A What should hidden variables explain?

[naima]

It is often said that the Bell's theorem precludes local hidden variables. From a "modern" point of view one should never deduce conclusions from the existence of outputs in non commutative measurements.
It seems that the derivations of this theorem use such results.
Is there a proof which uses only the $\lambda$ in the case of possible measurements?

 

[atyy]

Hidden variables should allow to say that the moon is there when we are not looking at it.

More mathematically, hidden variables should allow us to show that quantum mechanics can be obtained from a theory whose state space is a simplex, as it is in classical probability.

 

[haushofer]

Hidden variables should allow to say that the moon is there when we are not looking at it.

Isn't decoherence allowing us for that?

 

[naima]

I think that the usual proofs of Bell's theorem are dated. they were for those
who had doubts about QM. They thought that imposible measurements nevertheless had outputs. Assuming that the theorem tells them that this leads to the well known inequalities.
Time have changed. We cannot allow such assumptions to discard hiden variables.

 

[ShayanJ]

I think that the usual proofs of Bell's theorem are dated. they were for those
who had doubts about QM. They thought that imposible measurements nevertheless had outputs. Assuming that the theorem tells them that this leads to the well known inequalities.
Time have changed. We cannot allow such assumptions to discard hiden variables.

1) Bell's theorem doesn't rule out all hidden variable theories, it only rules out local hidden variable theories.
2) What impossible measurements are you talking about? All Bell assumes on the experimental side, is that there are two apparatus that can measure the spin of the two particles!

 

[naima]

Take for instance this good link written by DrChinese.
He writes:
"no matter which of the 8 scenarios which actually occur"
We know that with 2 particles we cannot measure a spin in 3 directions.
It is a prequantic idea.
If Bell's aim were to discard local hidden variables we cannot accept such arguments. Would you accept the use of absolute simultaneity in the proof?

 

[ShayanJ]

Take for instance this good link written by DrChinese.
He writes:
"no matter which of the 8 scenarios which actually occur"
We know that with 2 particles we cannot measure a spin in 3 directions.
It is a prequantic idea.
If Bell's aim were to discard local hidden variables we cannot accept such arguments. Would you accept the use of absolute simultaneity in the proof?

That's absurd. You're trying to defend hidden variables by denying them!
The whole point of hidden variables is that physical systems have properties and those properties have values, independent of the observer and whether s\he measures anything or not. If you don't like the assumption that we can talk about properties that can't be measured, then you actually don't like hidden variables!
And that's exactly what Bell's theorem is about! You seem to think that hidden variables are something different from this assumption and so you think if we put aside this assumption, we can retain hidden variables. But hidden variables are invented exactly for that purpose, to let people talk about properties of the physical systems independent of the observer, and Bell is showing that this assumption is incompatible with locality.

Also...where does absolute simultaneity play any role?

 

[naima]

You are partly right.
I do not like these hidden variables that would assign values to unmeasured properties. But when t'Hooft is looking for a cellular deterministic automat, he starts with initial values to find the outcome in an actual measurement.
I also call these positions hidden variables because one cannot know all them in the universe. There is no copyright for "hidden variables"

Of course absolute simultaneity plays no role here. I am sure that you would reject a theorem (would you read it entirely) if it supposed it?
But you accept that Bell can suppose that a spin can have values along different directions even if QM says that it is not possible.

 

[atyy]

Isn't decoherence allowing us for that?

No. Decoherence does not solve the measurement problem, unless additional assumptions are added - that is pretty much the standard view. Thus for example, many-worlds tries to add the assumption that more than one outcome occurs.

 

[ShayanJ]

I do not like these hidden variables that would assign values to unmeasured properties. But when t'Hooft is looking for a cellular deterministic automat, he starts with initial values to find the outcome in an actual measurement.

I think Bell's theorem encompasses that too. So such models should be non-local. And if his approaches aren't much different from Stephen Wolfram's, I think this shouldn't be surprising that they give rise to non-local models.

Of course absolute simultaneity plays no role here. I am sure that you would reject a theorem (would you read it entirely) if it supposed it?

So that was an analogy. OK, that depends but generally I won't be enthusiastic about it. I see what you're saying.

