Scholarpedia article on Bell's Theorem

[billschnieder]

Read my previous message about the "hypothetical theory". In that case, each \lambda would label each "type/group" of pairs (inside each group all the pairs will produce the exact same stochastic prediction, that is how this "theory" works).

Imagine there are 10 "groups" and the source produces (in average) 15% pairs from the first group, 10% pairs from the second type/group,...

If we have measured 2 million pairs with the a,b setting, in average 15% of this 2 million will be "of the first type/group", 10% of this 2 million will be "of the second type/group", ...

If we have measured 3 million pairs with the a,c setting, in average 15% of this 3 million will be "of the first type/group", 10% of this 3 million will be "of the second type/group", ...

That is why if the numbers are big enough, the lambda experimental distribution (even if they are not actually known) of relative frecuencies will be the same for any setting, because it will resemble more and more the theoretical distribution (15% from the first type, 10% from the second type...) that the Theory establishes.

For each setting a,b there will be millions of pairs from all the different "types" (lambdas), that is why later on you have to integrate wrt the lambdas (to take into account their own theoretical distribution that your "theory" establishes) to be able to get the predicted value of C(a,b).

I think you are the one not reading what I'm writing carefully. For the (a,b) pair you measured 2 million, for the (a,c) pair you measured 3 million, why do you think that is? What if type 1 photons are less likely to pass the constraints of the coincidence circuit when measured with the (a,b) setting pair than when measured with the (a,c) setting such that 1 million less of type 1 photons are actually considered (due to coincidence circutary) for the first experiment than for the second one. Do you then believe that increasing the number of photons measured for the (a,b) setting pair to 1 billion will remedy this deficiency? Do you still think it will be reasonable to assume in this situation that the distribution of your different types of photons is the same in all the experiments?

 

[mattt]

I think you are the one not reading what I'm writing carefully. For the (a,b) pair you measured 2 million, for the (a,c) pair you measured 3 million, why do you think that is?

Because we can choose what setting to use to each pair whatever we like (with a dice, asking my aunt by phone,...) so we can decide to mantain the a,c setting much more time than the a,b setting. It does not matter at all to what the theorem (CHSH theorem) is saying.

What if type 1 photons are less likely to pass the constraints of the coincidence circuit when measured with the (a,b) setting pair than when measured with the (a,c) setting such that 1 million less of type 1 photons are actually considered (due to coincidence circutary) for the first experiment than for the second one. Do you then believe that increasing the number of photons measured for the (a,b) setting pair to 1 billion will remedy this deficiency? Do you still think it will be reasonable to assume in this situation that the distribution of your different types of photons is the same in all the experiments?

For the moment I am talking about the ideal experimental case (not real yet) where there are not detection loopholes, i.e. all pair produced are detected.

In this ideal case, as long as you measure enough numbers of pairs in each setting conditions, the "hidden variable \lambda labeling each type of pair" will appear (in each setting) with the same relative frecuencies (unless you think the "source" knows in advance what setting are we going to choose later on, and the source "prepare" different types of pairs (in average) depending on his "knowledge" of what setting we will choose later for every pair).

 

[billschnieder]

My guess is that Bill is wrongly thinking of λ as referring to the state of the particles after some or all of the polarization measurements have been made. If that's what you think λ means, then -- no doubt -- ρ(λ) should probably depend on the polarizer orientations! But ... that's simply not what λ refers to.

I think I've come to the conclusion that you do not understand basic probability theory, from the kinds of things you write and from the way you misinterpret arguments.

We have a series of outcomes sets. For the setting pair (a,b) we have a distribution of all λ
values that the photons detected for the (a,b) setting HAD. That is what ρ(a,b,λ) means, the joint probability distribution of a,b and λ. It doesn't mean "after measurement" as you think. It simply means we are considering ONLY those lambdas corresponding to the pairs of photons that were actually measured for the (a,b) setting pair. ρ(a,b,λ) = ρ(λ) means that the distribution of photons actually measured and considered was not different when the (a,b) setting pair was used from any other setting pair. In other words, it means the setting pair did not favor some lambda values over others. If all photons are detected and considered there is no problem with saying that ρ(a,b,λ) = ρ(λ). But we know that this is not the case. Either you do not understand basic probability theory or you are just being dishonest here and deliberately trying to confuse people by misrepresenting my argument. Photons do not exist any longer post measurement so the suggestion that I'm talking about λ after measurement is outrageous.

Note another misunderstanding here. The modern experiments use 2-channel polarizers. It isn't true that a pair only gets counted if both particles "pass" their polarizers. Yes, there are detection efficiency issues in the experiments, but basically each photon is subjected to a measurement in which one detector clicks if the photon is "horizontal" (relative to the axis a) and a different detector clicks if the photon is instead "vertical" (relative to a). That is, each photon is detected either way. The alternative to "passing through the polarizer" is not getting absorbed and hence never seen and hence never counted, but is rather getting counted instead by the other detector.

And how is this different from what I'm saying?

Are you sayng "each photon pair emitted" is detected? Or are you saying "each photon that is detected is detected as a pair"? What do you think the role of coincidence counting is?
Do you or do not reject from consideration photons that do not meet the constraints of coincidence counting? A photon which is detected but it's counterpart was not detected is not considered. Don't you know that? If you can not understand simple things like this, I feel sorry for the students you teach.

