Bell experiment would somehow prove non-locality and information FTL?

[JesseM]

HEY JESSE, have you MISQUOTED wm. (Which certainly doesn't help his case at all.)

Yes, sorry, from the context I think it was pretty clear that I was quoting DrChinese and had just accidentally typed the wrong name. Fixed now.

 

[JesseM]

Even if you could somehow do this, you still would not disprove Bell's Theorem, which states (in my words):

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

Well; I just did ''this'' (the classical experiment).

As I pointed out earlier, your classical version will not work for arbitrary distances between Alice and the source and arbitrarily fast switching between measurement angles for Alice and Bob. In particular, if Alice and Bob randomly switch their measurement angles every x seconds, and their distance from the source is larger than x light-seconds (ensuring that by the time the source gets any signal about Alice's settings, she has already switched to a new random setting), then you won't be able to get any violation of the Bell inequality when you compare Alice and Bob's measurements at a given time, assuming you're doing a classical experiment involving non-entangled photons (or any classical object/signal, like classical EM waves or digital signals), and assuming there is a spacelike separation between each pair of measurements. Do you disagree?

 

[wm]

OK, just trying to make sure I understand the math here: Now, you said you were using the angles a = 0, b = 67.5, c = 45. So, calculating P(BC = S|bc), P(AC = D|ac) and P(AB = S|ab) explicitly, we have:

<CUT>

Now that I think I understand your proposal, I'll explain why it isn't a genuine violation of Bell's theorem in a followup post.

Jesse, I haven't checked all the details, but you seem to on the right path.

I just wanted to rush to you an issue that might be confusing much on this thread:

While I am confident that other work is a genuine violation of Bell's theorem, the proposal we are discussing is designed to violate Bell's theorem in the (limited) following way:

Peres, CHSH, etc give Bellian inequalities based on dichotomic + xor - outcomes. Such inequalities are often supported by ''simplistic'' classical examples (eg, down-hill skiers, dirty-socks). My proposal shows (I believe) that there are classical settings which breach their inequalities.

So Bell's Theorem is challenged to the extent that it leads to the common-garden Bellian-Inequalities.

There are multi-particle experimental results which call for more complex analysis. While I am happy to provide such, that was NOT the goal of the CLASSICAL experiment we are discussing. The one that you appear to be on top of.

Hope this clarifies somewhat, wm

 

[DrChinese]

But with the yoking, you're making the entanglement irrelevant so it can be viewed as a classical problem...

JesseM,

wm is saying there are no entangled photons. Ok, then there is nothing important to discuss, you end up coming back to a classical example with colored socks or similar. The issue is: QM, locality, realism and that is all anyone really cares about. I agree with your treatment of the issue but I do not see where any of it is going. Good luck helping wm.

wm,

Your understanding of the context and history of the Bell's Theorem is off, and you would be better served by additional research before you put forth your assertions. Bell's Theorem stands. Also, you seriously mischaracterize Bell's position. He was not sorry for creating his theorem, and as best I can tell evolved to believe in a non-local view of hidden variables.

I will continue to follow the thread, but will lay low for a while. I think wm's example has become so convoluted as to primarily touch on issues that are unproductive. I will chime again if I see something I might be able to add.

Cheers,

-DrC

 

[JesseM]

My proposal shows (I believe) that there are classical settings which breach their inequalities.

Sure, but Bell's theorem doesn't say the inequalities can never be violated in a classical universe, just there are certain specific conditions under which they won't be violated in a classical universe, such as the condition that there's a spacelike separation between the two measurements. If you look at an experiment which doesn't respect these conditions (as yours does not), violating the inequalities is quite trivial! For example, if Alice and Bob each ask me one of three yes-or-no questions every 10 minutes, and I only answer once I have heard both their questions, it'll be quite easy for me to make sure that I always give the same answer to both when they ask the same question, while giving different answers more than 1/3 of the time when they ask different questions. Your example doesn't challenge Bell's theorem any more than this one does.

 

[wm]

As I pointed out earlier, your classical version will not work for arbitrary distances between Alice and the source and arbitrarily fast switching between measurement angles for Alice and Bob. In particular, if Alice and Bob randomly switch their measurement angles every x seconds, and their distance from the source is larger than x light-seconds (ensuring that by the time the source gets any signal about Alice's settings, she has already switched to a new random setting), then you won't be able to get any violation of the Bell inequality when you compare Alice and Bob's measurements at a given time, assuming you're doing a classical experiment involving non-entangled photons (or any classical object/signal, like classical EM waves or digital signals), and assuming there is a spacelike separation between each pair of measurements. Do you disagree?

Given the above defined conditions, I fully agree.

BUT NOTE: You are adding conditions that were never intended to be addressed by the proposal. Please see a recent reply to vanesch: Alice sits beside her detector and the box, which are close-coupled (like much lab equipment), ON HER DESK. Bob is far away, etc.

This is another clarification which needs to be added to the proposal's specification to maintain its focus on a simple classical device.

Thanks, wm

 

[wm]

Sure, but Bell's theorem doesn't say the inequalities can never be violated in a classical universe, just there are certain specific conditions under which they won't be violated in a classical universe, such as the condition that there's a spacelike separation between the two measurements. If you look at an experiment which doesn't respect these conditions (as yours does not), violating the inequalities is quite trivial! For example, if Alice and Bob each ask me one of three yes-or-no questions every 10 minutes, and I only answer once I have heard both their questions, it'll be quite easy for me to make sure that I always give the same answer to both when they ask the same question, while giving different answers more than 1/3 of the time when they ask different questions. Your example doesn't challenge Bell's theorem any more than this one does.

I think that it does. Does your example breach the CHSH Inequality?

Please let me know, wm

 

[wm]

JesseM,

wm is saying there are no entangled photons.

Doc, enjoy your vacation; I hope you'll soon be back.

But for the record: I did not say so loosely ''there are no entangled photons''. From the very beginning I referred to Aspect's experiment (= entangled photons) ...

Ok, then there is nothing important to discuss, you end up coming back to a classical example with colored socks or similar. The issue is: QM, locality, realism and that is all anyone really cares about. I agree with your treatment of the issue but I do not see where any of it is going. Good luck helping wm.

I'm happy to acknowledge Jesse's help; help more than many (including Jesse) may appreciate.

wm,

Your understanding of the context and history of the Bell's Theorem is off, and you would be better served by additional research before you put forth your assertions. Bell's Theorem stands. Also, you seriously mischaracterize Bell's position. He was not sorry for creating his theorem, and as best I can tell evolved to believe in a non-local view of hidden variables.