But you accept that Bell can suppose that a spin can have values along different directions even if QM says that it is not possible.

But this is very different. There are two things that you still don't understand.

1) QM doesn't say that a spin can't have values along different directions. This is obvious from the Stern-Gerlach experiment where you can decompose the beam of particles along any two perpendicular directions that you desire. But I suppose that's not what you mean. I think you wanted to say, that QM says a spin can't have components along directions along which no measurement is being done on it. But QM doesn't say that either! Before Bell, there was only the thought experiment presented by EPR, and an equivalent version by Bohm and Aharonov. It was supposed to demonstrate that QM is incomplete. It was Bell's theorem that clearly demonstrated what QM has to say about such thought experiments and what are the true implications of them!

2) Have you ever tried to prove that $\sqrt{2}$ is an irrational number? Its usually done using proof by contradiction, you assume something is true and then show that it leads to contradiction and so you conclude that the assumption was false and so you have proved that the negation of the assumption is true. What would you say if after such a proof, someone criticizes your proof by saying that your proof isn't correct because your assumption leads to contradiction and so you didn't have the right to make that assumption?! You see? That's what you're doing here! Bell assumes local hidden variables and shows that they lead to contradiction and so they can't be correct. So I really don't understand what it is that you're criticizing!

 

[naima]

There can be alternate definitions of "hidden variables". (post 9)
Bell's proof is about the local hidden variables which assign values to unmeasured experiments.

 

[ShayanJ]

There can be alternate definitions of "hidden variables". (post 9)
Bell's proof is about the local hidden variables which assign values to unmeasured experiments.

Its the definition of hidden variables to assign values to unmeasured quantities. It doesn't matter whether they're local or non-local. Its just that, by Bell's theorem, local ones are in contradiction with QM.
And as far as I understand it, t'Hooft's approach is a hidden variable theory too!

 

[naima]

Ok that is your de finition. Can you give me links from good authors or textbooks that agree with it?
I do not read that in wikipedia. They talk about underlying parameters which would give deterministic outcomes for each actual measurement.

 

[ShayanJ]

Ok that is your de finition. Can you give me links from good authors or textbooks that agree with it?
I do not read that in wikipedia. They talk about underlying parameters which would give deterministic outcomes for each actual measurement.

I can't give any reference for that but that's pretty obvious.
One of the principles of QM is that the information that the wave-function of a quantum system gives you, is the maximum amount of information you can have about that system and there is simply nothing more to be known about it. And even when you have all there is to know about the quantum system, all you can predict is the probability distributions of values for different observables. This means that the maximum amount of information you can have about a quantum system, doesn't determine the values of its properties uniquely. That's what some people don't like, and the only way to change it in a way that you retain the power you had in classical physics where you could uniquely predict the value of any property you wanted(at least in principle), is to somehow distinguish between states that QM considers equivalent. And that can only be done by postulating that the information contained in the wave-function is not the maximum amount of information you can have about the system and there are some extra information that if we had access to, we could uniquely determine the values of different observables. Those extra information are called hidden variables. In fact if you don't use hidden variables in this way, they lose their meaning and there is no reason to have them in the theory. What else are they supposed to do?!

Anyway, I checked this paper by 't Hooft. It seems to me that he circumvents Bell's theorem by assuming superdeterminism, which means he assumes that the experimenters are not free to choose the settings on their measurement devices and this implies that because the settings on the devices are predetermined and so each device already "knows" what's going to happen to the other device, they can adjust themselves somehow that Bell's inequality is violated while still retaining both realism and locality!

 

[Truecrimson]

There can be alternate definitions of "hidden variables". (post 9)
Bell's proof is about the local hidden variables which assign values to unmeasured experiments.

But you can't rule out Many-Worlds and other theories where the definition of "local hidden variables" is modified because it gives the same predictions as quantum theory (in the domain where we can currently test).

And Bell's theorem is still immensely useful today in certifying that quantum devices can't be simulated by using shared randomness. The customers may believe in quantum theory, but they don't have to trust the devices' manufacturers.

