# What should hidden variables explain?

• A
Gold Member
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?

Gold Member
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
Gold Member
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!

Gold Member
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
Gold Member
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?

Gold Member
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.

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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
Gold Member
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!

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

ShayanJ
Gold Member
There can be alternate definitions of "hidden variables". (post 9)
Bell's proof is about the local hidden variables wich 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!

Gold Member
Ok that is your de finition. Can you give me links from good authors or text books 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
Gold Member
Ok that is your de finition. Can you give me links from good authors or text books 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!

dlgoff
There can be alternate definitions of "hidden variables". (post 9)
Bell's proof is about the local hidden variables wich 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.

Gold Member
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.

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.

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

Gold Member
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)

Truecrimson
Cough!

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.

Truecrimson
stevendaryl
Staff Emeritus
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.

DrClaude, maline and 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.

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

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.