I rely on the Mermin quote: ''To those for whom nonlocality is anathema, Bell's Theorem finally spells the death of the hidden variables program. But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible.'' (Mermin, Rev. Mod. Phys. 65 page 814, 1993) [emphasis added].

NB: IT WAS WRITTEN AFTER BELL'S DEATH.

I'd (of course) welcome a citation from an equivalent authority giving a dissident view.

With best regards, wm

 

[JesseM]

I think that it does. Does your example breach the CHSH Inequality?

Please let me know, wm

In general, any inequality that's violated in QM should also be easy to violate in a classical question-and-answer game like the one I described, where I get to hear both questions before giving my answers--all I have to do is translate the questions into settings of some hypothetical inequality-violating quantum experiment, calculate in my head what the probability is that each experiment will get a + or - according to quantum mechanics, and then make my probabilities of answering "yes" or "no" proportional to these calculated probabilities! Obviously an ordinary classical computer can calculate the probabilities in any quantum experiment, so no actual quantum effects are needed here.

But I can try to come up with a simpler strategy to violate a given inequality in a question-and-answer game, if you like. I looked up the CHSH inequality and could only find it in the wikipedia article which stated it in terms of the expectation values, I'd like to restate it in terms of probabilities to make it easier to think up with an example. In terms of expectation values, if both Alice and Bob have two measurement settings A and B which can each yield two results + or -, assigned values +1 and -1, then the CHSH inequality says:

-2 <= X <= 2

('<=' stands for 'smaller than or equal to', when I typed the actual symbol the board's softward replaced it with a question mark)

where

X = E(Alice measures A, Bob measures A) - E(Alice measures A, Bob measures B) + E(Alice measures B, Bob measures A) + E(Alice measures B, Bob measures B)

Converting expectation values into probabilities should give:

1. E(Alice measures A, Bob measures A) = P(S|aa) - P(D|aa)
2. E(Alice measures A, Bob measures B) = P(S|ab) - P(D|ab)
3. E(Alice measures B, Bob measures A) = P(S|ba) - P(D|ba)
4. E(Alice measures B, Bob measures B) = P(S|bb) - P(D|bb)

Since P(D|aa) = 1 - P(S|aa) and so forth, this simplifies to:

1. E(Alice measures A, Bob measures A) = 2*P(S|aa) - 1
2. E(Alice measures A, Bob measures B) = 2*P(S|ab) - 1
3. E(Alice measures B, Bob measures A) = 2*P(S|ba) - 1
4. E(Alice measures B, Bob measures B) = 2*P(S|bb) - 1

So, that should give X = 2*[P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab)] - 3

which means -2 <= X <= 2 is equivalent to:

1/2 <= P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab) <= 5/2

Would this be a valid formulation of one case of the CHSH inequality? (the case when Alice and Bob both choose between the same two measurements A and B--from the way it was formulated on wikipedia it seems to be more general than that) Let me know if I made an error in math or understanding. Also, what is being assumed about the results when both choose the same detector settings? Are P(S|aa) and P(S|bb) equal to 1, or 0? Or can I just assume any probability between 0 and 1 for each of the four? Once I'm clear on the equation for the CHSH inequality in terms of probabilities, and on what assumptions are being made about identical settings, I should be able to come up with some simple strategy for answering the questions in a way that the inequality is violated.

 

[JesseM]

Also, you seriously mischaracterize Bell's position. He was not sorry for creating his theorem, and as best I can tell evolved to believe in a non-local view of hidden variables.

I rely on the Mermin quote: ''To those for whom nonlocality is anathema, Bell's Theorem finally spells the death of the hidden variables program. But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible.'' (Mermin, Rev. Mod. Phys. 65 page 814, 1993) [emphasis added].

Isn't that exactly what DrChinese just said? (note the part I put in bold) Bell wasn't persuaded that hidden variables were impossible, so he looked for a non-local hidden variables theory (since he was not one 'for whom nonlocality is an anathema'), because Bell's theorem showed a local hidden variables theory wouldn't work.

 

[DrChinese]

I rely on the Mermin quote: ''To those for whom nonlocality is anathema, Bell's Theorem finally spells the death of the hidden variables program. But not for Bell. None of the no-hidden-variables theorems persuaded him that hidden variables were impossible.'' (Mermin, Rev. Mod. Phys. 65 page 814, 1993) [emphasis added].

You read this differently than I, I certainly have no argument with Mermin. Bell was a believer in non-local hidden variables at the end. There are other theorems which address the hidden variable issue other than Bell's Theorem, these usually address what is called non-contextual definiteness.

 

[MeJennifer]

... such as the condition that there's a spacelike separation between the two measurements.

But how can we exclude that there is no spacelike separation as you suggest?

For instance consider a quantum system with two photons each traveling in opposite direction with the speed of c. Now could we show they are causally disconnected? Both particles travel on a null path and hence are causally connected if we time reverse the path of each particle to the initial quantum state. Certainly photons do not violate time symmetry.

In this light think of Wheeler-Feynman like theories.

I am not claiming that that is actually the case, but can it be excluded?

 

[JesseM]

For instance consider a quantum system with two photons each traveling in opposite direction with the speed of c. Now could we show they are causally disconnected? Both particles travel on a null path and hence are causally connected if we time reverse the path of each particle to the initial quantum state. Certainly photons do not violate time symmetry.

That's not how spacelike vs. timelike separations work in relativity, it has nothing to do with "reversing paths", just on whether one event lies in the other's light cone, or whether (in an inertial coordinate system in flat spacetime) ds^2 = dx^2 + dy^2 + dz^2 - c^2*dt^2 is positive or negative. If two photons travel in opposite directions in flat spacetime, the two events of each photon being received by detectors will always have a spacelike separation. Two events with a spacelike separation that have a common cause are still "causally disconnected" in the sense that neither event can affect the other one, although they may both have been affected by some event which lies in both their past light cones, and they may both have an effect on some event which lies in both their future light cones.

 

[MeJennifer]

That's not how spacelike vs. timelike separations work in relativity, it has nothing to do with "reversing paths", just on whether one event lies in the other's light cone, or whether (in an inertial coordinate system in flat spacetime) ds^2 = dx^2 + dy^2 + dz^2 - c^2*dt^2 is positive or negative. If two photons travel in opposite directions in flat spacetime, the two events of each photon being received by detectors will always have a spacelike separation. Two events with a spacelike separation that have a common cause are still "causally disconnected" in the sense that neither event can effect the other one, although they may both have been affected by some event which lies in both their past light cones, and they may both have an effect on some event which lies in both their future light cones.