 

[naima]

And Bell's theorem is still immensely useful today in certifying that quantum devices can't be simulated by using shared randomness.

I did not think that this theorem was useful.Could you elaborate? thanks.

 

[Jilang]

Naima, I see where you are coming from. I could entertain there being a distinction between an unmeasured quantity and a measured one. But a predisposition to a certain measured quantity would of itself be a quantity.

 

[Truecrimson]

I did not think that this theorem was useful.Could you elaborate? thanks.

No problem. The fact that Bell tests depend only on measurement statistics and not where the statistics come from is the basis of device-independent quantum cryptography. Maximal violation of Bell-CHSH inquality also certifies that you have a singlet state and can be used to certify some quantum computation. These are in the section "Applications of quantum nonlocality" of this review article: http://arxiv.org/abs/1303.2849

 

[naima]

Thank you for the link Truecrimson.
I read in page 3 that the locality property
$P(a,b,\lambda) = P(a,\lambda) P(b,\lambda)$ leads the Bell's inequalities without requiring that the same lambda give outputs for noncommuting observations.
I understand now why Shyan could not give me a reference with this property of hidden variables. (nobody here reacted)

 

[Jilang]

Cough!

 

[Heinera]

Thank you for the link Truecrimson.
I read in page 3 that the locality property
$P(a,b,\lambda) = P(a,\lambda) P(b,\lambda)$ leads the Bell's inequalities without requiring that the same lambda give outputs for noncommuting observations.
I understand now why Shyan could not give me a reference with this property of hidden variables. (nobody here reacted)

Any proof of Bell's theorem (or similar theorems) assumes that a local realistic model has the property of counterfactual definiteness, i.e, it should be possible to use the model to compute a predicted outcome of any hypothetical experiment for which the model applies, irrespective of whether the experiment is performed or not. In particular, it should be possible to use the model to predict outomes for say both of two possible detector settings for one and the same lambda, even if only one setting can be applied at a time in an actual experiment. There is actually nothing in the paper linked by Truecrimson that contradicts this.

 

[stevendaryl]

Thank you for the link Truecrimson.
I read in page 3 that the locality property
$P(a,b,\lambda) = P(a,\lambda) P(b,\lambda)$ leads the Bell's inequalities without requiring that the same lambda give outputs for noncommuting observations.
I understand now why Shyan could not give me a reference with this property of hidden variables. (nobody here reacted)

It's not necessary to ASSUME that \lambda determines the outcomes of all measurements, but that is a mathematical conclusion.

You might start assuming that

• P_A(a,\lambda) = the probability of Alice getting spin up, given that the hidden variable has value \lambda, and that the orientation of Alice's detector is a
• P_B(b,\lambda) = the probability of Bob getting spin up, given that the hidden variable has value \lambda, and that the orientation of Bob's detector is b.

If we assume that every particle is detected, and that the spin is either spin-up or spin-down, then

• 1 - P_A(a,\lambda) = the probability of Alice getting spin down, given that the hidden variable has value \lambda, and that the orientation of Alice's detector is a
• 1 - P_B(b,\lambda) = the probability of Bob getting spin down, given that the hidden variable has value \lambda, and that the orientation of Bob's detector is b.

But in the case of the anti-correlated spin-1/2 twin pairs, you know that: If Alice measures spin-up at angle a, then Bob will definitely not measure spin-down at that angle. So there is zero probability that they both measure spin-up at angle a. That implies:

(1) P_A(a,\lambda) \cdot P_B(a, \lambda) = 0

But also, if Alice measures spin-down, then Bob definitely will NOT measure spin-down. That implies:
(2) (1-P_A(a,\lambda)) \cdot (1 - P_B(a, \lambda)) = 0

Together, (1) and (2) imply that

P_A(a,\lambda) = 0\ \&\ P_B(a,\lambda) = 1
or P_A(a,\lambda) = 1\ \&\ P_B(a,\lambda) = 0

That implies that the spin Alice measures for angle a is completely determined by \lambda, and similarly for Bob.

 

[Truecrimson]

$P(a,b,\lambda) = P(a,\lambda) P(b,\lambda)$ leads the Bell's inequalities without requiring that the same lambda give outputs for noncommuting observations.