 

[billschnieder]

Because we can choose what setting to use to each pair whatever we like (with a dice, asking my aunt by phone,...) so we can decide to mantain the a,c setting much more time than the a,b setting. It does not matter at all to what the theorem (CHSH theorem) is saying. For the moment I am talking about the ideal experimental case (not real yet) where there are not detection loopholes, i.e. all pair produced are detected.

From what you know classically about light, and what you know about the way experiments are actually performed using coincidence circuitary do you still think your "ideal case" is reasonable?

Do you see now that in real "non-ideal" experiments actually performed, you do not need a conspiracy for the distribution of lambda to be different from one setting pair to the next?

 

[ttn]

Imagine one of these distributions prediction outcomes is the following one:

$$P_{a,b}(A_1=1,A_2=1|\lambda_0)=0.9$$

$$P_{a,b}(A_1=1,A_2=-1|\lambda_0)=0.04$$

$$P_{a,b}(A_1=-1,A_2=1|\lambda_0)=0.03$$

$$P_{a,b}(A_1=-1,A_2=-1|\lambda_0)=0.03$$


This "theory" does NOT satisfy your factorizability condition (4), but now I don't see why someone who believes in "locality" would call this "theory" non-local.

Mattt, good question. It makes me so happy that somebody is actually reading the paper carefully and scrutinizing the arguments!

Basically I would say this. What are we trying to capture with the idea of "locality" here, specifically in the context of *not assuming determinism*, i.e., assuming that there is some irreducible randomness in the way the outcomes arise? I think what we want to capture is that once you specify the state of the particles, the probabilities for each possible outcome on one side shouldn't depend on what did or didn't randomly happen when the distant particle suffered some kind of measurement. Do we agree that that's what "locality" should mean in this context (or more precisely that locality requires this)?

If so, then it's easy to see why your 4 joint probabilities imply a kind of nonlocality. If you calculate for example the marginals

P(A1=+1) = .94

P(A1=-1) = .06

P(A2=+1) = .93

P(A2=-1) = .97

you can then calculate the conditional probabilities

P(A1=+1 | A2=+1) = P(A1=+1,A2=+1) / P(A2=+1) = .9 / .93 = .97

while

P(A1=+1 | A2=-1) = P(A1=+1,A2=-1) / P(A2=-1) = .04 / .97 = .04

So think about what that means: the probability of A1 coming up +1 is radically different depending on how the random outcome turned out on the other side, i.e., whether A2 came up +1 or -1. But the whole idea is that "A2 coming up +1 or -1" is something that just happened -- something that just popped newly into existence -- way the heck over there on the other side. That outcome shouldn't be able to influence the chances for A1 coming up +1 over here. So, it's a violation of locality.

Of course, all of this is just another way of saying that the joint probability should factorize once you condition everything on λ. So maybe if you didn't accept that you won't accept this either! But maybe it will help to think about the conditional probability depending on the distant outcome.

 

[ttn]

Is such a conclusion really inevitable? What if you had an acausal theory?

What is an "acausal theory"?

 

[lugita15]

What is an "acausal theory"?

Acausal means you have effects or phenomena occur without them having been caused by things either in the present or in the past. At best, all the argument proves is that assuming the predictions of quantum mechanics are correct, then causation implies nonlocal causation. But instead you can not have causation.

 

[zonde]

What is an "acausal theory"?

Oxymoron :rofl:

 

[lugita15]

Oxymoron :rofl:

Can you not have a theory in which events happen without anything causing them, and in which you just have probabilities of certain events happening at certain times?

 

[mattt]

From what you know classically about light, and what you know about the way experiments are actually performed using coincidence circuitary do you still think your "ideal case" is reasonable?

Do you see now that in real "non-ideal" experiments actually performed, you do not need a conspiracy for the distribution of lambda to be different from one setting pair to the next?

You are saying that the detectors may be biased with respect to some hidden mechanism (some hidden variable).

It may be, but I want to separate the following things in our discussion:

1) The mathematical theorem (CHSH Theorem), which is correct (its mathematical proof is correct).

2) What do its mathematical premises say about what we think of "locality" (do its mathematical premises codify what we exactly think of locality or not)

3) How close or how far are the real experiments with respect to the ideal experiment with 100% detection efficiency.

What I am treating (for the moment) with Travis is 1) (we both agree on that one, it is a mathematical theorem with a correct mathematical proof) and 2) (I still have doubts with respect to calling his "factorizability condition" a "necessary condition for locality").

I will post later more on 2)

 

[mattt]

Mattt, good question. It makes me so happy that somebody is actually reading the paper carefully and scrutinizing the arguments!

Basically I would say this. What are we trying to capture with the idea of "locality" here, specifically in the context of *not assuming determinism*, i.e., assuming that there is some irreducible randomness in the way the outcomes arise? I think what we want to capture is that once you specify the state of the particles, the probabilities for each possible outcome on one side shouldn't depend on what did or didn't randomly happen when the distant particle suffered some kind of measurement. Do we agree that that's what "locality" should mean in this context (or more precisely that locality requires this)?

That is exactly the crux of the matter for me.