I am not talking about the detectors.
I am talking about the time-reversed null path of each photon towards the initial quantum state.

 

[JesseM]

I am not talking about the detectors.
I am talking about the time-reversed null path of each photon towards the initial quantum state.

OK, but you were quoting my statement "such as the condition that there's a spacelike separation between the two measurements", which referred to the detection-events. Certainly the two entangled photons have a common origin point in spacetime, but the proof of Bell's theorem doesn't forbid this. In fact, Bell's theorem is based on assuming that the reason entangled particles show perfect correlation when measured on the same axis is because they were generated from a common source in either identical or opposite states, and then showing that you can't explain the violation of Bell inequalities when different axes are measured without violating locality.

 

[Anonym]

Since the 1990s the experiments have been definitive; no physicist doubts the existence of nonlocality.

Under request, I will show you my documents.

The question wether - prior to observation - a particle is in a defined state, is something unknowable. Knowing the state of the particle requires an observation and this observation alters the state of the particle.

Dear Karl, you have no idea what you are talking about. Your followers would send you to Solovki for the presented level of knowledge of quantum physics. At least, ask your friend Friedrich what is his point of view.

I just read your discussions here as a matter of curiosity. I have two basic questions:

1. I lost track from “old” literature that said that the spin of free electron can’t be measured (N.Bohr, L.Rosenfeld, N.F.Mott, H.S.W. Massey). What is the current status?
2. What is the practical purpose of all that entanglement stuff? If Bob want to put Alice in his bed, why not to do that locally?

Crosson version is only slightly more complicated:” Here is a description of EPR:Imagine that there is a pair of (literally) identical twin brothers who are interested in dating a pair of identical twin sisters. The brothers live together, but their dates live separate lives on opposite sides of town.” Again,why not to do that locally?

In separate Fock spaces.

 

[heusdens]

Under request, I will show you my documents.
Dear Karl, you have no idea what you are talking about. Your followers would send you to Solovki for the presented level of knowledge of quantum physics. At least, ask your friend Fridrich what is his point of view.

I just read your discussions here as a matter of curiosity. I have two basic questions:

1. I lost track from “old” literature that said that the spin of free electron can’t be measured (N.Bohr, L.Rosenfeld, N.F.Mott, H.S.W. Massey). What is the current status?
2. What is the practical purpose of all that entanglement stuff? If Bob want to put Alice in his bed, why not to do that locally?

Crosson version is only slightly more complicated:” Here is a description of EPR:Imagine that there is a pair of (literally) identical twin brothers who are interested in dating a pair of identical twin sisters. The brothers live together, but their dates live separate lives on opposite sides of town.” Again,why not to do that locally?

In separate Fock spaces.

Greetings to you.

The discoveries of the material sciences had lead to progress in the human understanding of the world and in technology, but the working class is hardly any better of.

Although I and Friedrich are very unfamiliar with this field of knowledge since we studied it, we can not escape from telling that these new discoveries do not conlict with dialectical materialism. Science has found what we already expected to find, that the material world is unlimitedless.

Althoug we must say, some interpretations of the quantum mechanics, which have made statements that seem to reject materialism. This interpretation however can however not be made, since the senseous experiment of the material world are so to say "acts of the flesh" and not "acts of the mind".

"Motion is the mode of existence of matter. Never anywhere has there been matter without motion, or motion without matter, nor can there be."

"Change of form of motion is always a process that takes place between at least two bodies, of which one loses a definite quantity of motion of one quality (e.g. heat), while the other gains a corresponding quantity of motion of another quality (mechanical motion, electricity, chemical decomposition).

"Dialectics, so-called objective dialectics, prevails throughout nature, and so-called subjective dialectics (dialectical thought), is only the reflection of the motion through opposites which asserts itself everywhere in nature, and which by the continual conflict of the opposites and their final passage into one another, or into higher forms, determines the life of nature."

Fredrick Engels
Dialectics of Nature

But dialectical materialism insists on the approximate relative character of every scientific theory of the structure of matter and its properties; it insists on the absence of absolute boundaries in nature, on the transformation of moving matter from one state into another, that from our point of view [may be] apparently irreconcilable with it, and so forth.

Vladimir Lenin
Materialism and Empirio-criticism Karl.

 

[heusdens]

Here's another example of brilliant dialectical thougt, which predicts that the attraction of gravity is only one-sided approach, and that even gravity must have both sides (attraction and repulsion). This kind of gravity repulsive force is conceived of in current cosmological models in the form of cosmological inflation.

Gravity as the most general determination of materiality is commonly accepted. That is to say, attraction is a necessary property of matter, but not repulsion. But attraction and repulsion are as inseparable as positive and negative, and hence from dialectics itself it can already be predicted that the true theory of matter must assign as important a place to repulsion as to attraction, and that a theory of matter based on mere attraction is false, inadequate, and one-sided. In fact sufficient phenomena occur that demonstrate this in advance. If only on account of light, the ether is not to be dispensed with. Is the ether of material nature? If it exists at all, it must be of material nature, it must come under the concept of matter. But it is not affected by gravity. The tail of a comet is granted to be of material nature. It shows a powerful repulsion. Heat in a gas produces repulsion, etc.

* * *​

Attraction and gravitation. The whole theory of gravitation rests on saying that attraction is the essence of matter. This is necessarily false. Where there is attraction, it must be complemented by repulsion. Hence already Hegel was quite right in saying that the essence of matter is attraction and repulsion.[194] And in fact we are more and more becoming forced to recognise that the dissipation of matter has a. limit where attraction is transformed into repulsion,. and conversely the condensation of the repelled matter has a limit where it becomes attraction.

* * *​

The transformation of attraction into repulsion and vice versa is mystical in Hegel, but in substance he anticipated by it the scientific discovery that came later. Even in a gas there is repulsion of the molecules, still more so in more finely-divided matter, for instance in the tail of a comet, where it even operates with enormous force. Hegel shows his genius even in the fact that he derives attraction as something secondary from repulsion as something preceding it: a solar system is only formed by the gradual preponderance of attraction over the originally prevailing repulsion. – Expansion by heat=repulsion. The kinetic theory of gases.

