Understanding Bell's mathematics

[Gordon Watson]

From: Is action at a distance possible as envisaged by the EPR Paradox.

You asked if the mathematical legitimacy of Bell's theorem is irrefutable. The mathematical form of Bell's theorem is the Bell inequalities, and they are irrefutable. Their physical meaning, however, is debatable.

In order to determine the physical meaning of the inequalities we look at where they come from, Bell's locality condition, P(AB|H) = P(A|H)P(B|H).

Then we can ask what you asked and we see that:
1. A and B are correlated in EPR settings.
2. Bell uses P(AB|H) = P(A|H)P(B|H)
3. P(AB|H) = P(A|H)P(B|H) is invalid when A and B are correlated.

Conclusion: The form, P(AB|H) = P(A|H)P(B|H), cannot possibly model the experimental situation. This is the immediate cause of violation of BIs based on limitations imposed by this form.

What does this mean?

P(AB|H) = P(A|H)P(B|H) is the purported locality condition. Yet it is first the definition of statistical independence. The experiments are prepared to produce statistical dependence via the measurement of a relationship between two disturbances by a joint or global measurement parameter in accordance with local causality.

Bell inequalities are violated because an experiment prepared to produce statistical dependence is being modeled as an experiment prepared to produce statistical independence.

Bell's theorem says that the statistical predictions of qm are incompatible with separable predetermination. Which, according to certain attempts (including mine) at disambiguation, means that joint experimental situations which produce (and for which qm correctly predicts) entanglement stats can't be viably modeled in terms of the variable or variables which determine individual results.

Yet, per EPR elements of reality, the joint, entangled, situation must be modeled using the same variables which determine individual results. So, Bell rendered the lhv ansatz in the only form that it could be rendered in and remain consistent with the EPR meaning of local hidden variable.

Therefore, Bell's theorem, as stated above by Bell, and disambiguated, holds.

Does it imply nonlocality -- no.

This is not correct because it is not what Bell says. You are mixing up his separability formula (Bell's 2), which has a different meaning. Bell is simply saying that there are 2 separate probability functions which are evaluated independently. They can be correlated, there is no restiction there and in fact Bell states immediately following that "This should equal the Quantum mechanical expectation value..." which is 1 when the a and b settings are the same. (This being the fully correlated case.)

I thought this needed a new thread (to stop a hi-jack), with an emphasis on Bell's mathematics please.

 

[DrChinese]

OOO goooood.

Bell covers plenty of ground in his paper. As I said, he wrote it for a certain audience. He believed they would be able to follow his line of reasoning. So it is necessary to follow Bell using the strongest arguments. He could see that the basic idea was important: that the EPR result (QM was not complete) was now incompatible with the kind of world they envisioned. We call that world Local Realistic. So we should talk about the requirements - per Bell - for a local realistic world.

Bell used 2 central ideas:

a) that the setting used for Alice did not change the result for Bob (and vice versa). This is often called the separability requirement. Bell said said this was to restore locality. This is his (2).
b) that there should be definite outcomes possible for counterfactual measurement settings. This is introduced after his (14). He says "It follows that c is another unit vector..." but what he really means is: "Assume c is another unit vector...".

Both of the above are needed to get the result, but they can be expressed a variety of ways.

 

[Gordon Watson]

You asked if the mathematical legitimacy of Bell's theorem is irrefutable. The mathematical form of Bell's theorem is the Bell inequalities, and they are irrefutable. Their physical meaning, however, is debatable.

In order to determine the physical meaning of the inequalities we look at where they come from, Bell's locality condition, P(AB|H) = P(A|H)P(B|H).

Then we can ask what you asked and we see that:
1. A and B are correlated in EPR settings.
2. Bell uses P(AB|H) = P(A|H)P(B|H)
3. P(AB|H) = P(A|H)P(B|H) is invalid when A and B are correlated.

Conclusion: The form, P(AB|H) = P(A|H)P(B|H), cannot possibly model the experimental situation. This is the immediate cause of violation of BIs based on limitations imposed by this form.

What does this mean?

P(AB|H) = P(A|H)P(B|H) is the purported locality condition. Yet it is first the definition of statistical independence. The experiments are prepared to produce statistical dependence via the measurement of a relationship between two disturbances by a joint or global measurement parameter in accordance with local causality.

