Is there any hope at all for Locality?

[stevendaryl]

I have no idea what you are saying but objectivity is the independence of measured values from the experimental setup. In the MWI, macro objects exist as parts of a giant wavefunction that splits into classical-like worlds. A classical-like world is different from a classical world where strict causality determines the properties of macro objects.

No, I wouldn't think that objectivity has anything to do with measurements. Objectivity is the existence of a unique external world that is independent of observers. In an objective model, a measurement is certainly revealing something pre-existing about the world, but there's no reason to assume that there is a one-to-one correspondence between what's true in the world and what we measure.

 

[audioloop]

Objectivity is the existence of a unique external world that is independent of observers.

Riiiiiiiiiight !

 

[Maui]

No, I wouldn't think that objectivity has anything to do with measurements. Objectivity is the existence of a unique external world that is independent of observers. In an objective model, a measurement is certainly revealing something pre-existing about the world, but there's no reason to assume that there is a one-to-one correspondence between what's true in the world and what we measure.

Then macro causality goes out the window which was my point from the beginning.

 

[Maui]

Riiiiiiiiiight !

I don't see your point but if it's about there being an outside reality that is different form what we observe (which is kind of obvious), there is some bad news for classical causality which can only exist in a purely classical setting and which can be found neither in experiment nor in theory.

What stevendaryl talks about is a classical-LIKE(or classically-consistent) world, not a classical world. In classical-like worlds macro causality would be secondary and emergent/apparent and when people usually say 'objective reality' they mean classical world, not classical-like world that comes about via decoherence + collapse/world splitting.

 

[meBigGuy]

@mfb Why would you assume it must interact with us? First tell me of something that exists that interacts (or will interact) with *NOTHING*, and how you know it exists. Then we can discuss that. Assuming anyone actually thinks *WE* must observe something for it to exist is a mis-interpretation of what is being said.

@Nugatory " This is not a crazy philosophical thought, but a reasonable position suggested by hard experimental fact." You will get no argument from me on that. Even with your edit, it is, to me, a profound concept.

@bhobba Don't be so difficult (is it intentional?). Ok, thought experiment. You are saying that the Albert's Socks thought experiment illustrated nothing worth thinking about. Your terse "No" answer indicates that you feel that it was not a meaningful mental exercise that illustrated a subtlety behind the reality of entanglement? I am nicely asking you to elaborate on that opinion.

 

[audioloop]

From this we inductively conclude that there is no valid theory that preserves locality.

or like says Gisin:Quantum Nonlocality: How Does Nature Do It?
http://www.alice.id.tue.nl/references/gisin-2009.pdf

"The quantum correlations are indeed coming from outside space-time in the sense that there is no story inside space-time that can describe them"

i prefer to say from other layer of reality/nature.
or Lorenzo:
Are quantum correlations genuinely quantum?
http://arxiv.org/abs/1205.0878

"It is shown that the probabilities for the spin singlet can be reproduced through classical resources, with no communication between the distant parties, by using merely shared (pseudo-)randomness"

 

[meBigGuy]

Objectivity is the existence of a unique external world that is independent of observers.

I am way out of my league here, so be forgiving. I'd like to pin down what is meant by "observers" in QM discussions. To me it infers any interaction within space/time. But that makes your above definition of objectivity somewhat empty since we established that (via Albert's socks) that there is no existence without observations (interactions within space/time). In light of that, is objectivity simply the existence of a unique external world that can be observed (and contains observers)?

 

[audioloop]

I am way out of my league here, so be forgiving. I'd like to pin down what is meant by "observers" in QM discussions. To me it infers any interaction within space/time. But that makes your above definition of objectivity somewhat empty since we established that (via Albert's socks) that there is no existence without observations (interactions within space/time). In light of that, is objectivity simply the existence of a unique external world that can be observed (and contains observers)?

independent of observers/interactions.

Reality is the state of things/objects as they actually exist.

 

[bhobba]

Don't be so difficult (is it intentional?). Ok, thought experiment. You are saying that the Albert's Socks thought experiment illustrated nothing worth thinking about. Your terse "No" answer indicates that you feel that it was not a meaningful mental exercise that illustrated a subtlety behind the reality of entanglement? I am nicely asking you to elaborate on that opinion.

Its not intentional. Its just in this sort of stuff needs precision of thought.

You asked:

But, with respect to the analogy itself, did you see it as a meaningful mental exercise that illustrates a subtlety behind the reality of entanglement?