I'm also not sure why the paper I gave convinced you of that. Maybe it's the part where they say the hidden variables don't have to be constant throughout different runs? It's a random variable so its values can fluctuate, but it's still the same variable $\lambda$ that influences the outcomes of noncommuting measurements.

 

[naima]

I have a problem with my computer. I'll be back next week.

 

[Ilja]

We know that with 2 particles we cannot measure a spin in 3 directions.
It is a prequantic idea.
If Bell's aim were to discard local hidden variables we cannot accept such arguments. Would you accept the use of absolute simultaneity in the proof?

Of course, if the existence of absolute simultaneity would be derived from the assumptions as part of the proof.

As it is in this case with the "hidden variables".

The "hidden variables" are derived from the EPR principle of reality, and the observable fact that if Alice and Bob measure in the same direction, they get a 100% correlated result. The EPR principle: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, there exists an element of physical reality corresponding to this physical quantity.

 

[Ilja]

This means that the maximum amount of information you can have about a quantum system, doesn't determine the values of its properties uniquely. That's what some people don't like, and the only way to change it in a way that you retain the power you had in classical physics where you could uniquely predict the value of any property you wanted(at least in principle), is to somehow distinguish between states that QM considers equivalent. And that can only be done by postulating that the information contained in the wave-function is not the maximum amount of information you can have about the system and there are some extra information that if we had access to, we could uniquely determine the values of different observables. Those extra information are called hidden variables. In fact if you don't use hidden variables in this way, they lose their meaning and there is no reason to have them in the theory. What else are they supposed to do?!

The aim is simply to get a meaningful, non-mystical description of reality. Some hidden variable theories have randomness, thus, do not propose any type of additional control, and even deterministic dBB theory assumes (derives) a quantum equilibrium which one cannot leave, and which gives the same uncertainty as QT. So, your picture of the hidden variable proponent is a strawman.

 

[ShayanJ]

The aim is simply to get a meaningful, non-mystical description of reality. Some hidden variable theories have randomness, thus, do not propose any type of additional control, and even deterministic dBB theory assumes (derives) a quantum equilibrium which one cannot leave, and which gives the same uncertainty as QT. So, your picture of the hidden variable proponent is a strawman.

I never said such theories differ with QM in predictions. In dBB particles always have definite positions, but the randomness and other features that make its predictions indistinguishable from QM are given to it by other means!

 

[Ilja]

My criticism was directed against your picture of the hidden variable proponent.

The hidden variable proponent wants to understand what happens. A very natural thing. To use some simple models of what could possible happen in reality is a standard way to reach this aim.

In general, I think it is not very good to speculate about the motives of those who prefer other choices. To demonstrate why, I can present a similar unfavorable picture of the "relativists". They observe that all such realistic models violate Einstein causality. This is something they are not ready to accept, and if there is a theorem which proves that every realistic model has these properties, all the worse for reality - he will give up realism, once it does not fit his relativistic belief. And, once there is no longer any reality outside, there is no longer any need to study realistic models which violate the "relativistic belief".

One can, then, extend this to comparisons of "relativists" with various religious believers. A strawman? I'm afraid to discuss this in Physics Forums would not be appropriate. (If you want, let's discuss this, say, at http://ilja-schmelzer.de/forum/.) But I hope you get the point that I, as a proponent of hidden variable theories, feel equally uncomfortable with such speculations about my motivation.

 

[ShayanJ]

My criticism was directed against your picture of the hidden variable proponent.

The hidden variable proponent wants to understand what happens. A very natural thing. To use some simple models of what could possible happen in reality is a standard way to reach this aim.

In general, I think it is not very good to speculate about the motives of those who prefer other choices. To demonstrate why, I can present a similar unfavorable picture of the "relativists". They observe that all such realistic models violate Einstein causality. This is something they are not ready to accept, and if there is a theorem which proves that every realistic model has these properties, all the worse for reality - he will give up realism, once it does not fit his relativistic belief. And, once there is no longer any reality outside, there is no longer any need to study realistic models which violate the "relativistic belief".