Bell inequalities are violated because an experiment prepared to produce statistical dependence is being modeled as an experiment prepared to produce statistical independence.

Bell's theorem says that the statistical predictions of qm are incompatible with separable predetermination. Which, according to certain attempts (including mine) at disambiguation, means that joint experimental situations which produce (and for which qm correctly predicts) entanglement stats can't be viably modeled in terms of the variable or variables which determine individual results.

Yet, per EPR elements of reality, the joint, entangled, situation must be modeled using the same variables which determine individual results. So, Bell rendered the lhv ansatz in the only form that it could be rendered in and remain consistent with the EPR meaning of local hidden variable.

Therefore, Bell's theorem, as stated above by Bell, and disambiguated, holds.

Does it imply nonlocality -- no.

This is not correct because it is not what Bell says. You are mixing up his separability formula (Bell's 2), which has a different meaning. Bell is simply saying that there are 2 separate probability functions which are evaluated independently. They can be correlated, there is no restiction there and in fact Bell states immediately following that "This should equal the Quantum mechanical expectation value..." which is 1 when the a and b settings are the same. (This being the fully correlated case.)

DrC, ThomasT.

You both appear to agree that Bell uses P(AB|H) = P(A|H).P(B|H) in his work.

I cannot see how EPR studies using that formula could be serious. If H includes a hidden variable for each particle, that formula gives P(AB|H) = P(A|H).P(B|H) = (1/2).(1/2) = 1/4.

Can you direct me to an example where Bell uses P(AB|H) = P(A|H).P(B|H) in his work, please?

 

[alxm]

I just read the Bell paper for the first time (woo!), and the way it looks to me, his [2] does not imply "P(AB|H) = P(A|H).P(B|H)" at all.

That would imply that A and B weren't correlated and as DrChinese said, they can be as correlated as they want, through their mutual dependence on lambda - any number of hidden variables you're free to pick. And yes, Bell does explicitly address this.

You can only get P(AB|H) = P(A|H).P(B|H) from that if you throw out the hidden-variable dependence. They'd naturally be uncorrelated then. The point of [2] is that A is independent of B's detector settings. Apart from that, they can be as mutually interdependent as one chooses.

 

[Gordon Watson]

I just read the Bell paper for the first time (woo!), and the way it looks to me, his [2] does not imply "P(AB|H) = P(A|H).P(B|H)" at all.

That would imply that A and B weren't correlated and as DrChinese said, they can be as correlated as they want, through their mutual dependence on lambda - any number of hidden variables you're free to pick. And yes, Bell does explicitly address this.

You can only get P(AB|H) = P(A|H).P(B|H) from that if you throw out the hidden-variable dependence. They'd naturally be uncorrelated then. The point of [2] is that A is independent of B's detector settings. Apart from that, they can be as mutually interdependent as one chooses.

Thank you alxm.

So Bell needs to do something like this -

P(AB|Hab$$\lambda$$) = P(A|Hab$$\lambda$$).P(B|Hab$$\lambda$$A) = P(A|Ha$$\lambda$$).P(B|Hab$$\lambda$$A).

Does he need to? Does he do it?

Thank you.

 

[DrChinese]

Thank you alxm.

So Bell needs to do something like this -

P(AB|Hab$$\lambda$$) = P(A|Hab$$\lambda$$).P(B|Hab$$\lambda$$A) = P(A|Ha$$\lambda$$).P(B|Hab$$\lambda$$A).

Does he need to? Does he do it?

Thank you.

Again, Bell was writing for a small audience who could be counted on to understand his points. So if it is read as a general piece for a general audience, it will end up having a somewhat different meaning than intended.

There has been a lot of discussion around Bell's (2) over the years. Some try to read it literally. He is attempting to express the idea that the detector setting for Alice does not affect the result at Bob, and vice versa. He says those words. Now, my question is: how might you express it better mathematically than his (2)?

Second, does he need it? Well, I don't think so. You can express everything you need - in my opinion - with something akin to the following:

a) P(A, B) = P(A, B, C) + P(A, B, ~C)
b) P(A, B, C) + P(~A, B, C) + ...(the other 6 permutations) = 1

So I get this directly from Bell, although I realize some don't see these points. I mean, really, what does it matter? Once you see the line of reasoning, you can express it different ways. I have a page where I show that, based on the above, some cases should have a -10% chance of occurance - an absurd result. Mermin has a great way of expressing it too, I have a page on that approach as well. Or you can follow the master, Bell.