Its simply a thought experiment illustrating some of the weirdness associated with QM. Entanglement is an interesting and subtle phenomena and as a mental exercise illustrating that its worthwhile. But its nothing beyond that.

Thanks
Bill

 

[stevendaryl]

I am way out of my league here, so be forgiving. I'd like to pin down what is meant by "observers" in QM discussions. To me it infers any interaction within space/time.

No, to me, an observer is something that can make notes about the world. Not every interaction is an observation.

To say that something is independent of observers just means that it exists whether or not there is anything conscious noting its existence. You know, a tree falling in a forest with no-one around.

 

[stevendaryl]

Are quantum correlations genuinely quantum?
http://arxiv.org/abs/1205.0878

"It is shown that the probabilities for the spin singlet can be reproduced through classical resources, with no communication between the distant parties, by using merely shared (pseudo-)randomness"

I'll take a look at that, but it seems to be claiming to have shown something that other people have shown to be impossible.

 

[bhobba]

I am way out of my league here, so be forgiving. I'd like to pin down what is meant by "observers" in QM discussions. To me it infers any interaction within space/time. But that makes your above definition of objectivity somewhat empty since we established that (via Albert's socks) that there is no existence without observations (interactions within space/time). In light of that, is objectivity simply the existence of a unique external world that can be observed (and contains observers)?

In QM an observer is any device, assumed to behave in a classical way, that registers something that occurred in the quantum domain. An example would be a particle detector. It clicks, flashes, or does something here in the classical macro world, when a subatomic particle is detected. In the standard Copenhagen interpretation the existence of such devices is assumed and a fundamental part of that interpretation. Some people get hung up on the usual meaning in everyday usage of observer as requiring a conscious observer and run into all sorts of semantic difficulties - that's another example of the precision of thought and expression required when dealing with this stuff I mentioned previously. I remember watching this movie What The Bleep Do We Know Anyway that was full of that sort of rubbish to the point it was embarrassing.

QM is a theory about observations on quantum systems - it is not a theory about what its doing, existing or anything like that when its not observed. The quantum state is simply a codification of the probabilities of the results of observations if you were to observe it. It tells you nothing other than that. Basically it is, at its fundamental level, a theory about systems interactions with other systems where those other systems change in some way that registers here in the commonsense, classical macro world. It changes nothing about that commonsense world out there - it exists independent of us, it is there when we are not looking and all the usual stuff we take for granted. QM in no way changed that.

The existence of a commonsense macro world is assumed but since everything is quantum this division is a blemish. What you want is a fully quantum theory of measurement. Great progress has been made along these lines (particularly in the area of decoherence) but some issues do remain - although what we do know have clarified a lot of things.

Thanks
Bill

 

[stevendaryl]

I'll take a look at that, but it seems to be claiming to have shown something that other people have shown to be impossible.

The author is doing something kind of weird, it seems to me. In his model, (section 7.1) he's assuming that each experimenter's choice of detector setting is causally influenced by a random hidden variable shared by the experimenters.

 

[bhobba]

It seems circular to me. The law of large numbers doesn't say that relative frequency approaches probability, it says that the set of sequences for which this doesn't happen has measure zero. Why does measure zero mean it doesn't happen? Any actual run has probability zero.

I don't want to sidetrack this thread - this issue really needs a thread of its own. And if you decide to start one don't do it in the probability section of Physic's Forums because they work with mathematical probability which is fully described with zero problems by the Kolmogorov axioms without the nasty subtleties of actually applying it.

However I want to simply add your view is fairly common but does leave me scratching my head. Books like Fellers - Introduction To Probability completely avoid circularity by starting from the Kolmogorov axioms and via the Law of Large numbers derived rigorously from those axioms connects it to the frequency interpretation with no circularity. The Law of Large Numbers says as the number of trials increases the proportion of outcomes being the same as the probability approaches 1 - using some definition of convergence such as convergence in probability. Now the assumption is since it converges to it a sufficiently large number exists that for all practical purposes is the same ie is one. There is no circularity - simply a standard assumption made in applied math all the time.

Thanks
Bill

 

[bhobba]

No, to me, an observer is something that can make notes about the world. Not every interaction is an observation.

To say that something is independent of observers just means that it exists whether or not there is anything conscious noting its existence. You know, a tree falling in a forest with no-one around.