I think I would not demand so much (in a non-deterministic hidden variable theory that pretends to predict the outcomes of these type of experiments) to be able to call it "local".

What I would require from that "theory" (to be able to call it "local") is a bit less, and is the following:

"The probabilities for each possible outcome on one side should not depend on what observable they decide to measure on the other side"

But there may be statistical dependence in the outcomes for a given setting here and a given setting there, why not?, after all if you measure two random variables, for example height in the individuals of a given ordered sample and weight in the individuals of another different (but equal in size) given ordered sample, there may be a statistical dependence between them (it may be the case that the first person of the first sample is twin-brother of the first person of the second sample, the second person of the first sample is twin-brother of the second person of the second sample...you know, they are "entangled" and that is why there is a statistical dependence :) ) ; what would be really strange (really "non-local") is that the statistical outcomes of the height of the first sample were different depending on what (weight or eye-colour) we decide to measure in the individuals of the second sample.

So, imagine my previous "Theory"'s predictions are like this:

$$P_{a,b}(A_1=1,A_2=1|\lambda_0)=0,9$$

$$P_{a,b}(A_1=1,A_2=-1|\lambda_0)=0,04$$

$$P_{a,b}(A_1=-1,A_2=1|\lambda_0)=0,03$$

$$P_{a,b}(A_1=-1,A_2=-1|\lambda_0)=0,03$$


$$P_{a,c}(A_1=1,A_2=1|\lambda_0)=0,89$$

$$P_{a,c}(A_1=1,A_2=-1|\lambda_0)=0,05$$

$$P_{a,c}(A_1=-1,A_2=1|\lambda_0)=0,04$$

$$P_{a,c}(A_1=-1,A_2=-1|\lambda_0)=0,02$$


and you know what I mean for the rest of setting cases...


In this "Theory"'s predictions, the predicted outcomes distribution for A_1 given the setting "a", does NOT depend on what setting ("b" or "c") they decide to use on the other side.

But this "theory" does NOT satisfy your "factorizability condition (4) ".

Your "CHSH theorem" does not apply to this "theory", but I would call this theory "local" (in a certain intuitive meaning of "local" I have).

 

[ttn]

Acausal means you have effects or phenomena occur without them having been caused by things either in the present or in the past. At best, all the argument proves is that assuming the predictions of quantum mechanics are correct, then causation implies nonlocal causation. But instead you can not have causation.

Can you give an example of the kind of theory you have in mind?

I can't tell if you mean merely that the theory might be stochastic/nondeterministic (which is of course completely fine, and the theorem still applies straightforwardly) -- or whether instead you have something in mind in which there is no cause at all for a given event, according to a theory, not even the causes of its (irreducible) probability having been such and such, etc. At the end of the day I will agree with zonde that, if you mean the latter, it's an oxymoron -- there's no such thing as that. (Even if a theory says some things are not determined, it still has to say something about what the probabilities *depend on* -- otherwise it would be saying nothing at all, it would give you no handle at all for saying what was going to happen, there would be no way at all to test it or decide if it was confirmed or refuted, etc.) So that's why I'd like you to describe a concrete example of what kind of thing you have in mind.

 

[ttn]

Your "CHSH theorem" does not apply to this "theory", but I would call this theory "local" (in a certain intuitive meaning of "local" I have).

I don't understand your "intuitive meaning of local". It seems to be based on the idea of there being some kind of distinction between two kinds of events (in, remember, an irreducibly stochastic theory):

(a) events like a person rotating a polarizer

and

(b) events like a certain photon emerging from the "horizontal" port of a polarizer

In the stochastic kind of theory (b) is supposed to be something that couldn't have been predicted in advance -- something new pops into the world right there at the location of the distant apparatus when (b) happens. But how is this any different from (a)? Something new pops into the world over there when the person changes the orientation of the polarizer. So your claimed intuition is that it is OK (I mean, consistent with locality) for the probabilities over here, for this particle, to depend on space-like separated (b)-type events, but not on space-like separated (a)-type events. But I don't understand what the difference between (a) and (b) is even supposed to be. One is microscopic and one is macroscopic? One is controllable by humans and one isn't? Such distinctions shouldn't be showing up in our formulation of fundamental ideas like locality.

Probably you would be interested to read the section on "Bell's definition of locality" which purports to be such a formulation. (Or see my recent AmJPhys paper on this.) You'll see that the idea is basically that the probability for a certain event should not depend on anything physically real at spacelike separation, once everything physically real in (a suitable part of) the past light cone has been specified. If you think carefully about that, I think you will agree that it perfectly captures the idea that there should be no faster-than-light causal influences. And the point here is that, obviously, it rules out your proposed dependencies on (b)-type events every bit as much as it rules out dependencies on (a)-type events, basically because, in fundamental terms, (a) and (b) are exactly the same: they refer to some physically real thing/happening over there.

 

[ttn]

In this "Theory"'s predictions, the predicted outcomes distribution for A_1 given the setting "a", does NOT depend on what setting ("b" or "c") they decide to use on the other side.