One can, then, extend this to comparisons of "relativists" with various religious believers. A strawman? I'm afraid to discuss this in Physics Forums would not be appropriate. (If you want, let's discuss this, say, at http://ilja-schmelzer.de/forum/.) But I hope you get the point that I, as a proponent of hidden variable theories, feel equally uncomfortable with such speculations about my motivation.

I have no idea what you're talking about!

 

[Ilja]

I didn't like your speculation about "That's what some people don't like" from https://www.physicsforums.com/goto/post?id=5491731#post-5491731. It ended with "In fact if you don't use hidden variables in this way, they lose their meaning and there is no reason to have them in the theory. What else are they supposed to do?!", which I interpreted as an argument that your speculation about the motives of hidden variable proponents is the only possible one. So, it is a claim about what all proponents of hidden variables don't like.

I'm a proponent of hidden variable theories, and I do not care at all about what I'm supposed not to like. So I have objected. Sorry if I have completely misunderstood you.

 

[naima]

I have nothing against the possibility of hidden vatiables.
According to me everytime a measurement is DONE, a hidden variable would correspond to a set of datas from which an algorithm or procedure or map would give the outcome of the measurement.
It is not the idea that given these datas there is a map which give an output to any measurement.

There is also a big problem in the methodology used to discard hidden variables.
Take the paper written by DrChinese.
He shows very well how Bell uses non observed "observables".
Take the EPR experiment with two particles. At each shot the spin is measured along two different directions chosen into A B and C.
We begin with 3 counters (equal to zero) i call them AB BC and CA
if say the directions are C and A and the spins match i add 1 to CA else i add nothing.
I can consider at the end the mean value of AB + BC + CA: if there were N shots i divide their sum by N.
Let us now look at the way one can discard hidden variables:
We consider now that AB BC CA are observables having values at each shot. there are 8 possibility so DrChinese introduce a table wit 8 lines.Suppose that the directions are A and B and that the spins do not match. one is up and the other down. What about the spin along C? it will match with A or B. so we will add 1 to BC or CA. A each shot AB + BC +CA is increased. Of course here at the end we divide by 3N. the result is > 1/3
In the reality we can have sequences of three shots that do not match. and the observed meanvalue < 1/3 so we say that hidden variables are discarded.
How can we accept such biased methodology to derive anything?
Take the proof of Kochen theorem in the WIKI it also use such tableaux.
I thank DrChinese for the quality of his paper.

 

[Ilja]

There is also a big problem in the methodology used to discard hidden variables.
Take the paper written by DrChinese.
He shows very well how Bell uses non observed "observables".
...
In the reality we can have sequences of three shots that do not pair. and the observed meanvalue < 1/3 so we say that hidden variables are discarded.
How can we accept such biased methodology to derive anything?
Take the proof of Kochen theorem in the WIKI it also use such tableaux.
I thank DrChinese for the quality of his paper.

If you mean http://drchinese.com/David/Bell_Theorem_Easy_Math.htm, it is fine, but has not prevented you from not understanding a central point: Namely the point that all three values A, B, C have to be predefined is not an arbitrary hidden variable assumption, but a conclusion, derived from the EPR argument. And that we do not reject hidden variables in general, but only those Einstein-local realistic theories, where we can use the EPR argument to derive that the results have to be predefined even if not measured.

Bell was at that time aware that there exists a hidden variable theory - de Broglie-Bohm theory - which gives all the quantum predictions. His aim was not at all to disprove hidden variables - he was at that time essentially the only proponent of dBB theory. It was to show that the attack against dBB theory that it needs a (hidden) preferred frame was unjustified, because every realistic interpretation needs a preferred frame. It was, I would guess, unimaginable at that time that people would be readily give up realism and causality and all this only to preserve the (metaphysical) idea that relativistic symmetry is fundamental.

 

[naima]

We find the words "local" and "locality" in the DrChinese paper but is there a locality argument in the way to fill the table.
We have a game with two possible results: 1 and 1/3.
Do we need an experiment to show that Nature with its possible 0 results is not like that?