So when I see someone trying to say that P(AB) = P(A)*P(B) I usually think they have already missed the starting line (I am talking to ThomasT here). What he is really saying is that if you consider A and B as settings, P(AB) must equal the quantum mechanical value AND it must be able to be decomposed into P(A) and P(B). This is pretty much the same thing as my a) and b) above.

 

[DrChinese]

I just read the Bell paper for the first time (woo!), and the way it looks to me, his [2] does not imply "P(AB|H) = P(A|H).P(B|H)" at all.

That would imply that A and B weren't correlated and as DrChinese said, they can be as correlated as they want, through their mutual dependence on lambda - any number of hidden variables you're free to pick. And yes, Bell does explicitly address this.

You can only get P(AB|H) = P(A|H).P(B|H) from that if you throw out the hidden-variable dependence. They'd naturally be uncorrelated then. The point of [2] is that A is independent of B's detector settings. Apart from that, they can be as mutually interdependent as one chooses.

It's an interesting read, isn't it? There is a lot hidden in the paper, and because of the intended audience it sometimes takes an extra read (or three) to appreciate the full value. He really circled things from a lot of perspectives. If I had had the great insight Bell had, I would never have been able to dot the i's and cross the z's like he did.

 

[Gordon Watson]

Again, Bell was writing for a small audience who could be counted on to understand his points. So if it is read as a general piece for a general audience, it will end up having a somewhat different meaning than intended.

There has been a lot of discussion around Bell's (2) over the years. Some try to read it literally. He is attempting to express the idea that the detector setting for Alice does not affect the result at Bob, and vice versa. He says those words. Now, my question is: how might you express it better mathematically than his (2)?

Second, does he need it? Well, I don't think so. You can express everything you need - in my opinion - with something akin to the following:

a) P(A, B) = P(A, B, C) + P(A, B, ~C)
b) P(A, B, C) + P(~A, B, C) + ...(the other 6 permutations) = 1

So I get this directly from Bell, although I realize some don't see these points. I mean, really, what does it matter? Once you see the line of reasoning, you can express it different ways. I have a page where I show that, based on the above, some cases should have a -10% chance of occurance - an absurd result. Mermin has a great way of expressing it too, I have a page on that approach as well. Or you can follow the master, Bell.

So when I see someone trying to say that P(AB) = P(A)*P(B) I usually think they have already missed the starting line. What he is really saying is that if you consider A and B as settings, P(AB) must equal the quantum mechanical value AND it must be able to be decomposed into P(A) and P(B). This is pretty much the same thing as my a) and b) above.

Thank you DrC, but this seems confusing.

A and B considered "settings"? Aren't they specific outcomes? P(two settings) is not making sense to me.

Agreed that detector setting a should not influence outcome B, and b ... A. Agreed that P(AB|H) must equal the QM value. We agree on the essentials.

Then your requirement that P(AB|H) be decomposed into P(A|H) and P(B|H) puts you and your mathematics outside the EPR context (because EPR outcomes are correlated).

Granted there may be other ways of equating with Bell's conclusion, can you help me stay with Bell's mathematics please? This thread is about understanding Bell's mathematics.

 

[zonde]

Then your requirement that P(AB|H) be decomposed into P(A|H) and P(B|H) puts you and your mathematics outside the EPR context (because EPR outcomes are correlated).

It is assumed that correlation appears because of H (LHV assumption). And then it is correct that P(AB|H) can be decomposed into P(A|H) and P(B|H).

Where you deviate from experimental situation is that P(A|H)+P(~A|H)=1 is not correct. To get that you have to normalize P(A|H)+P(~A|H) to 1. Like that k*P(A|H)+k*P(~A|H)=1.

Now you can write k3*P(AB|H)=k1*P(A|H)*k2*P(B|H). And there you invoke fair sampling assumption by equating k1=k2=k and k3=k^2.
But that's my line about unfair sampling and it might be a bit to the side from your question.