By the way, as you probably know already, ordinary QM is a theory of just the type you describe here. How does it make the right predictions for the correlations? Well, it says that the state of the particle pair is originally something (the usual QM wf) but then after the first measurement is made on one of the particles, the state *changes* -- the wf collapses -- and you need to use this new updated state to calculate the new probabilities for the other particle. That is, in effect, the state of the second particle (the thing that determines what the possibilities/probabilities are for a subsequent measurement on it) *changes* as a result of the first measurement. To me, that is a blatant case of non-locality: something happening here depends on what happened over there.

You disagree?

 

[harrylin]

[..] ordinary QM [..] says that the state of the particle pair is originally something (the usual QM wf) but then after the first measurement is made on one of the particles, the state *changes* -- the wf collapses -- and you need to use this new updated state to calculate the new probabilities for the other particle. That is, in effect, the state of the second particle (the thing that determines what the possibilities/probabilities are for a subsequent measurement on it) *changes* as a result of the first measurement. To me, that is a blatant case of non-locality: something happening here depends on what happened over there.
You disagree?

Stated like that, it is certainly wrong - as you should know: such probabilities can be affected with infinite speed, and even back in time. As Jaynes explained that has nothing to do with spooky action at a distance; and while Jaynes doubted it, I do think that Bell understood that. Thus I hope that you do not make a similar statement in your article.

PS: you seem to have missed my post #205

 

[martinbn]

By the way, as you probably know already, ordinary QM is a theory of just the type you describe here. How does it make the right predictions for the correlations? Well, it says that the state of the particle pair is originally something (the usual QM wf) but then after the first measurement is made on one of the particles, the state *changes* -- the wf collapses -- and you need to use this new updated state to calculate the new probabilities for the other particle. That is, in effect, the state of the second particle (the thing that determines what the possibilities/probabilities are for a subsequent measurement on it) *changes* as a result of the first measurement.

This seems imprecise at the least. There is no first and second particle while they are entangled. It is meaningless to talk about the state of the second particle while they are entangled.

To me, that is a blatant case of non-locality: something happening here depends on what happened over there.

But is it really like that. To me it sounds more like something that happened here is corralated with something that happened there.

 

[ttn]

The introduction mentions correctly, but with lack of clarification, that EPR formulated their argument in terms of position and momentum and Bohm reformulated it in terms of spin. The clarification that I think would be good is if the qualitative difference is important or not, and why not: position vs. momentum is directly and obviously related to the uncertainty principle, but up spin vs down spin not, or at least not in the same way. Maybe it's somewhere further in the article, but then still a mention of it there would be useful.

I don't think there is any important difference, qualitative or otherwise. Bohm's version in terms of spin is just more directly related to Bell's theorem and the experimental tests of the inequality. But the argument is the same, whether you use x/p or s_x/s_y or whatever. Note that s_x and s_y (I am thinking of spin 1/2 here, but if you want to think about photons you should say instead "polarization along some axis" and "polarization along some non-parallel axis") don't commute and so there is an uncertainty principle relation between them the same way there is between x and p.

 

[ttn]

Stated like that, it is certainly wrong - as you should know: such probabilities can be affected with infinite speed, and even back in time. As Jaynes explained that has nothing to do with spooky action at a distance; and while Jaynes doubted it, I do think that Bell understood that. Thus I hope that you do not make a similar statement in your article.

I can't follow what you're saying. I'm just supposed to accept that "such probabilities can be affected with infinite speed, and even back in time"? As to Bell, I can assure you that his views match my own on this -- see in particular his (last) article, "la nouvelle cuisine", where all of this is laid out very carefully and clearly.

It's true that using ordinary/orthodox QM as an example can muddy the waters, because it is never so clear whether we're supposed to take the wave function as a physically real thing or not, etc. The fundamental point, though, is what I was explaining a few posts back in response to mattt. The idea of "locality" or equivalently "local causality" presupposes that there is some physically real stuff out there and that things happening with the stuff at one place affect what happens to the stuff at some other place. That is the basic context in which "locality" is even meaningful. The issue is then: do these influences go slower than c always, or do they sometimes go faster than c? The difficult thing is to formulate that idea crisply, and in a way that doesn't presuppose (for example) the sorts of anthropocentric concepts (like "signalling" or "controllability") that clearly don't belong in a formulation of a fundamental idea like this. Bell managed to do this; see his "la nouvelle cuisine" paper, or the "Bell's definition..." section of our article, or my recent AmJPhys paper.

I have spent a long long time thinking carefully about Bell's formulation, and I'm frankly not that interested in arguing with people who haven't even considered it but instead just look at some equations and say "this looks like correlation to me, not causality". So I will shift the burden back onto the skeptics. If you think Bell's formulation is no good (in the sense that it diagnoses as "nonlocal" things that are clearly in fact "local") tell me exactly what's wrong with it, and tell me how *you* would formulate the idea of "locality" instead.

 

[ttn]

This seems imprecise at the least. There is no first and second particle while they are entangled. It is meaningless to talk about the state of the second particle while they are entangled.

I agree, it's horribly imprecise. But the reason is because ordinary QM is terribly vague about what really exists. I think what you say here (about no particle pair existing as long as they're entangled, etc.) is quite controversial, even as a statement about what ordinary QM says, but I'll concede that it's not at all clear what, exactly, ordinary QM says. Note though that, even if we interpret things your way, where "there is no first and second particle while they are entangled", then evidently the first particle pops newly into existence when a measurement is made on the distant second particle. So surely that counts as an example of nonlocal causation -- the (perhaps last-second) decision to make a polarization measurement over there, caused a whole particle to pop suddenly into existence over here!