Good way of expressing it - must keep it in mind.

And indeed the standard Copenhagen interpretation assumes the existence of a world out there just like that eg trees falling in a forest make a sound regardless of if there is anyone there to hear it.

Thanks
Bill

 

[stevendaryl]

The Law of Large Numbers says as the number of trials increases the proportion of outcomes being the same as the probability approaches 1 - using some definition of convergence such as convergence in probability. Now the assumption is since it converges to it a sufficiently large number exists that for all practical purposes is the same ie is one. There is no circularity - simply a standard assumption made in applied math all the time.

I guess we would need to state explicitly what is being claimed in order to say whether it's circular or not. Certainly the proof of the law of large numbers isn't circular. It's a proof from the axioms of probability. But what I'm saying is that the law of large numbers by itself doesn't imply that one must treat probability 0 events as impossible.

Let's define for a sequence of coin flips:

F_n = \dfrac{H_n}{n}

where H_n is the number of heads in the first n coin flips.

To say that F_n \rightarrow \frac{1}{2} is to say that:

For every \epsilon > 0, there is a number \hat{n} such that for all n > \hat{n}, |F_n - \frac{1}{2}| < \epsilon. That's true for some sequences of coin flips, and false for other sequences of coin flips.

What the weak law of large numbers says is that the probability that |F_n - \frac{1}{2}| < \epsilon goes to 1 as n \rightarrow \infty

But the issue is whether probability 1 means "certain" and probability 0 means "impossible". The weak law of large numbers by itself doesn't say that.

The strong law of large numbers states that

The strong law of large numbers states that the sample average converges almost surely to the expected value.

But the meaning of "almost surely" is "true for all sequences except for a set of measure 0". Once again, the issue is why does having a set of measure 0 mean "impossible"?

The nice thing about probability zero is that it's consistent to treat any finite (or countable) set of probability zero events as if they were impossible. So it's a consistent way to reason. This is in contrast with an assumption saying "I'm going to ignore events that have only a 1% chance of happening." That's actually not a consistent way to reason, because if you wait long enough, events of probability 1/100 will happen.

 

[bhobba]

But what I'm saying is that the law of large numbers by itself doesn't imply that one must treat probability 0 events as impossible.

But the meaning of "almost surely" is "true for all sequences except for a set of measure 0". Once again, the issue is why does having a set of measure 0 mean "impossible"?

OK. I think I see where you are coming from. In applying the Kolmogorov Axioms they do not force you to associate a probability of zero with an event not in the event space and hence impossible. Its a reasonable assumption we make since its just as consistent with it in the event space as not in it. Welcome to applied math. But why anyone would want to make an issue it of it is beyond me. I used to get worked up about this kind of stuff a bit during my ubdergrad days but my teachers eventually cured me of it by pointing out the morass you end up in otherwise. I see a lot of that sort of thing with people discussing QM - you want to scream - why look at it that way and make things harder for yourself. Sometimes you get the distinct impression they take a perverse delight in it.

As you correctly point out the difference between the strong law and weak law of large numbers is the type of convergence - weak is convergence in probability - strong is almost assuredly. But from an applied viewpoint is not really relevant - the simple assumption is we can find a n large enough that is so close to one for all practical purposes it can be taken as one. Its the same sort of thing you see with instantaneous velocity - that's impossible as well but times so short exist it's the same for all practical purposes. People seem to accept that but for some reason not for probability - don't quite know why.

So to me the issue in applying probability is not one of circularity - its simply one of reasonable assumptions like we do in many areas of applied math.

I am ignoring issues with zero probability in continuous and infinite event spaces which is a whole new can of worms

Thanks
Bill

 

[stevendaryl]

OK. I think I see where you are coming from. In applying the Kolmogorov Axioms they do not force you to associate a probability of zero with an event not in the event space and hence impossible. Its a reasonable assumption we make since its just as consistent with it in the event space as not in it. Welcome to applied math. But why anyone would want to make an issue it of it is beyond me.

I'm not making an issue about anyone ASSUMING probability zero events never happen. I'm making an issue about the claim that that's somehow provably true. It's not. So in complaining about MWI that it doesn't justify the Born interpretation, my reaction is: what would it even mean to justify it?