 

[ShayanJ]

I have nothing against the possibility of hidden vatiables.
According to me everytime a measurement is DONE, a hidden variable would correspond to a set of datas from which an algorithm or procedure or map would give the outcome of the measurement.
It is not the idea that given these datas there is a map which give an output to any measurement.

There is also a big problem in the methodology used to discard hidden variables.
Take the paper written by DrChinese.
He shows very well how Bell uses non observed "observables".
Take the EPR experiment with two particles. At each shot the spin is measured along two different directions chosen into A B and C.
We begin with 3 counters (equal to zero) i call them AB BC and CA
if say the directions are C and A and the spins match i add 1 to CA else i add nothing.
I can consider at the end the mean value of AB + BC + CA: if there were N shots i divide their sum by N.
Let us now look at the way one can discard hidden variables:
We consider now that AB BC CA are observables having values at each shot. there are 8 possibility so DrChinese introduce a table wit 8 lines.Suppose that the directions are A and B and that the spins do not match. one is up and the other down. What about the spin along C? it will match with A or B. so we will add 1 to BC or CA. A each shot AB + BC +CA is increased. Of course here at the end we divide by 3N. the result is > 1/3
In the reality we can have sequences of three shots that do not match. and the observed meanvalue < 1/3 so we say that hidden variables are discarded.
How can we accept such biased methodology to derive anything?
Take the proof of Kochen theorem in the WIKI it also use such tableaux.
I thank DrChinese for the quality of his paper.

We find the words "local" and "locality" in the DrChinese paper but is there a locality argument in the way to fill the table.
We have a game with two possible results: 1 and 1/3.
Do we need an experiment to show that Nature with its possible 0 results is not like that?

What is your question? Can you state it clearly?
Non of the things you say make sense to me!

 

[naima]

Read the sentences with a ? at the end!
Your "What is the question," is absurd but i understand that it looks like a question.

 

[ShayanJ]

How can we accept such biased methodology to derive anything?

What bias are you talking about?

 

[naima]

I repeat once again.
In the paper we have 8 lines and 3 counters. each of them can be increased at each shot. their sum always increases at less by 1/3. and at the end we compute the mean value.
We say then let us see if Nature agrees.
We also need 3 counters and at each shot we increase one of them if we have ++ or -- for the random directions. At the end it is possible that they still contain 0 ans that the mean value is null.
Do you think that there is no methodology problem when you compare the mean values?

 

[andrewkirk]

Its the definition of hidden variables to assign values to unmeasured quantities.

I thought the definition of 'hidden variables' in this context was just that they contain information that is not contained in the quantum representation (ket or sum of operators). In Bell's words it implies the existence of 'a more complete specification of the state', where the additional information is represented by the symbol $\lambda$.

As I understand his paper, Bell does not say that $\lambda$ must be unmeasurable. Quite likely, if there were such non-local hidden variables, as in Bohm's theory, there would be no current way to measure them. But I am not aware of any theorem that says that such additional information is in principle unmeasurable. Is there such a theorem? Can we perhaps obtain such a theorem as a corollary of the No-Communication Theorem, as being able to measure non-local quantities could conceivably open the door to FTL communication.

 

[Zafa Pi]

It is often said that the Bell's theorem precludes local hidden variables. From a "modern" point of view one should never deduce conclusions from the existence of outputs in non commutative measurements.
It seems that the derivations of this theorem use such results.
Is there a proof which uses only the $\lambda$ in the case of possible measurements?

With the following clarifications every thing Shyan has said makes sense to me.
First there are various Bell type theorems and second there are Bell inequalities. Let me give a concrete example:

Theorem: Let Ah, At, Bh, Bt each be +1 or -1. (these values may come about via some random process such as coin flipping)
If Ah•Bh = 1 then P(At•Bt = -1) ≤ P(At•Bh = -1) + P(Ah•Bt = -1).

Proof: P(At•Bt = -1) = P(At•Bt•Ah•Bh = -1) = P(At•Bh•Bt•Ah = -1) =
P({At•Bh = -1 and Bt•Ah = 1} or {At•Bh = 1 and Bt•Ah = -1}) =
P(At•Bh = -1 and Bt•Ah =1) + P(At•Bh = 1 and Bt•Ah = -1) ≤
P(At•Bh = -1) + P(Ah•Bt = -1) QED

The Theorem is a valid piece of mathematics. Within the Theorem there is an Inequality.