Don't take this seriously or argue with it, though. The real point is just that although, yes, there are lots of different "stories" that different people might want to tell to go along with the ordinary QM math, all of these will involve nonlocality. But who cares about ordinary QM. It means a million different things to different people. Much better to forget about it and just formulate locality in a way that is crisp and clear and which doesn't involve any of the vague muddles that afflict ordinary QM. That's what Bell's formulation does, and as soon as you see that, you can see that *no matter how you interpret it exactly* ordinary QM is nonlocal.

But is it really like that. To me it sounds more like something that happened here is corralated with something that happened there.

The whole point of having a clear and neutral and general formulation of "locality" (I mean, one that isn't tied to the proprietary concepts of any particular candidate theory) is that we don't have to know or care whether the particular story someone chooses to tell to go along with a particular theory is true or not. So, you can say it sounds like whatever you want, but if Bell's definition of locality is not respected, it's not respected, no matter what you think it sounds like. Of course, maybe Bell's definition of "locality" is wrong. If you think so, tell me how it's wrong, what's wrong with it, and what should replace it.

 

[martinbn]

I agree, it's horribly imprecise. But the reason is because ordinary QM is terribly vague about what really exists. I think what you say here (about no particle pair existing as long as they're entangled, etc.) is quite controversial, even as a statement about what ordinary QM says, but I'll concede that it's not at all clear what, exactly, ordinary QM says.

I should probably make myself clearer. I didn't mean to suggest that things don't exist objectively. All I was saying is that when two particles are entangled it is meaningless to talk about the state of each particle. I am not saying they don't exist. Also if the particles are identical it is meaningless to say the first and the second, entangled or not, all that can be said is there are two particles.

Note though that, even if we interpret things your way, where "there is no first and second particle while they are entangled", then evidently the first particle pops newly into existence when a measurement is made on the distant second particle. So surely that counts as an example of nonlocal causation -- the (perhaps last-second) decision to make a polarization measurement over there, caused a whole particle to pop suddenly into existence over here!

See, you agree to view things that way, and then the next sentence you go back to 'first' and 'second'! Why not say that there is one quantum mechanical system, consisting of two entangled particles. We make two measurements on that same system, the results are correlated. The correlations are very different than what you can have in classical mechanics.

Don't take this seriously or argue with it, though. The real point is just that although, yes, there are lots of different "stories" that different people might want to tell to go along with the ordinary QM math, all of these will involve nonlocality. But who cares about ordinary QM. It means a million different things to different people. Much better to forget about it and just formulate locality in a way that is crisp and clear and which doesn't involve any of the vague muddles that afflict ordinary QM. That's what Bell's formulation does, and as soon as you see that, you can see that *no matter how you interpret it exactly* ordinary QM is nonlocal.

Well, I am trying to understand. It may take time for me. So far I am no sure why you are so attached to the word non-local.

The whole point of having a clear and neutral and general formulation of "locality" (I mean, one that isn't tied to the proprietary concepts of any particular candidate theory) is that we don't have to know or care whether the particular story someone chooses to tell to go along with a particular theory is true or not. So, you can say it sounds like whatever you want, but if Bell's definition of locality is not respected, it's not respected, no matter what you think it sounds like. Of course, maybe Bell's definition of "locality" is wrong. If you think so, tell me how it's wrong, what's wrong with it, and what should replace it.

I don't think a definition can be wrong or true! It can be useful or not. I don't have any objections about the definition. I am just not sure whether the name is good, because it carries different connotations for different people.

 

[ttn]

Why not say that there is one quantum mechanical system, consisting of two entangled particles. We make two measurements on that same system, the results are correlated. The correlations are very different than what you can have in classical mechanics.

Of course I agree with what you say here, in the sense that it is a kind of minimalist statement of at least part of what's happening. But the whole point under discussion, when the discussion is about "locality", is to probe a bit deeper and not just say "they're correlated, end of discussion" but instead to ask how those correlations arose and in particular whether there was any nonlocal causation at work. Sure, you can just bury your head in the sand and refuse to talk about it. But refusing to talk about it is hardly the same as somehow proving there was no nonlocality!

I don't think a definition can be wrong or true! It can be useful or not. I don't have any objections about the definition. I am just not sure whether the name is good, because it carries different connotations for different people.

Well, then maybe "definition" is the wrong word. Call it a "formulation". The point is that we are trying to capture, in a mathematically precise way, an idea that we have a reasonably clear intuitive sense of -- roughly, all of the causal influences on a given event should be inside the event's past light cone. The difficult thing is precisely to formulate this in a way that gets at *causal influences* rather than mere correlations.

Seriously, reading "la nouvelle cuisine" is a good idea here.

 

[billschnieder]

I can't follow what you're saying. I'm just supposed to accept that "such probabilities can be affected with infinite speed, and even back in time"?

Yes. Because this is basic probability theory, this is what Jaynes wrote:

As his words above show, Bell took it for granted that a conditional probability P(X|Y) expresses a physical causal influence, exerted by Y on X . But we show presently that one cannot even reason correctly in so simple a problem as drawing two balls from Bernoulli's Urn, if he interprets probabilities in this way. Fundamentally, consistency requires that conditional probabilities express logical inference, just as Harold Jeffreys saw.
...