(from Dialectics of Nature)

 

[Anonym]

Dear Karl,

One very famous physicist (level compatible with A. Einstein) wrote:” We are facing here the perennial question whether we physicists do not go beyond our competence when searching for philosophical truth.”

The philosophy is far beyond my competence. I can’t argue against you. I may only ask you a question that was interested me during all my life: Roots of the dialectical materialism lies in the ancient Indian philosophy (niaa and vaisheishika, sorry if I write not correctly). How the closely related to them dialectical materialism was translated to deterministic applications of Solovki,Gulag and similarly in other countries?

Personally, I hate the dialectical materialism. I was supposed to spend several years in jail only for absence of knowledge of the dialectical materialism. Unbelievable courage of idealistic philosophy professor which did not know me at all saved me.

From Dialectics of Nature:

“Gravity as the most general determination of materiality is commonly accepted. That is to say, attraction is a necessary property of matter, but not repulsion. But attraction and repulsion are as inseparable as positive and negative, and hence from dialectics itself it can already be predicted that the true theory of matter must assign as important a place to repulsion as to attraction, and that a theory of matter based on mere attraction is false, inadequate, and one-sided. In fact sufficient phenomena occur that demonstrate this in advance. If only on account of light, the ether is not to be dispensed with. Is the ether of material nature? If it exists at all, it must be of material nature, it must come under the concept of matter. But it is not affected by gravity. The tail of a comet is granted to be of material nature. It shows a powerful repulsion. Heat in a gas produces repulsion, etc.
* * *

Attraction and gravitation. The whole theory of gravitation rests on saying that attraction is the essence of matter. This is necessarily false. Where there is attraction, it must be complemented by repulsion. Hence already Hegel was quite right in saying that the essence of matter is attraction and repulsion.[194] And in fact we are more and more becoming forced to recognise that the dissipation of matter has a. limit where attraction is transformed into repulsion,. and conversely the condensation of the repelled matter has a limit where it becomes attraction.
* * *

The transformation of attraction into repulsion and vice versa is mystical in Hegel, but in substance he anticipated by it the scientific discovery that came later. Even in a gas there is repulsion of the molecules, still more so in more finely-divided matter, for instance in the tail of a comet, where it even operates with enormous force. Hegel shows his genius even in the fact that he derives attraction as something secondary from repulsion as something preceding it: a solar system is only formed by the gradual preponderance of attraction over the originally prevailing repulsion. – Expansion by heat=repulsion. The kinetic theory of gases.”

I do not see any difference compare with writings of Northerdamus. However, “The whole theory of gravitation rests on saying that attraction is the essence of matter” is completely wrong statement. The whole theory of gravitation rests on the universally valid experimental result that the inertial mass is identical to the gravitation mass.
By the way, my physical intuition says to me that the gravitation is only attractive similarly to the strong interaction.

Dany.

P.S. It is written in your reference: Frederick Engels (1873-1886).
Does it mean that FE lived only 13 years?

 

[DrChinese]

Dear Karl,

One very famous physicist (level compatible with A. Einstein) wrote:” We are facing here the perennial question whether we physicists do not go beyond our competence when searching for philosophical truth.”

The philosophy is far beyond my competence...

Dear Anonym,

This material is far off thread and belongs elsewhere, perhaps in the philosophy section. You should probably review the forum posting guidelines.

Regards,

-DrC

 

[wm]

Dear Karl,

<SNIP>

The transformation of attraction into repulsion and vice versa is mystical in Hegel, but in substance he anticipated by it the scientific discovery that came later. Even in a gas there is repulsion of the molecules, still more so in more finely-divided matter, for instance in the tail of a comet, where it even operates with enormous force. Hegel shows his genius even in the fact that he derives attraction as something secondary from repulsion as something preceding it: a solar system is only formed by the gradual preponderance of attraction over the originally prevailing repulsion. – Expansion by heat=repulsion. The kinetic theory of gases.”

<SNIP>

[Emphasis added above] Some light relief; of more than passing interest; in partial explanation of a current pre-occupation; welcome to Comet McNaught:

http://news.nationalgeographic.com/news/2007/01/070118-comet.html

 

[mgelfan]

... Two events with a spacelike separation that have a common cause are still "causally disconnected" in the sense that neither event can affect the other one ...

The statistics accumulated via spacelike separated events in EPR-Bell experiments are not independent of each other. An initial detection at either A or B affects the sample space of the other. That is, a detection at one end causes the sample space at the other end to be modified from a random to a non-random sampling (of the incident optical disturbances) for that pairing.

This (dependence of A upon B, and/or vice versa) is why the mathematical formulation chosen to represent locality (ie., factorability) does not represent locality -- at least regarding the way in which EPR-Bell tests are usually prepared.

Quantum theory assumes that the analyzers/filters at A and B have, in effect, analyzed/filtered the same incident optical disturbance for paired detection attributes.

Once this assumption is made, and preparations are made to insure that disturbances emitted in opposite directions during the same transition/emission interval are being paired via coincidence circuitry, then the quantum mechanical predictions just follow from elementary classical optics, more or less.

No spooky action at a distance. No nonlocality.

 

[wm]

I think that it does. Does your example breach the CHSH Inequality?

Please let me know, wm

In general, any inequality that's violated in QM should also be easy to violate in a classical question-and-answer game like the one I described, where I get to hear both questions before giving my answers--all I have to do is translate the questions into settings of some hypothetical inequality-violating quantum experiment, calculate in my head what the probability is that each experiment will get a + or - according to quantum mechanics, and then make my probabilities of answering "yes" or "no" proportional to these calculated probabilities! Obviously an ordinary classical computer can calculate the probabilities in any quantum experiment, so no actual quantum effects are needed here.

Jesse, I'm happy for you to explore any realistic option that you can think of. I want to encourage you to do so.

For it seems to me that Bell's theorem [as reflected in the vast literature on Bellian inequalities (BI)] is, in general, FALSE. And it seems to me that this falsity derives from the associated REALISM.

That is, to be clearer: Models satisfying a Bellian inequality will be NOT be satisfactory as a general representation of realism. For they will be based on the subset thereof -- known as Bell realism (or naive realism; or strong realism ... ).

But I can try to come up with a simpler strategy to violate a given inequality in a question-and-answer game, if you like.

Well I think that this approach would be closer to the spirit of the BI literature and my question re CHSH. (Otherwise, you may as well stand beside the experiment and call the results; or feed the results directly to the computer; as I read the above?)

