which is published in a VERY respectable journal, and has MANY citations. Pay particular attention to Eq. (2.1).

 

[Demystifier]

In the case of eq. 42 the orthonormal sets is chosen with eigenvectors of the operator A corresponding to the quantity to be measured; so this orhonormal sets does not depend on the vector v, i.e. the wave function of the electron.

But in (42) v does not need to be the wave function of the electron before the measurement. Instead, v_i are BASIS states in which the wave function of the electron before the measurement can be expanded.