BERNOULLI'S URN REVISITED
Define the propositions:
I = Our urn contains N balls, identical in every respect except that M of them are red, the remaining N , M white. We have no information about the location of particular balls in the urn. They are drawn out blindfolded without replacement."
Ri = Red on the i'th draw, i = 1, 2, ..."
Successive draws from the urn are a microcosm of the EPR experiment. For the first draw, given only the prior information I , we have

(16) P(R1|I)  = M/N

Now if we know that red was found on the first draw, then that changes the contents of the urn for
the second:

(17) P(R2|R1, I)  = (M - 1)/(N - 1)

and this conditional probability expresses the causal influence of the first draw on the second, in just the way that Bell assumed. But suppose we are told only that red was drawn on the second draw; what is now our probability for red on the first draw? Whatever happens on the second draw cannot exert any physical influence on the condition of the urn at the first draw; so presumably one who believes with Bell that a conditional probability expresses a physical causal influence, would say that

P(R1|R2, I)  = P(R1|I) .

But this is patently wrong; probability theory requires that

(18) P(R1,R2, I)  = P(R2 |R1, I)

This is particularly obvious in the case M = 1; for if we know that the one red ball was taken inthe second draw, then it is certain that it could not have been taken in the first. In (18) the probability on the right expresses a physical causation, that on the left only an inference. Nevertheless, the probabilities are necessarily equal because, although a later draw cannot physically affect conditions at an earlier one, information about the result of the second draw has precisely the same effect on our state of knowledge about what could have been taken in the first draw, as if their order were reversed.

...

It might be thought that this phenomenon is a peculiarity of probability theory. On the contrary; it remains true even in pure deductive logic; for if A implies B, then not-B implies not-A. But if we tried to interpret "A implies B" as meaning A is the physical cause of B", we could hardly accept that "not-B" is the physical cause of "not-A". Because of this lack of contraposition, we cannot in general interpret logical implication as physical causation, any more than we can conditional probability. Elementary facts like this are well understood in economics (Simon & Rescher, 1966; Zellner, 1984); it is high time that they were recognized in theoretical physics.

 

[billschnieder]

More quick debunking of Bill:
...
A can depend on the local polarizer setting (a) and the state of the particles (λ) ... but it cannot depend on the distant polarizer setting (b). That is, A = A(a,λ). Likewise, B = B(b,λ). (B cannot depend on the setting that is distant to it.) That is locality.

This demonstrates a serious misunderstanding. A(a,λ) is a function not a probability. A function maps values from it's domain to a *single* value in its codomain. If A(a1,λ1) gives one value at one point and a different value at a different point, it means you have not defined your function correctly (check the definition of a function). In other words there may be other variables such that the function A(a,λ,x), which would result in a single outcome for every element in it's domain. So just because you write A(a,λ) does not mean the outcomes depend ONLY on (a,λ), they will depend on variables of type x as well. If it is your claim that every variable of the type x has already been merged into λ, then a *function* A(a,λ), must necessarily produce a single outcome for every element (ai,λi) of it's domain.

Furthermore, as explained in the Jaynes quote in my previous post, dependence does not mean causation. The codomain of A(a,λ) can depend on b if b is used to restrict or select the domain on which to apply the function.

 

[ttn]

I've read Jaynes on Bell. I generally like Jaynes. But Jaynes is simply dead wrong when he writes, for example,

Bell took it for granted that a conditional probability P(X|Y) expresses a physical causal influence, exerted by Y on X

Bell did *not* take this for granted *at all*. For example, he explicitly states that the issue motivating his paper "la nouvelle cuisine" is the problem of sharply formulating and distinguishing "these notions, of cause and effect on the one hand, and of correlation on the other."

Anybody who says that Bell's formulation of local causality is merely the requirement that two events not be statistically correlated, or who says that he simply took for granted that "statistical dependence" (as can be expressed for example in terms of certain conditional probabilities) implies "causal dependence", or anything like that, is simply *wrong*. They obviously *have not read Bell carefully* and *don't understand Bell's formulation of locality* and generally *don't know what they're talking about* on this front.

Anybody with the slightest confusion or uncertainty or skepticism about this is urged to read Bell (esp. "la nouvelle cuisine") for themselves. If you don't have that handy (it's in the second, but not the first, edition of "speakable and unspeakable") Bell's views are summarized and discussed in (especially) sections IV and V of my paper "Bell's concept of local causality" (recently published in AmJPhys) that is available here for free:

http://arxiv.org/abs/0707.0401

 

[zonde]

Can you not have a theory in which events happen without anything causing them, and in which you just have probabilities of certain events happening at certain times?

Theory has to provide explanation for something but if you say that events happen without any cause to me it seems like antithesis of explanation. So it actually does not fit together with the second part about probabilities.

 

[billschnieder]

I've read Jaynes on Bell. I generally like Jaynes. But Jaynes is simply dead wrong when he writes, for example,
Bell did *not* take this for granted *at all*. For example, he explicitly states that the issue motivating his paper "la nouvelle cuisine" is the problem of sharply formulating and distinguishing "these notions, of cause and effect on the one hand, and of correlation on the other."