I looked up the CHSH inequality and could only find it in the wikipedia article which stated it in terms of the expectation values, I'd like to restate it in terms of probabilities to make it easier to think up with an example. In terms of expectation values, if both Alice and Bob have two measurement settings A and B which can each yield two results + or -, assigned values +1 and -1, then the CHSH inequality says:

-2 <= X <= 2

('<=' stands for 'smaller than or equal to', when I typed the actual symbol the board's softward replaced it with a question mark)

where

X = E(Alice measures A, Bob measures A) - E(Alice measures A, Bob measures B) + E(Alice measures B, Bob measures A) + E(Alice measures B, Bob measures B)

I think CHSH is more general than that. More like:

(CHSHI) |<AB>+<BC>+<CD> - <DA>| </= 2

Converting expectation values into probabilities should give:

1. E(Alice measures A, Bob measures A) = P(S|aa) - P(D|aa)
2. E(Alice measures A, Bob measures B) = P(S|ab) - P(D|ab)
3. E(Alice measures B, Bob measures A) = P(S|ba) - P(D|ba)
4. E(Alice measures B, Bob measures B) = P(S|bb) - P(D|bb)

Since P(D|aa) = 1 - P(S|aa) and so forth, this simplifies to:

1. E(Alice measures A, Bob measures A) = 2*P(S|aa) - 1
2. E(Alice measures A, Bob measures B) = 2*P(S|ab) - 1
3. E(Alice measures B, Bob measures A) = 2*P(S|ba) - 1
4. E(Alice measures B, Bob measures B) = 2*P(S|bb) - 1

So, that should give X = 2*[P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab)] - 3

which means -2 <= X <= 2 is equivalent to:

1/2 <= P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab) <= 5/2

Would this be a valid formulation of one case of the CHSH inequality?

In that the central expression can equal 2, then you've confirmed CHSH.

(the case when Alice and Bob both choose between the same two measurements A and B--from the way it was formulated on wikipedia it seems to be more general than that)

Yes; the relation you need to target is:

(CHSHI) |<AB>+<BC>+<CD> - <DA>| </= 2.

Let me know if I made an error in math or understanding. Also, what is being assumed about the results when both choose the same detector settings? Are P(S|aa) and P(S|bb) equal to 1, or 0? Or can I just assume any probability between 0 and 1 for each of the four?

Your task (as I see it) is to define an experiment in your own terms; then rebut CHSH. Alternatively or in parallel: Use my experiment to rebut CHSH and see if that helps in you building your own model.

Once I'm clear on the equation for the CHSH inequality in terms of probabilities, and on what assumptions are being made about identical settings, I should be able to come up with some simple strategy for answering the questions in a way that the inequality is violated.

I would say rather: The specific experiment provides the Probability for identical settings; no further assumptions needed, as I see it.

Hope this helps, wm

 

[JesseM]

Jesse, I'm happy for you to explore any realistic option that you can think of. I want to encourage you to do so.

For I claim that Bell's theorem [as reflected in the vast literature on Bellian inequalities(BI)] is, in general, FALSE. AND (according to me): This falsity derives from the associated REALISM.

You misunderstand me, I'm not "exploring realistic options" for QM, I'm just pointing out that it's a quite trivial observation that you can violate the Bell inequalities classically when you violate some of the conditions on the experiment stipulated by Bell theorem, but that this is of course not a violation of Bell's theorem. Bell's theorem says, in effect, "under conditions X,Y,Z it is impossible for a local realist theory to reproduce the Bell inequalities". All you have shown is a statement to the effect of "if I am allowed to set up an experiment which does not respect condition Y, then I can violate the Bell inequalities with a local realist theory." But this is completely obvious--it's the whole reason they bothered to stipulate those conditions!

All I am pointing out with my examples is how trivial it is that you can violate the Bell inequalities classically if you (the 'source') are aware of what questions you'll be asked ('measurements') before you have to give answers (the 'results' of each measurement). But suppose me and a friend have to travel at close to the speed of light in opposite directions, and then the events of our each being asked a question have a spacelike separation, with the two experimenters picking their questions at random at that moment. In this case, there is no prearranged strategy me and the friend can use to come up with answers that are always the same (or opposite) when asked the same question, but which violate the Bell inequalities when asked different questions, unless we have some kind of faster-than-light psychic connection, or have precognitive abilities that allow us to know what the experimenters will ask in advance, or are able to split ourselves and the experimenters into multiple copies and decide which copies of experimenter 1 are matched with which copies of experimeter 2 later. That's what Bell's theorem is telling you, it doesn't say a thing about there being a problem with violating the Bell inequalities in circumstances where the source can be informed in advance what one or both of the measurements will be.

Well I think that this approach would be closer to the spirit of the BI literature and my question re CHSH. (Otherwise, you may as well stand beside the experiment and call the results; or feed the results directly to the computer; as I read the above?)

No, the strategy of just calculating what quantum mechanics would predict for the probabilities, and basing your answers on these probabilities, can only work if you know what both of the measurements are before making your calculation. Therefore, as I said above, it won't work if you and a friend must travel apart to both experimenters, and the events of being asked the two questions have a spacelike separation, and you each have to answer before there's been time for a signal to pass from one to the other informing each what question the other was asked.

I think CHSH is more general than that. More like:

(CHSHI) |<AB>+<BC>+<CD> - <DA>| </= 2

I realize that, but that's why I said at the end of my last post I was just trying to come up with a specific case: "Would this be a valid formulation of one case of the CHSH inequality? (the case when Alice and Bob both choose between the same two measurements A and B--from the way it was formulated on wikipedia it seems to be more general than that)"

1/2 <= P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab) <= 5/2

Would this be a valid formulation of one case of the CHSH inequality?

In that the central expression can equal 2, then you've confirmed CHSH.

Actually, now that I think about it I think there must be an error either in my understanding or the way it's stated on wikipedia. Even if I respect all the conditions of Bell's theorem, it seems like it wouldn't be too hard to violate this inequality--all I have to do is send one object/signal in state {A = +1, B = +1} to Alice and another in state {A = -1, B = -1} to Bob, and then it's guaranteed that the probability of them getting the same result is 0, no matter what they measure; thus the sum of all those P(S|xx)'s is 0, which is smaller than 1/2. And in terms of expectation values, the product of Alice and Bob's answer is guaranteed to be -1, and the sum (-1) - (-1) + (-1) + (-1) is -3, which of course is smaller than -2. edit: D'oh! Curse you, simple arithmetic! See post #240 below.