 

[Gordon Watson]

It is assumed that correlation appears because of H (LHV assumption). And then it is correct that P(AB|H) can be decomposed into P(A|H) and P(B|H).

Where you deviate from experimental situation is that P(A|H)+P(~A|H)=1 is not correct. To get that you have to normalize P(A|H)+P(~A|H) to 1. Like that k*P(A|H)+k*P(~A|H)=1.

Now you can write k3*P(AB|H)=k1*P(A|H)*k2*P(B|H). And there you invoke fair sampling assumption by equating k1=k2=k and k3=k^2.
But that's my line about unfair sampling and it might be a bit to the side from your question.

Thank you zonde.

This information appears to be to the side of my question on Bell's mathematics.

H stands for the conditions under which we are analyzing AB.

 

[zonde]

H stands for the conditions under which we are analyzing AB.

I am quite sure this is upside down.
It's H that we are analyzing under conditions A and B.
H stands for photons (I prefer to keep closer to real experiments) but A and B stands for polarizer or PBS settings.

 

[Gordon Watson]

I am quite sure this is upside down.
It's H that we are analyzing under conditions A and B.
H stands for photons (I prefer to keep closer to real experiments) but A and B stands for polarizer or PBS settings.

Isn't A an outcome from Alice's detector which Alice had set at angle a? B an outcome from Bob's detector which Bob had set at angle b? With photons A can be G (green light) or R (red light), B can be G' (green) or R' (red). So combined outcomes can be GG', GR', RG', RR' and we are interested in their probability under condition H?

So the settings are a and b which are angles?

 

[ThomasT]

Then your requirement that P(AB|H) be decomposed into P(A|H| and P(B|H) puts you and your mathematics outside the EPR context (because EPR outcomes are correlated).

Right. This is what the probability analog (in which A and B are specific outcomes) to Bell's ansatz is saying. The EPR context comprises only two joint settings, or two values for |a-b|. These are the only joint settings where there's a correlation between A and B . For these two joint settings the form P(AB|H) = P(A|H) P(B|AH) doesn't reduce to P(AB|H) = P(A|H) P(B|H). (And this doesn't imply ftl transmission, because the information that allows us to write P(AB|) = P(A|H) P(B|AH) contingent on certain joint settings is in the experimental preparation which is in the past light cones of both observers.) For all other values of |a-b|, P(AB|H) = P(A|H) P(B|AH) reduces to P(AB|H) = P(A|H) P(B|H). (That is, when there's no correlation between A and B, then P(AB|H) = P(A|H) P(B|H) obtains.)

The requirement (Bell's task), per EPR, was to model the joint, entangled, state in terms of parameters which determine individual results. That is, in order to model separable predetermination, per EPR, Bell had to model the joint measurement situation in a separable form as a combination of the individual situations which are determined by the hidden parameter(s), H. The use of the probability analogs is an attempt to support Bell's result by showing that the EPR requirement entails a contradiction between the reality of the experimental situation and the form that any, EPR constrained, local realistic model has to be rendered in.

The 'physical' reason why it can't be done is because the joint measurement context involves different parameters than the individual contexts. The RELATIONSHIP between counter-propagating, entangled, photons is NOT what determines the individual results -- but it IS what determines the joint results. Hence, there's a dilemma if it's required that local realistic models of joint, entanglement, situations be rendered in terms of individual results.

Thus, it can be understood that inequalities based on certain (EPR) modelling constraints don't (and can't) represent the actual joint experimental situations or preparations designed to produce entanglement. That's why the experimental results (and qm predictions) exceed the limits set by inequalities so constrained -- and not because Nature is nonlocal or because the results which determine individual results aren't predetermined and separately determining those individual results. (Keeping in mind that separable predetermination IS compatible with the qm description of the individual measurement situations.)

Further, it's been shown that the joint, entangled, situation CAN be viably modeled as a nonseparable situation involving predetermined (eg., via emission), realistic (but not in the EPR sense of determining individual results), global hidden parameters.