Anybody who says that Bell's formulation of local causality is merely the requirement that two events not be statistically correlated, or who says that he simply took for granted that "statistical dependence" (as can be expressed for example in terms of certain conditional probabilities) implies "causal dependence", or anything like that, is simply *wrong*. They obviously *have not read Bell carefully* and *don't understand Bell's formulation of locality* and generally *don't know what they're talking about* on this front.

No. You don't know what you are talking about:

In Bell's Bertlmann's socks paper (http://cdsweb.cern.ch/record/142461/files/198009299.pdf), page 15, second paragraph, he says:

To avoid the inequality, we could allow P1 in (11) to depend on b or P2 to depend on a. That is to say we could admit the signal at one end as a causal influence at the other end.

This is the Bell quote Jaynes was referring to:

It would be very remarkable if b proved to be a causal factor for A, or a for B ; i.e., if P(A|a,λ) depended on b or P(B|b,λ) depended on a. But
according to quantum mechanics, such a dilemma can happen. Moreover, this peculiar long range infuence in question seems to go faster than light"

From J. S. Bell 1987, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press.

 

[ttn]

This demonstrates a serious misunderstanding. A(a,λ) is a function not a probability. A function maps values from it's domain to a *single* value in its codomain. If A(a1,λ1) gives one value at one point and a different value at a different point, it means you have not defined your function correctly (check the definition of a function). In other words there may be other variables such that the function A(a,λ,x), which would result in a single outcome for every element in it's domain. So just because you write A(a,λ) does not mean the outcomes depend ONLY on (a,λ), they will depend on variables of type x as well. If it is your claim that every variable of the type x has already been merged into λ, then a *function* A(a,λ), must necessarily produce a single outcome for every element (ai,λi) of it's domain.

There is some irony here, since earlier I was trying to be careful to keep the possibility of non-determinism live, where Bill insisted on notation that implied determinism. Now he's telling me that my capitulation to his notation demonstrates a serious misunderstanding on my part.

For the record: A(a,λ) does presuppose both locality and determinism. Yes, this "A" is not a probability. Bill thinks that there are more variables ("x") in addition to λ that might help determine the outcome. But this just shows that he -- like Jaynes and Jarrett, incidentally -- have not studied/understood Bell carefully. Bell is absolutely clear that "λ" denotes a *complete* specification of the state of the physical stuff in question (the particle pair, or whatever). So, indeed, as Bill puts it, "every variable of the type x has already been merged into λ". And then it is of course true that "a *function* A(a,λ) must necessarily produce a single outcome for every element ... of its domain." But what is the problem supposed to be? If the problem is only that this presumes determinism, OK, great, let's go back to the more general case (not assuming determinism) where we speak instead of the probabilities for the two possible values of A. The idea then (as I've been discussing with mattt and others) is that the probability for A having a certain outcome should not depend on "nonlocal stuff" (like the setting b of the distant apparatus or the outcome B of the distant measurement). That's what "locality" means in the context of a general stochastic, not-necessarily-deterministic, theory. And there is no problem deriving the inequality still. See the article for details.

 

[ttn]

No. You don't know what you are talking about

I have read every single paper Bell ever wrote, probably at least 10 times each. I have published my own papers about Bell's views on things. It's frankly absurd to say I don't know what I'm talking about here.

Bill, have you read Bell's paper "la nouvelle cuisine"? yes or no.

It would be very remarkable if b proved to be a causal factor for A, or a for B ; i.e., if P(A|a,λ) depended on b or P(B|b,λ) depended on a. But according to quantum mechanics, such a dilemma can happen. Moreover, this peculiar long range infuence in question seems to go faster than light

You can't understand Bell by lifting two sentences out of context. If you go back and study the whole paper, indeed his whole body of work, you will then understand that λ is not just any old thing anybody feels like, and the P's above are not (are you listening Jaynes??) subjective probabilities, i.e., not in any sense based on somebody's *incomplete* information about things. They are instead the probabilities assigned by some candidate fundamental micro-theory -- assigned (by hypothesis) on the basis of *complete specification of the physical state in some appropriate region of space-time in the past of the event in question*. If these probabilities are actually different, depending on something going on non-locally, that can *not* be interpreted in terms of "updated information", etc. That is the whole *point* -- Bell's formulation of locality is *specifically designed* to distinguish causation from "mere correlation such that the conditional probability depends on some distant thing*.

You can say over and over again that Bell just assumed these were the same thing, blah blah blah, but the fact is that you haven't read Bell, you are just taking somebody else's (Jaynes') word for it, and you really don't know what you're talking about. Go read Bell (or, if absolutely necessary, me) until you understand what Bell actually tried to do, and then tell me what you think is wrong with it. Don't just deny that he even tried to do what he in fact did do, simply on the grounds that you haven't bothered to look at it for yourself.

 

[Delta Kilo]

No. You don't know what you are talking about:

In Bell's Bertlmann's socks paper (http://cdsweb.cern.ch/record/142461/files/198009299.pdf), page 15, second paragraph, he says:

To avoid the inequality, we could allow P1 in (11) to depend on b or P2 to depend on a. That is to say we could admit the signal at one end as a causal influence at the other end.