I looked on arxiv.org for stuff on the CHSH inequality and I found this paper which I think gives the answer to the problem. Instead of the way it's written on wikipedia, in eq. 24 it states the CHSH inequality as:

|E(a,b) - E(a,b')| + |E(a',b') + E(a',b)| <= 2

where a and a' are Alice's two choices of experimental settings, and b and b' are Bob's. You can see that when stated this way, if the expectation value is -1 in each case, then it becomes |0| + |-2| <= 2, which works.

Anyway, it's still not hard to violate this inequality in a question-and-answer game where I know both questions before I have to give an answer. All I have to do is make sure to always give the same answer in cases (a,b), (a',b'), and (a',b), so the expectation value for these is 1, and always give different answers in case (a,b'), so the expectation value for this case is -1. Again, this just shows that it's extremely trivial to violate various Bell inequalities classically when you don't respect the conditions of Bell's theorem, but as I said, this is clearly not a disproof of Bell's theorem, since the theorem is only about the impossibility of violating the inequalities classically when all the conditions are respected.

 

[wm]

Jesse; sorry; I was in middle of editing that post when you replied; essentially no change to technical content though. So:

You misunderstand me, I'm not "exploring realistic options" for QM, I'm just pointing out that it's a quite trivial observation that you can violate the Bell inequalities classically when you violate some of the conditions on the experiment stipulated by Bell theorem, but that this is of course not a violation of Bell's theorem. Bell's theorem says, in effect, "under conditions X,Y,Z it is impossible for a local realist theory to reproduce the Bell inequalities". All you have shown is a statement to the effect of "if I am allowed to set up an experiment which does not respect condition Y, then I can violate the Bell inequalities with a local realist theory." But this is completely obvious--it's the whole reason they bothered to stipulate those conditions!

OK. So while you're working to classically rebut CHSH or a similar Bellian Inequality, it would be good to get those X, Y, and Z clearly expressed in your terms. That will certainly help us to understand which of them are breached.

All I am pointing out with my examples is how trivial it is that you can violate the Bell inequalities classically if you (the 'source') are aware of what questions you'll be asked ('measurements') before you have to give answers (the 'results' of each measurement). But suppose me and a friend have to travel at close to the speed of light in opposite directions, and then the events of our each being asked a question have a spacelike separation, with the two experimenters picking their questions at random at that moment. In this case, there is no prearranged strategy me and the friend can use to come up with answers that are always the same (or opposite) when asked the same question, but which violate the Bell inequalities when asked different questions, unless we have some kind of faster-than-light psychic connection, or have precognitive abilities that allow us to know what the experimenters will ask in advance, or are able to split ourselves and the experimenters into multiple copies and decide which copies of experimenter 1 are matched with which copies of experimeter 2 later. That's what Bell's theorem is telling you, it doesn't say a thing about there being a problem with violating the Bell inequalities in circumstances where the source can be informed in advance what one or both of the measurements will be. No, the strategy of just calculating what quantum mechanics would predict for the probabilities, and basing your answers on these probabilities, can only work if you know what both of the measurements are before making your calculation.

Is this correct??

Because my simple example appears to BEAT this both requirement. The source in my example has no knowledge of (nor access to) Bob's setting at any stage.

Therefore, as I said above, it won't work if you and a friend must travel apart to both experimenters, and the events of being asked the two questions have a spacelike separation, and you each have to answer before there's been time for a signal to pass from one to the other informing each what question the other was asked. I realize that, but that's why I said at the end of my last post I was just trying to come up with a specific case: "Would this be a valid formulation of one case of the CHSH inequality? (the case when Alice and Bob both choose between the same two measurements A and B--from the way it was formulated on wikipedia it seems to be more general than that)" Actually, now that I think about it I think there must be an error either in my understanding or the way it's stated on wikipedia. Even if I respect all the conditions of Bell's theorem, it seems like it wouldn't be too hard to violate this inequality--all I have to do is send one object/signal in state {A = +1, B = +1} to Alice and another in state {A = -1, B = -1}, and then it's guaranteed that the probability of them getting the same result is 0, no matter what they measure; thus the sum of all those P(S|xx)'s is 0, which is smaller than 1/2. And in terms of expectation values, the product of Alice and Bob's answer is guaranteed to be -1, and the sum (-1) - (-1) + (-1) + (-1) is -3, which of course is smaller than -2.

I looked on arxiv.org for stuff on the CHSH inequality and I found this paper which I think gives the answer to the problem. Instead of the way it's written on wikipedia, in eq. 24 it states the CHSH inequality as:

|E(a,b) - E(a,b')| + |E(a',b') + E(a',b)| <= 2

where a and a' are Alice's two choices of experimental settings, and b and b' are Bob's. You can see that when stated this way, if the expectation value is -1 in each case, then it becomes |0| + |-2| <= 2, which works.

Anyway, it's still not hard to violate this inequality in a question-and-answer game where I know both questions before I have to give an answer. All I have to do is make sure to always give the same answer in cases (a,b), (a',b'), and (a',b), so the expectation value for these is 1, and always give different answers in case (a,b'), so the expectation value for this case is -1.

This not quite clear to me: You say it's quite easy to breach CHSH in ...

Have you yet done that?

Again, this just shows that it's extremely trivial to violate various Bell inequalities classically when you don't respect the conditions of Bell's theorem, but as I said, this is clearly not a disproof of Bell's theorem, since the theorem is only about the impossibility of violating the inequalities classically when all the conditions are respected.

The XYZ elements that you mention will help us here. Especially if they are in your own words. And I don't mean that extremes like FTL, psychic, magic etc have to be included in such a specification. Just common-sense boundary conditions.

PS: It is still not clear to me that the paper you cited helps on this. ''The events of type C+-/ii are not supposed to be influenced by the measuring operations Li and Rj. ...''.

In my model C+- is not so influenced, is it? The +- there being random and beyond the control of Alice and Bob?

wm

 

[JesseM]

Bell's theorem says, in effect, "under conditions X,Y,Z it is impossible for a local realist theory to reproduce the Bell inequalities". All you have shown is a statement to the effect of "if I am allowed to set up an experiment which does not respect condition Y, then I can violate the Bell inequalities with a local realist theory." But this is completely obvious--it's the whole reason they bothered to stipulate those conditions!

OK. So while you're working to classically rebut CHSH or a similar Bellian Inequality, it would be good to get those X, Y, and Z clearly expressed in your terms. That will certainly help us to understand which of them are breached.