 

[Gordon Watson]

Right. This is what the probability analog (in which A and B are specific outcomes) to Bell's ansatz is saying. The EPR context comprises only two joint settings, or two values for |a-b|. These are the only joint settings where there's a correlation between A and B . For these two joint settings the form P(AB|H) = P(A|H) P(B|AH) doesn't reduce to P(AB|H) = P(A|H) P(B|H). (And this doesn't imply ftl transmission, because the information that allows us to write P(AB|) = P(A|H) P(B|AH) contingent on certain joint settings is in the experimental preparation which is in the past light cones of both observers.) For all other values of |a-b|, P(AB|H) = P(A|H) P(B|AH) reduces to P(AB|H) = P(A|H) P(B|H). (That is, when there's no correlation between A and B, then P(AB|H) = P(A|H) P(B|H) obtains.)

The requirement (Bell's task), per EPR, was to model the joint, entangled, state in terms of parameters which determine individual results. That is, in order to model separable predetermination, per EPR, Bell had to model the joint measurement situation in a separable form as a combination of the individual situations which are determined by the hidden parameter(s), H. The use of the probability analogs is an attempt to support Bell's result by showing that the EPR requirement entails a contradiction between the reality of the experimental situation and the form that any, EPR constrained, local realistic model has to be rendered in.

The 'physical' reason why it can't be done is because the joint measurement context involves different parameters than the individual contexts. The RELATIONSHIP between counter-propagating, entangled, photons is NOT what determines the individual results -- but it IS what determines the joint results. Hence, there's a dilemma if it's required that local realistic models of joint, entanglement, situations be rendered in terms of individual results.

Thus, it can be understood that inequalities based on certain (EPR) modelling constraints don't (and can't) represent the actual joint experimental situations or preparations designed to produce entanglement. That's why the experimental results (and qm predictions) exceed the limits set by inequalities so constrained -- and not because Nature is nonlocal or because the results which determine individual results aren't predetermined and separately determining those individual results. (Keeping in mind that separable predetermination IS compatible with the qm description of the individual measurement situations.)

Further, it's been shown that the joint, entangled, situation CAN be viably modeled as a nonseparable situation involving predetermined (eg., via emission), realistic (but not in the EPR sense of determining individual results), global hidden parameters.

Looking at the mathematics only, this seems confused.

In EPR-Bell settings, the formula P(AB|H) = P(A|H).P(B|HA) holds for any value of a and any value of b. So it holds for any value of |a - b| or (a - b). In my opinion.

 

[zonde]

Isn't A an outcome from Alice's detector which Alice had set at angle a? B an outcome from Bob's detector which Bob had set at angle b? With photons A can be G (green light) or R (red light), B can be G' (green) or R' (red). So combined outcomes can be GG', GR', RG', RR' and we are interested in their probability under condition H?

So the settings are a and b which are angles?

If we talk about photons and analyze them with polarizers then event A would be that Alice's photon with hidden variable H was detected after passing polarizer that is set at an angle $$\vec{a}$$.
So if we write $$P(A\cap B|H\vec{a}\vec{b})=P(A|H\vec{a})P(B|H\vec{b})$$ would it be unambiguous now?

In case of Bell A is two different events i.e. A=+1 when spin up was detected and A=-1 when spin down was detected. So it messes up things a bit if you refer to A as an event with some probability while meaning it in context of Bell paper. Maybe that's the reason of confusion? Were you taking H as either +1 or -1 condition?

 

[Gordon Watson]

If we talk about photons and analyze them with polarizers then event A would be that Alice's photon with hidden variable H was detected after passing polarizer that is set at an angle $$\vec{a}$$.
So if we write $$P(A\cap B|H\vec{a}\vec{b})=P(A|H\vec{a})P(B|H\vec{b})$$ would it be unambiguous now?

In case of Bell A is two different events i.e. A=+1 when spin up was detected and A=-1 when spin down was detected. So it messes up things a bit if you refer to A as an event with some probability while meaning it in context of Bell paper. Maybe that's the reason of confusion? Were you taking H as either +1 or -1 condition?

Thank you zonde for your attempts at clarification. I totally support them. However (IMHO) there a some hidden subtleties and confusions (that need amendment) in what you wrote. My cuts and pastes and amendments and extensions are --

1. If we talk about photons and analyze them with polarizers then event A would be that, under condition H, Alice's photon was detected after passing Alice's polarizer that is oriented $$\vec{a}$$.

1a. A will be signaled by an outcome in Alice's analyzer, either G or R; that's how we know the photon is detected. A = {G, R}? So we can discuss P(G|H$$\vec{a}$$) = 0.5, and P(R|H$$\vec{a}$$) = 0.5; H defining the conditions.