There's a thought-experiment that sort-of relates to MWI. Imagine a world that is completely deterministic except for coin flips, and as far as anyone knows, there is no way to predict what the result will be. Let's assume that the way it works is that whenever anyone begins a coin flip, God stops time and makes two copies of the world, in the exact same state. In one of the copies, he let's the coin flip yield "heads" and in the other copy, he let's it yield "tails".

This metaphysics is completely deterministic, like MWI. But for people living in one of the copies of the world, it will appear that coin flips are non-deterministic. Are those people justified in assuming that a coin flip has probability \frac{1}{2} of resulting in "heads"? I think they would be. But is that probability derivable from the metaphysics?

But suppose instead of God making two copies, with one copy having result "heads" and the other copy having the result "tails", he makes three copies, two having result "heads" and only one having the result "tails"? Does that change the probability?

On the one hand, you could say that in this new metaphysics, a coin has a \frac{2}{3} chance of resulting in "heads" and a \frac{1}{3} chance of resulting in "tails". On the other hand, to the people living in one of the copies, it can't possibly make any difference what happens in a world that isn't their world. If they were previously justified in assuming "heads" has a probability of \frac{1}{2}, it is certainly permitted for them to continue to make that assumption.

I used to get worked up about this kind of stuff a bit during my ubdergrad days but my teachers eventually cured me of it by pointing out the morass you end up in otherwise.

I have no problems with people just saying: I'm not going to get into it, I'm going to assume X, to make things simpler. But I do have problems with people failing to realize that X is an assumption, not a necessary truth, and that reasonable people could make a different assumption.

I see a lot of that sort of thing with people discussing QM - you want to scream - why look at it that way and make things harder for yourself. Sometimes you get the distinct impression they take a perverse delight in it.

It's called trying to understand. People understand things by looking at them from lots of different angles.

As you correctly point out the difference between the strong law and weak law of large numbers is the type of convergence - weak is convergence in probability - strong is almost assuredly. But from an applied viewpoint is not really relevant - the simple assumption is we can find a n large enough that is so close to one for all practical purposes it can be taken as one. Its the same sort of thing you see with instantaneous velocity - that's impossible as well but times so short exist it's the same for all practical purposes. People seem to accept that but for some reason not for probability - don't quite know why.

As I said, I don't have any problem with making the assumption that relative frequencies approach probabilities. I just want it to be clear that it's an assumption, not a necessary truth. It simplifies our reasoning to make it (in some cases, we can't do much reasoning at all without it, or something similar). But to me, it strongly suggests Bayesianism. Probabilities are, at least partially, subjective.

 

[mfb]

@mfb Why would you assume it must interact with us?

I don't assume that. On the contrary, my point is that this is not a useful requirement for the existence of objects.

First tell me of something that exists that interacts (or will interact) with *NOTHING*, and how you know it exists.

How is that related to the discussion?

Assuming anyone actually thinks *WE* must observe something for it to exist is a mis-interpretation of what is being said.

I heard exactly that argument so many times...

 

[morrobay]

For locality with probability and statistics see:www.arxiv.org/pdf/quant-ph/0007005v2.pdf
For locality with geometric explanation see: www.iisc.ernet.in/currsci/jul252000/UNNIKRISHNAN.pdf
Regarding the inequality spin violations , I would like to hear from the particle physicists on whether
there is an explanation in terms of the spin changes of the particle from its interaction with the
detectors magnetic field. Sometimes it seems like non-locality is a sacred cow.

 

[bhobba]

For locality with probability and statistics see:www.arxiv.org/pdf/quant-ph/0007005v2.pdf

Yes indeed it does seem that entanglement with its strange correlations is the key thing that separates standard probability theory from QM:
http://arxiv.org/pdf/0911.0695v1.pdf

Thanks
Bill

 

[meBigGuy]

I'd like to move forward to refine, my definition of observer. I still say that an observer is anything that interacts with an observed (or symmetrical observer) within space/time. A particle collides with another, or, a photon is absorbed. Consciousness, awareness, note-taking, detection, are all superfluous to observation in a QM sense. I think you can just say that energy must change (exchange?). I specify within space-time because a photon exists on it's long trek to my eye. (Feynman talked about that).

The Albert's-socks discussion makes a significant point about the existence of "that which is not observed", and that point is key to entanglement. The difference between "preparing the dresser" and "arranging the socks", if you will.

 

[bhobba]

I'd like to move forward to refine, my definition of observer. I still say that an observer is anything that interacts with an observed (or symmetrical observer) within space/time.