Sure. I'm not a great expert on all the technical details of Bell's theorem, but these are the conditions I'm aware of, with #2 being the one violated in your example:

1. spacelike separation between the two measurement-events

2. source has no foreknowledge of either of the measurement choices on each trial; state of signals/objects sent out by source is statistically independent of measurement settings. In a universe obeying local realism, this condition can be guaranteed by setting things up so that the time between randomly choosing a measurement setting for a given trial and finishing the measurement period for that trial is smaller than the time it would take for a signal moving at the speed of light to travel from the measurement apparatus to the source and back.

3. only a single definite outcome to each measurement--this rules out "many-worlds" type solutions

There may also be additional conditions for specific inequalities derived for specific types of experiments--for example, some inequalities depend on the assumption that Alice and Bob are both choosing between an identical set of binary measurements, and that whenever they choose the same measurement, they always get identical (or opposite, depending on the experiment) results. The CHSH inequality is the most generally-applicable one I've seen, since it doesn't depend on this sort of assumption.

All I am pointing out with my examples is how trivial it is that you can violate the Bell inequalities classically if you (the 'source') are aware of what questions you'll be asked ('measurements') before you have to give answers (the 'results' of each measurement). But suppose me and a friend have to travel at close to the speed of light in opposite directions, and then the events of our each being asked a question have a spacelike separation, with the two experimenters picking their questions at random at that moment. In this case, there is no prearranged strategy me and the friend can use to come up with answers that are always the same (or opposite) when asked the same question, but which violate the Bell inequalities when asked different questions, unless we have some kind of faster-than-light psychic connection, or have precognitive abilities that allow us to know what the experimenters will ask in advance, or are able to split ourselves and the experimenters into multiple copies and decide which copies of experimenter 1 are matched with which copies of experimeter 2 later. That's what Bell's theorem is telling you, it doesn't say a thing about there being a problem with violating the Bell inequalities in circumstances where the source can be informed in advance what one or both of the measurements will be. No, the strategy of just calculating what quantum mechanics would predict for the probabilities, and basing your answers on these probabilities, can only work if you know what both of the measurements are before making your calculation.

Is this correct??

What specifically are you asking about?

Because my simple example appears to BEAT this both requirement. The source in my example has no knowledge of (nor access to) Bob's setting at any stage.

But what I said was "it doesn't say a thing about there being a problem with violating the Bell inequalities in circumstances where the source can be informed in advance what one or both of the measurements will be." It's true that in your experiment the source does not have foreknowledge of both measurements, but foreknowledge of "one or both" of the measurements is violating a condition of Bell's theorem.

Anyway, it's still not hard to violate this inequality in a question-and-answer game where I know both questions before I have to give an answer. All I have to do is make sure to always give the same answer in cases (a,b), (a',b'), and (a',b), so the expectation value for these is 1, and always give different answers in case (a,b'), so the expectation value for this case is -1.

This not quite clear to me: You say it's quite easy to breach CHSH in ...

Have you yet done that?

Yes, that's what I just did in the quoted paragraph. Alice can ask me question a or a', Bob can ask me b or b'; so my strategy is that if the questions they ask are (a,b) or (a',b') or (a',b), then I give the same answer to both questions ('yes, yes' or 'no, no'), but if they ask me the questions (a,b') then I give different answers to both questions ('yes, no' or 'no, yes'). With "yes" assigned value +1 and "no" assigned value -1, and the result of each trial being the product of the two answers, this means E(a,b) = 1, E(a',b') = 1, E(a',b) = 1, and E(a,b') = -1. So, |E(a,b) - E(a,b')| + |E(a',b') + E(a', b)| = |1 - (-1)| + |1 + 1| = |2| + |2| = 4, violating the CHSH inequality which says that |E(a,b) - E(a,b')| + |E(a',b') + E(a', b)| <= 2.

The XYZ elements that you mention will help us here. Especially if they are in your own words. And I don't mean that extremes like FTL, psychic, magic etc have to be included in such a specification. Just common-sense boundary conditions.

Well, see above. Also, it's not actually necessary for me to have no-FTL as one of the "X,Y,Z conditions" since I summarized Bell's theorem as "under conditions X,Y,Z it is impossible for a local realist theory to reproduce the Bell inequalities", and "impossible for a local realist theory" already presupposes we are talking only about theories that say FTL is impossible.

PS: It is still not clear to me that the paper you cited helps on this. ''The events of type C+-/ii are not supposed to be influenced by the measuring operations Li and Rj. ...''.

In my model C+- is not so influenced, is it? The +- there being random and beyond the control of Alice and Bob?

wm

In http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Aquant-ph%2F0312176 , the i's are variables whose value can be anyone of the three orientation settings, which are labeled 1,2,3. As mentioned on p.4, C is a "common cause" which is put forth to explain the correlations you see when Bob and Alice both pick the same orientation setting. And as they say, "in the case of a perfect correlation no generality is achieved by allowing for a more than two-valued common cause variable"--for example, if Bob and Alice always get opposite results when they both choose axis 2, then the signals/objects emitted by the source must either be of type C+-/22 (meaning the properties of the object/signal are such that it is predetermined that if Bob and Alice both choose setting 2, Alice gets a + and Bob gets a -) or of type ~C+-/22 (the properties of the object/signal are such that it is predetermined that if Bob and Alice both choose setting 2, Alice gets a - and Bob gets a +) on the subset of trials where they both pick setting 2. Likewise, every pair of objects/signals emitted by the source must either be of type C+-/11 or ~C+-/11 (predetermined to get either Alice +/Bob - or Alice -/Bob + if they both choose setting 1) on the subset of trials where they both pick setting 1, and every pair must either be of type C+-/33 or ~C+-/33 (predetermined to get either Alice +/Bob - or Alice -/Bob + if they both choose setting 3) on the subset of trials where they both pick setting 3.

If the source has no prior knowledge of either one's settings before it emits the signals/objects, then if every pair of signals/objects is of type C+-/22 or ~C+-/22 on the subset of trials where they both pick 2, then it must also be true that every pair is of one type or the other on all trials. Your example is more complicated, because the source has foreknowledge of Alice's setting; if she picks 2, then the polarized light emitted by the source is at one of two possible angles such that if they are both on setting 2, then they're guaranteed to get either +- or -+ (I realize that in your example, they were originally guaranteed to get either ++ or -- on the same setting, but I hope you don't that I'm modifying your example to match the convention of the paper, which could be done practically by having the polarizer on the end of the source pointing at Bob always be at a 90 degree angle from the one on the end of the source pointing at Alice). However, if polarized light at either one of these same two angles were measured when they were both on setting 1 or 3, there would be some nonzero probability of getting all four results +-, -+, ++ and --, so in the case where Alice picks 2 we can't say the signal must either be of type C+-/11 or ~C+-/11, and likewise we can't say the signal must either be of type C+-/33 or ~C+-/33.