1b. B will similarly be a signaled outcome, either G' or R' at Bob's analyzer. B = {G', R'}? So we can discuss P(G'|H$$\vec{b}$$) = 0.5, and P(R'|H$$\vec{b}$$) = 0.5.

2. So if zonde wrote

$$P(GG'|H\vec{a}\vec{b})=P(G|H\vec{a})P(G'|H\vec{b})$$,

that would be unambiguous.

2a. But it would equal 0.25 and be unhelpful. Because H specifies the conditions and the conditions are EPR-Bell conditions and under such conditions G and G' are correlated.

2b. $$P(GG'|H\vec{a}\vec{b}) = P(G|H\vec{a}\vec{b})P(G'|H\vec{a}\vec{b}G) = P(G|H\vec{a})P(G'|H\vec{a}\vec{b}G)$$ would be OK.

3. Was I taking H as either +1 or -1 condition? No. H specifies the general overall total conditions.

 

[zonde]

1a. A will be signaled by an outcome in Alice's analyzer, either G or R; that's how we know the photon is detected. A = {G, R}? So we can discuss P(G|H$$\vec{a}$$) = 0.5, and P(R|H$$\vec{a}$$) = 0.5; H defining the conditions.

No, analyzer consisting of polarizer and detector does not give you G and R. It just gives you "clicks" time after time.
If you want G and R case then we have to switch to analyzer consisting of PBS and two detectors. Then we can have "click" in G detector or we can have "click" in R detector.

Next H is defining certain but unknown conditions that are shared between Alice and Bob. So these are conditions that have causal connection with source. Conditions that are causally related only to either Alice or Bob are not included in H.
P(G|H$$\vec{a}$$) does not have to be 0.5 because we talk about certain (but unknown) value of H.
If H is polarization of photon clearly in case where $$\vec{a}$$ is perfectly aligned with that polarization of photon we should have probability of 1 for G signal (or 0 depending which output we define as G).

You can say that P(G|$$\vec{a}$$)=0.5 but that too only after normalization.

3. Was I taking H as either +1 or -1 condition? No. H specifies the general overall total conditions.

That's not good either.
H is only shared conditions or more trivially speaking it's polarization of individual photon.
If you want to include still something else this must be described as additional variable individually for Alice and Bob (like $$\vec{a}$$ and $$\vec{b}$$).

 

[Gordon Watson]

[1] No, analyzer consisting of polarizer and detector does not give you G and R. It just gives you "clicks" time after time.
If you want G and R case then we have to switch to analyzer consisting of PBS and two detectors. Then we can have "click" in G detector or we can have "click" in R detector.

[2] Next H is defining certain but unknown conditions that are shared between Alice and Bob. So these are conditions that have causal connection with source. Conditions that are causally related only to either Alice or Bob are not included in H.


[3] P(G|H$$\vec{a}$$) does not have to be 0.5 because we talk about certain (but unknown) value of H.

[4] If H is polarization of photon clearly in case where $$\vec{a}$$ is perfectly aligned with that polarization of photon we should have probability of 1 for G signal (or 0 depending which output we define as G).

[5]. You can say that P(G|$$\vec{a}$$)=0.5 but that too only after normalization.


[6] That's not good either.
H is only shared conditions or more trivially speaking it's polarization of individual photon.
If you want to include still something else this must be described as additional variable individually for Alice and Bob (like $$\vec{a}$$ and $$\vec{b}$$).

I see your NO-s and NOT-s et cetera and suspect you are wrong or confused in each case.

1. You say NO ... DOES NOT ... IF ... ? My G/R polarizer-analyzers use pure Iceland spar so G or R for Alice, G' or R' for Bob, works quite OK.

2. Aren't I the one that introduced H? My H includes $$\vec{a}$$ and $$\vec{b}$$, but there's no problem pulling them out of H (if you wish and when it helps).

3. You talk about that if you wish. I choose not to. Makes no sense (to me).

4. Makes no sense with my H. Time to bring in your own Z, maybe?

5. Aren't probabilities normalized by definition? They're not the same as raw experimental frequencies.

6. See 2 above.

I think you are making too many assumptions about my notation and approach. Time to bring in your own for me to follow?