You can define it as anything you like, but the usual definition of an observation is its an interaction that leaves a mark here in the macro world. An observer is anything capable of doing it. QM is a theory about the outcomes of observations as defined the way I did it. It is not a theory about the way you defined it because QM does not tell us anything until it is actually observed, by which is meant a mark is left.

I think it would be useful for you to learn about QM from a modern textbook such as Ballentine - QM - A Modern Development where it is developed from 2 axioms - one of which is based on the definition I gave previously.

Thanks
Bill

 

[harrylin]

[..] The correlations in Bell's theorem imply that Alice measuring spin along a certain axis has an instantaneous effect on the probability distribution of the results of Bob's measurement. So retreating into the indeterminacy of the Copenhagen interpretation does not appear to have allowed us to preserve locality since an instantaneous effect has occurred across a spacelike interval.[..]

No instantaneous physical effect at a distance has to occur at all for an instantaneous effect on the probability distribution somewhere else (according to us here), as Bell also clarified in the introduction of his "Bertlmann's socks" paper.
As you likely realize, the issue is much trickier than that.

 

[stevendaryl]

For locality with probability and statistics see:www.arxiv.org/pdf/quant-ph/0007005v2.pdf

What the authors are saying here doesn't seem correct to me. Bell's "vital assumption" is that for the EPR twin-pair experiment, the result of the measurement at one detector does not depend on the settings of the distant detector. The authors say that Bell's inequality does not depend on this assumption, but that seems completely wrong.

Let's define two functions

F^{(1)}_a(\lambda, m_1, m_2)
F^{(2)}_a(\lambda, m_1, m_2)

[edit:]
(where m_1 is the detector settings at the first detector, and m_2 is the detector settings at the other detector) as follows:

F^{(1)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda < \frac{1}{2}.
= -1 if \frac{1}{2} \leq \lambda \leq 1.

F^{(2)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda < \frac{1}{2} sin^2(\frac{\theta}{2}).
= -1 if \frac{1}{2} sin^2(\frac{\theta}{2})\leq \lambda \leq 1.

where \theta is the angle between the two detector orientations. That's a perfectly deterministic (although non-local) "hidden variable" theory that reproduces exactly the predictions of QM and violates Bell's inequality.

 

[meBigGuy]

"You can define it as anything you like, but the usual definition of an observation is its an interaction that leaves a mark here in the macro world." "QM does not tell us anything until it is actually observed, by which is meant a mark is left"

Are you saying that QM tells us nothing about an electron that has not yet been measured?

Again, I want to better understand the nuances of "preparing the dresser" and "arranging the socks". The socks exist, but have not been observed *yet*. QM tells us nothing about them?

 

[harrylin]

What the authors are saying here doesn't seem correct to me. Bell's "vital assumption" is that for the EPR twin-pair experiment, the result of the measurement at one detector does not depend on the settings of the distant detector. The authors say that Bell's inequality does not depend on this assumption, but that seems completely wrong.

Let's define two functions

F^{(1)}_a(\lambda, m_1, m_2)
F^{(2)}_a(\lambda, m_1, m_2)

[edit:]
(where m_1 is the detector settings at the first detector, and m_2 is the detector settings at the other detector) as follows:

F^{(1)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda < \frac{1}{2}.
= -1 if \frac{1}{2} \leq \lambda \leq 1.

F^{(2)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda < \frac{1}{2} sin^2(\frac{\theta}{2}).
= -1 if \frac{1}{2} sin^2(\frac{\theta}{2})\leq \lambda \leq 1.

where \theta is the angle between the two detector orientations. That's a perfectly deterministic (although non-local) "hidden variable" theory that reproduces exactly the predictions of QM and violates Bell's inequality.

Likely you mean the inverse of the second function, but that's a detail. If I see it correctly, the probabilities are not the same for your two functions. While you refer to the same λ, the effective hidden parameter functions are different because they are different functions of that λ.The authors say that what Bell's inequality depends on, is that the random variables are defined on the same probability space.

BTW, it seems to me that on p.14 the first time that they use "predetermined", they mean "pre-existing" or "fixed". They explain it better further on.

 

[bhobba]

Are you saying that QM tells us nothing about an electron that has not yet been measured?

Exactly. Other than probabilities of what you would get if you were to measure it - nothing.

I will say it again. QM is a theory about the results of interactions with other systems where those systems leave a mark here in the common sense macro world. What properties it has otherwise it is silent about.