No, Bill, you lost it here. You resorted to pulling quotes out of context. Here is the entire paragraph:

So the quantum correlations are locally inexplicable. To avoid the inequality, we could allow P1 in (11) to depend on b or P2 to depend on a. That is to say we could admit the signal at one end as a causal influence at the other end. For the set-up described this would be not only a mysterious long-range influence - a non-locality or action at a distance in the loose sense - but one propagating faster than light (because Cδ<<L) - a non-locality in the stricter and more indigestible sense.

See, just from the sentence you've quoted, it appears that we can allow P1 to depend on b etc. But when reading the entire paragraph, it is clear that meaning is exactly opposite: we cannot do it without giving up locality in a big way.

++ungood.

 

[billschnieder]

No, Bill, you lost it here. You resorted to pulling quotes out of context. Here is the entire paragraph:See, just from the sentence you've quoted, it appears that we can allow P1 to depend on b etc. But when reading the entire paragraph, it is clear that meaning is exactly opposite:

Huh? The full quote you just provided supports my point. Bell assumes that dependence implies mysterious long range influence. He assumes that we should not allow them to depend on each other because he thinks allowing such would mean a long range physical influence. This is what Jaynes is criticizing.

we cannot do it without giving up locality in a big way.

++ungood.

Jaynes point is exactly the fact that we can do it. Simply because dependence does not mean causation. This is the error you still continue to make as evident in your statement. In basic probability theory, you CAN make them depend on each other WITHOUT giving up locality.

As usual you respond without understanding what it is that was said in the first place.

 

[harrylin]

I've read Jaynes on Bell. I generally like Jaynes. But Jaynes is simply dead wrong when he writes, for example[..]

Yes we discussed that in this forum; and thus I wrote in my comment to you that "while Jaynes doubted it, I do think that Bell understood that." However your reply gives the impression that you don't understand it despite having read Jaynes (but perhaps it's just a misunderstanding about words), and for some strange reason you even reacted on my comment about your statement as if it was criticism on the way Bell formulated his theorem. If you cite the passage of Bell that you think is the same as what you said, we can verify it.

 

[harrylin]

I don't think there is any important difference, qualitative or otherwise. Bohm's version in terms of spin is just more directly related to Bell's theorem and the experimental tests of the inequality. But the argument is the same, whether you use x/p or s_x/s_y or whatever. Note that s_x and s_y (I am thinking of spin 1/2 here, but if you want to think about photons you should say instead "polarization along some axis" and "polarization along some non-parallel axis") don't commute and so there is an uncertainty principle relation between them the same way there is between x and p.

OK thanks for the clarification! As I have read some papers (but sorry, I forgot which) in which this point was criticised, a footnote of that kind could be useful.
But I still don't get it: If we know the position more accurately, we know the momentum less accurately. I would say that it is quite different with spin: if we know one spin more accurately, we also know the other spin more accurately and I'm not aware that there is anything that we would know less accurately. Perhaps you can elaborate?

 

[lugita15]

If we know the position more accurately, we know the momentum less accurately. I would say that it is quite the opposite with spin: if we know one spin more accurately, we also know the other spin more accurately. Perhaps you can elaborate?

If we know the spin along one direction more accurately, then we know the spin along another direction less accurately. Position and momentum bear the exact same relation to each other that (say) the x-component of spin angular momentum has to the y-component of spin angular momentum.

 

[DrChinese]

I would say that it is quite different with spin: if we know one spin more accurately, we also know the other spin more accurately and I'm not aware that there is anything that we would know less accurately. Perhaps you can elaborate?

Just to elaborate on lugita15's accurate answer:

For a spin 1/2 electron, there are 3 spin components which are mutually perpendicular (i.e. 3 axes): sx, sy, sz. If you know sx, then sy and sz are completely indeterminate. Any observation which involves some mixture of sx and sy (let's say 45 degrees between) will likewise be a mixture of a known outcome and an indeterminate one. And so on for all angles.

For a spin 1 linear polarized photon, the same kind of mixtures are possible (of known and indeterminate) but the indeterminate component is 45 degrees away from the known one (rather than 90 degrees as is the case with an electron).

All of this is because spin components do not commute, just as p and q do not commute. One of the advantages of working with spin observations is that you can easily obtain these combinations by simply rotating a part of the apparatus.

 

[billschnieder]

For the record: A(a,λ) does presuppose both locality and determinism. Yes, this "A" is not a probability. Bill thinks that there are more variables ("x") in addition to λ that might help determine the outcome. But this just shows that he -- like Jaynes and Jarrett, incidentally -- have not studied/understood Bell carefully. Bell is absolutely clear that "λ" denotes a *complete* specification of the state of the physical stuff in question (the particle pair, or whatever).

You are arguing with yourself here. This contradicts what you said previously:

... the λ refers to the state of the particles *before any measurements are made on them*. That, of course, is precisely what λ means in all the derivations. But why in the world should ρ(λ) -- the distribution of states of an ensemble of particles that have just been shot toward some polarizers -- depend on the orientation of the polarizers?

So which one is it. Does λ represent the COMPLETE specification of the physical state relevant for the outcome (including all hidden particle AND instrument properties), or does it represent ONLY the "state of the particles shot toward some polarizers"? Make up your mind already.