The numbers Ah, At, Bh, Bt could come about in the usual fashion:
#1 The physical set up for the Theorem:
Alice and Bob are 2 light minutes apart, and Eve is half way between them. Alice
has a fair coin (see probability appendix) and a device. Her device has 2 buttons
labeled h and t, a port to receive a signal from Eve. The device also has a screen
that will display “Eve’s signal received” when a signal from Eve is received. It will
also display either +1 or -1 if one of the buttons is pushed. Bob has the same
equipment and shows the same values, tho the internal workings of his device
may be different.
#2 The following experiment is performed:
Eve simultaneously sends a light signal to each of Alice and Bob. When Alice’s
device indicates Eve’s signal has been received she flips her coin. If it comes up
heads she pushes button h, otherwise button t, and then notes what the screen
displays. What Alice does takes less than 30 seconds. The same goes for Bob.
Here we assume locality - no faster than light influences.
#3 Notation:
If Alice flipped a head and pushed button h, we let Ah be the value her screen
would show. So Ah = 1 or -1 and is the result of some objective physical process.
Similarly we let At be the value if she had flipped a tail. We let Bh and Bt be the
analogous values for Bob.

Eve could send each of Alice and Bob a photon from an entangled pair so that
Ah•Bh = 1, and P(At•Bt = -1) = 3/4, while P(At•Bh = -1) + P(Ah•Bt = -1) = 1/2.
(the state of the pair is √½(|00> +|11>), Ah and Bh measure at 0 degrees, At at 30,º and Bt at -30º)

The Inequality is false. How could this be? No aspect of reality or a physical theory will invalidate a mathematical theorem. The hypothesis of the Theorem has all 4 numbers Ah, At, Bh, Bt existing at once and shows up in the proof. In each experiment, however only two of the numbers are observed, e,g. At and Bt. The existence of the other two is inferred by counterfactual definiteness = hidden variables = realism, and it is that inference which must be questioned (assuming locality).

 

[Ilja]

We find the words "local" and "locality" in the DrChinese paper but is there a locality argument in the way to fill the table.
We have a game with two possible results: 1 and 1/3.
Do we need an experiment to show that Nature with its possible 0 results is not like that?

The "locality" (Einstein causality would be better) argument of this article is

But there was a price to pay for such this experimental setup: we must add a SECOND assumption. That assumption is: A measurement setting for one particle does not somehow affect the outcome at the other particle if those particles are space-like separated. This is needed because if there was something that affected Alice due to a measurement on Bob, the results would be skewed and the results could no longer be relied upon as being an independent measurement of a second attribute. This second assumption is called "Bell Locality" and results in a modification to our conclusion above. In this modified version, we conclude: the predictions of any LOCAL Hidden Variables theory are incompatible with the predictions of Quantum Mechanics. Q.E.D.

My attempt to explain it is http://ilja-schmelzer.de/realism/game.php. And I think that one should recognize what one gives up if one rejects it: The EPR criterion of reality: If we can, without distorting the system in any way, predict with certainty the result of the experiment, then there exists a corresponding element of reality.

 

[ShayanJ]

I thought the definition of 'hidden variables' in this context was just that they contain information that is not contained in the quantum representation (ket or sum of operators). In Bell's words it implies the existence of 'a more complete specification of the state', where the additional information is represented by the symbol $\lambda$.

As I understand his paper, Bell does not say that $\lambda$ must be unmeasurable. Quite likely, if there were such non-local hidden variables, as in Bohm's theory, there would be no current way to measure them. But I am not aware of any theorem that says that such additional information is in principle unmeasurable. Is there such a theorem? Can we perhaps obtain such a theorem as a corollary of the No-Communication Theorem, as being able to measure non-local quantities could conceivably open the door to FTL communication.

I didn't mean they should be unmeasurable. naima was criticizing that Bell's theorem rules out only hidden variable theories that assign values to unmeasured observables, not unmeasurable observables. So he was actually criticizing that Bell's theorem rules out only hidden variables theories that assume counterfactual definiteness, but actually hidden variables are there exactly to retain counterfactual definiteness! That's why they were proposed!