Now a question for you. You do understand that all a state tells you is the possible outcomes of an observable and their probabilities?

This sock thing you keep on talking about is simply a thought experiment illustrating the correlations of entangled systems. This is that some systems are such that if you measure the state of one part of the system you know automatically the state of another part of the system and the result you would get if you were to measure it, even if that other part is on the other side of the universe. This is one of the weird aspects of entanglement. It changes nothing about what a system state tells us, which is about the results of observations.

Here is a way of looking at QM at a foundational level that may make it clearer. Suppose you have a system and some observational apparatus that has n possible outcomes and you associate a number with each of the outcomes. This is represented by a vector of size n with n numbers yi. To bring this out write it as Ʃyi |bi>. Now we have a problem. The |bi> are freely chosen so nature can not depend on them. We need a way to represent it that does not depend on that. The way QM gets around it is to replace the |bi> by |bi><bi| giving Ʃyi |bi><bi|. This is defined as the observable of the observational apparatus. It says to each such apparatus there is a Hermitian operator whose eigenvalues are the possible outcomes of the observation. This is the first axiom of QM. The second axiom says the expected outcome of such an observation is Tr(PR) where R is the observable of the observation and P is a positive operator of unit trace called the state of the system. This can be proven by what is known as Gleason's Theorem if we assume something called non contextuality that you can read up on if you wish. So while the second axiom is not implied by the first it is strongly suggested by it - depending on exactly what you think of non-contextuality.

It is an interesting fact that all of QM is contained in those two axioms. To get the detail see Ballentine's book. The point here though is right at its foundations it is a theory about one and only one thing - the outcome of observations as indicated by observational apparatus.

Thanks
Bill

 

[stevendaryl]

Likely you mean the inverse of the second function, but that's a detail. If I see it correctly, the probabilities are not the same for your two functions. While you refer to the same λ, the effective hidden parameter functions are different because they are different functions of that λ.The authors say that what Bell's inequality depends on, is that the random variables are defined on the same probability space.

BTW, it seems to me that on p.14 the first time that they use "predetermined", they mean "pre-existing" or "fixed". They explain it better further on.

What I described is a single hidden variable, \lambda, that is shared by the two experimenters. It's the same probability space. But Alice's result is a different function of \lambda than Bob's result. That must be the case, because if Alice and Bob both measure the spin in the same direction, one of them will get spin-up, and the other will get spin-down. They can't possibly have the same dependence on \lambda if they get opposite results for the same \lambda.

 

[stevendaryl]

What the authors are saying here doesn't seem correct to me. Bell's "vital assumption" is that for the EPR twin-pair experiment, the result of the measurement at one detector does not depend on the settings of the distant detector. The authors say that Bell's inequality does not depend on this assumption, but that seems completely wrong.

Let's define two functions

F^{(1)}_a(\lambda, m_1, m_2)
F^{(2)}_a(\lambda, m_1, m_2)

[edit:]
(where m_1 is the detector settings at the first detector, and m_2 is the detector settings at the other detector) as follows:

F^{(1)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda &lt; \frac{1}{2}.
= -1 if \frac{1}{2} \leq \lambda \leq 1.

F^{(2)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda &lt; \frac{1}{2} sin^2(\frac{\theta}{2}).
= -1 if \frac{1}{2} sin^2(\frac{\theta}{2})\leq \lambda \leq 1.

where \theta is the angle between the two detector orientations. That's a perfectly deterministic (although non-local) "hidden variable" theory that reproduces exactly the predictions of QM and violates Bell's inequality.

[edit: I made a mistake in defining the two functions. Here's what I should have written:]

F^{(1)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda &lt; \frac{1}{2}.
= -1 if \frac{1}{2} \leq \lambda \leq 1.

F^{(2)}_a(\lambda, m_1, m_2) = +1 if 0 \leq \lambda &lt; \frac{1}{2} sin^2(\frac{\theta}{2}).
= -1 if \frac{1}{2} sin^2(\frac{\theta}{2})\leq \lambda \leq \frac{1}{2} (1 + sin^2(\frac{\theta}{2})).
= +1 if \frac{1}{2} (1 + sin^2(\frac{\theta}{2})) \leq \lambda \leq 1

This gives
P(A) = P(B) = \frac{1}{2}
P(A \wedge B) = P(\neg A \wedge \neg B) = \frac{1}{2}sin^2(\theta)