So I guess you're right that it's not exactly the no-conspiracy condition you're violating, since they state the no-conspiracy condition in a way that assumes some prior conditions which you've already violated. In particular, you're violating the condition that the "common cause variable" that they introduce on p. 2, whose value represents all the properties of the signal/object emitted by the source that are relevant to the probabilities of different outcomes (in your case, the common cause is the polarized light emitted at Alice and Bob by the source, and the possible values of q for the common cause variable V_q would just be the possible polarization angles of the light emitted by the source), "should not be correlated with the measurement choices" as they say in the second paragraph on p. 3. If this condition is respected, then if it's true that the common cause variable is always of type C+-/22 or ~C+-/22 on trials where they both choose setting 2, then it must be of one of these two types on all trials; but if you violate this condition then it won't necessarily work that way any more, as your example shows.

This paragraph also has a reference to this paper on common causes as an explanation for EPR-type results, and note that it includes the same sort of condition on p. 5:

6) Also, because the choices of the measurements are free in the sense that there is no mysterious conspiracy between the things that determine the choices of the measurements and those that determine the outcomes, one can assume that the measurement choices are independent of the common cause. ... These findings are partly read off from the empirical data (6) or they are straightforward consequences of the prohibition of superluminal causation.

And again, one consequence of the "prohibition of superluminal causation" is that if the time between randomly choosing a measurement setting and completing the measurement with that setting is smaller than the time it would take a light signal to travel from the measurement apparatus to the source and back, then the choice of setting on a given trial must (assuming a local classical theory) be statistically independent of the properties of whatever signals/objects the measurement apparatus is receiving from the source for the duration of that trial.

 

[vanesch]

See above, I think it works if you just assume that the probability Bob sees a photon get through is equal to cos^2(angle of source polarization - angle of Bob's polaroid), and likewise that the probability Alice sees a photon get through is equal to cos^2(angle of source polarization - angle of Alice's polaroid), with the condition that the source is "yoked" to Alice's polaroid so there's a 50% chance it's parallel to hers and a 50% chance it's at 90 degrees relative to hers. Of course, the problem is that this yoked condition actually violates one of the basic assumptions behind Bell's theorem, namely that any properties of what it emits, "hidden" or otherwise, should be statistically independent of Alice and Bob's choice of detector settings on each trial.

Uh, you mean:

Alice sets her polarizer to an absolute angle (say, with the sight line to Sirius) th_Alice, and then the SOURCE is rotated (with the box) such that the source only sends out light pulses OR perfectly PARALLEL to th_Alice, or PERFECTLY perpendicular to Alice, but nothing in between (in other words, this is not an isotropic source) ?

And this is supposed to prove that Bell's theorem is erroneous ?

That's not very LOCAL as a setup !

 

[vanesch]

Notice that when I was stating the assumptions behind Bell's theorem in post #133, I included the bolded part below: When I wrote this I was thinking of a well-known loophole in Bell's theorem, which I think would be explicitly ruled out with a statistical independence condition in any fully rigorous proof of the theorem. The loophole is that if the source is somehow able to "anticipate" the detector settings Alice and Bob choose ahead of time, then it can adjust the hidden variables based on this in such a way that Bell inequalities can be violated.

Yes, that's so-called "superdeterminism". The idea is that whatever physics is going to determine the settings of the polarizers (say, the state of your brain when you decide to turn the polarizers etc...), in a deterministic universe, this is entirely determined on a sufficiently remote spacelike surface in the past, which can also influence the source. Of course any explicit mechanism by which there is a common origin to the polarization of the light pulses from the source and your brain deciding which polarizing angle to choose, is rather unknown.

The problem with this view is that if you stick to it, there is no way to test any law of nature, because every correlation observed in nature can be simply due to a "common cause in the past". Consider a pharmaceutical company testing a new drug: it is not because it has been administered "blindly" to 10000 patients which all got cured, and a placebo was administered to 10000 patients, of which 95% died, that one can conclude that the drug is effective: a common cause in the past could be such that the process which made the "random blind choice" of the first 10000 patients was 100% correlated with a specific condition of their illness which made them get better.

 

[JesseM]

Uh, you mean:

Alice sets her polarizer to an absolute angle (say, with the sight line to Sirius) th_Alice, and then the SOURCE is rotated (with the box) such that the source only sends out light pulses OR perfectly PARALLEL to th_Alice, or PERFECTLY perpendicular to Alice, but nothing in between (in other words, this is not an isotropic source) ?

Right, that is wm's proposed setup.

And this is supposed to prove that Bell's theorem is erroneous ?

That's not very LOCAL as a setup !

Yeah, I'm trying to convince wm that this is obviously not a valid disproof of Bell's theorem for that reason.

 

[JesseM]

Actually, now that I think about it I think there must be an error either in my understanding or the way it's stated on wikipedia. Even if I respect all the conditions of Bell's theorem, it seems like it wouldn't be too hard to violate this inequality--all I have to do is send one object/signal in state {A = +1, B = +1} to Alice and another in state {A = -1, B = -1} to Bob, and then it's guaranteed that the probability of them getting the same result is 0, no matter what they measure; thus the sum of all those P(S|xx)'s is 0, which is smaller than 1/2. And in terms of expectation values, the product of Alice and Bob's answer is guaranteed to be -1, and the sum (-1) - (-1) + (-1) + (-1) is -3, which of course is smaller than -2.

Uhhh, me are stupid, I just realized that (-1) - (-1) + (-1) + (-1) is -2, not -3. This also means that my equation 1/2 <= P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab) <= 5/2 should actually be 0 <= P(S|aa) + P(S|ba) + P(S|bb) - P(S|ab) <= 2. But it doesn't affect my argument about the CHSH inequality being easy to violate when you are allowed to violate the condition that the source has no foreknowledge of what measurements will be made, though, since both the form of the CHSH inequality in the wikipedia article and the form I was using in my argument (namely |E(a,b) - E(a,b')| + |E(a',b') + E(a',b)| <= 2) are presumably correct.