 

[Ilja]

naima was criticizing that Bell's theorem rules out only hidden variable theories that assign values to unmeasured observables, not unmeasurable observables. So he was actually criticizing that Bell's theorem rules out only hidden variables theories that assume counterfactual definiteness, but actually hidden variables are there exactly to retain counterfactual definiteness! That's why they were proposed!

The position that Bell's theorem rules out "only" hidden variables that assume counterfactual definiteness missed, of course, the whole point of Bell's theorem, which uses, in its first part, the EPR argument to prove that there has to be counterfactual definiteness for this particular experiment if we assume Einstein causality.

But hidden variable theories are certainly not proposed to retain some counterfactual definiteness. Instead, proponents of dBB as well as other hidden variable theories feel quite comfortable with these proposals, despite the fact that these proposals do not have counterfactual definiteness.

 

[naima]

I would agree that hidden variables theory (as i see them) do not imply
counterfactual definiteness.
My criticism to the Bell theorem is that it gives a scientific status to counterfactuality. We examine a proposed model which is not even falsifiable. Is there a statistical experiment proposed by their defenders to verify it?
If i write that 97.5 % of the ghosts in NY know that God is a green Dragon, who has to prove or to disprove it?

 

[Ilja]

Sorry, but it is reverse. Bell's inequality indeed examines something which, before this proof, was thought to be completely unfalsifiable: The thesis that relativistic symmetry is not only restricted to observables (something we can test, and do test, and what is not questioned by the established hidden variable theories) but more, namely that it is a fundamental insight, and that there cannot exist a preferred frame.

And the result of Bell's theorem is that we now have a possibility to distinguish, by observation, theories where relativistic symmetry is fundamental from theories where relativistic symmetry is only an approximate, non-fundamental symmetry. For the first class, we can prove Bell's inequalities, for the second class we cannot.

Counterfactuality is only an intermediate step of Bell's proof. It is derived from the EPR argument and Einstein causality.

 

[haushofer]

Just my two cents: Hidden variable theories are most useful in an attempt to provide a clear ontology to QM, like the Bohmian interpretation.

 

[naima]

And the result of Bell's theorem is that we now have a possibility to distinguish, by observation, theories where relativistic symmetry is fundamental from theories where relativistic symmetry is only an approximate, non-fundamental symmetry. For the first class, we can prove Bell's inequalities, for the second class we cannot.

You will have to explain.
Take the electron Dirac theory.
According to you, the Bell theorem can give you a tool: By observation, you will be able to prove the inequality if the theory is relativistic. (maybe it is the reverse, the violation of the inequality?)
Are you thinking of the Aspect experiment?
Take any other quantic theory with a non relativistic lagrangian.
We are in the second class of theories.
Will you use the same device to prove something? and to prove what? non violation?

 

[Ilja]

You will have to explain.
Take the electron Dirac theory.
According to you, the Bell theorem can give you a tool: By observation, you will be able to prove the inequality if the theory is relativistic. (maybe it is the reverse, the violation of the inequality?)
Are you thinking of the Aspect experiment?

By observation of the violation of Bell's inequality (Aspect or better) I'm able to prove that we have a theory where Bell's theorem cannot be proven. In a fundamentally relativistic theory, it can be proven. So, the theory cannot be fundamentally relativistic.

 

[naima]

Can you give me an example of two quantic theories in the opposite cases?
How can the same observation device tell you which is relativistic?

 

[Ilja]

No, because I don't know any quantum theory which would allow to prove Bell's inequality.

What observation can tell is that a theory is not fundamentally relativistic. If cannot tell if it is.

 

[naima]

And the result of Bell's theorem is that we now have a possibility to distinguish, by observation, theories where relativistic symmetry is fundamental from theories where relativistic symmetry is only an approximate, non-fundamental symmetry. For the first class, we can prove Bell's inequalities, for the second class we cannot.

How can you say that there are two classes of theories without an example in each class?
We are in a quantum physics forum not in thermodynamics. Please could you give me examples in quantum theory?