All we want is an agreed notation that leads us to agree on Bell's mathematics.

 

[zonde]

I see your NO-s and NOT-s et cetera and suspect you are wrong or confused in each case.

1. You say NO ... DOES NOT ... IF ... ? My G/R polarizer-analyzers use pure Iceland spar so G or R for Alice, G' or R' for Bob, works quite OK.

I am not sure I understand how your analyzer works. Can you describe it a bit more? Where is photon when your analyzer gives G and where is photon when your analyzer gives R?

2. Aren't I the one that introduced H? My H includes $$\vec{a}$$ and $$\vec{b}$$, but there's no problem pulling them out of H (if you wish and when it helps).

If you mean that you introduced H when you wrote
P(AB|H)=P(A|H)P(B|H)
then my objections still hold. It's because you use the same H for both Alice and Bob. But they are spatially separated so they can't be described with the same conditions.
If you write something like P(AB|HH')=P(A|H)P(B|H') then yes you can introduce H and H' as you wish.

 

[Gordon Watson]

I am not sure I understand how your analyzer works. Can you describe it a bit more? Where is photon when your analyzer gives G and where is photon when your analyzer gives R?

For G (or +1), photon is absorbed in the ordinary ray detector. For R (or -1), photon is absorbed in extraordinary ray detector. (Easy to make. I supply DrC, Mermin, Clauser, Aspect, Zeilinger. You want some

If you mean that you introduced H when you wrote
P(AB|H)=P(A|H)P(B|H)
then my objections still hold. It's because you use the same H for both Alice and Bob. But they are spatially separated so they can't be described with the same conditions.
If you write something like P(AB|HH')=P(A|H)P(B|H') then yes you can introduce H and H' as you wish.

No problem to make you happy?

1. With EPR-Bell common condition H, Alice controls orientation a, sees R or G, assumes z has arrived. Bob controls orientation b, sees R' or G', assumes z has arrived.

2. z is Bell's lambda for your photon example.

3. I write formula. You give answer:

P(G|H) = ?

P(G|Ha) = ?

P(G|Haz) = ?

P(G|Hazb) = ?

P(G|HazbG') = ?

Repeat for R replacing G ... ... ... ...

P(G'|H) = ?

P(G'|Hb) = ?

P(G'|Hbz) = ?

P(G'|Hbza) =

P(G'|HazbG) =

Repeat for R' replacing G' ... ... ... ...

P(GG'|H) = ?

P(GG'|Ha) = ?

P(GG'|Haz) = ?

P(GG'|Hazb) = ?

P(GG'|HazbR') = ?

Repeat for R' replacing G' ... ... ... ...

et cetera

You happy?

 

[ThomasT]

Looking at the mathematics only, this seems confused.

In EPR-Bell settings, the formula P(AB|H) = P(A|H).P(B|HA) holds for any value of a and any value of b. So it holds for any value of |a - b| or (a - b). In my opinion.

P(AB|H) = P(A|H).P(B|HA) reduces to P(AB|H)=P(A|H)P(B|H) for all settings except |a-b| = 0 and 90 degrees. Anyway, sorry for the delay in replying, but I've been thinking about EPR-Bell from a different perspective. Also, rereading lots of threads and papers. So, I'll just be an occasional observer of this thread.

My little excursion into (simplified) probability notation was just to make a point that I thought might be important at the time, but which I currently don't think is the crux of the problem with interpretations of Bell's theorem. He made an assumption about the meaning of the realism (EPR) part of local realism that's even subtler than what the parsing of his locality condition revealed about that -- and it renders BIs physically insignificant except as possible 'entanglement' measures.

If you have some specific questions re the math in Bell's paper, why not just reproduce (either here or in the math forum) the stuff that you're not sure about and one of the advisors or mentors (or Zonde, or me if I happen to be around) can give you a straightforward answer?

--------------------------

By the way, I noticed your question in the nonlocality thread re why should nonlocality be invoked when entanglement setups produce slightly different, but still similar, correlations compared to nonentanglement setups. It's a good question. Indeed, it would seem more logical to look at the similarities between the two and conclude that the two situations are evolving according to the same physical principles and that the former is simply a special case of the latter. The reason that people opt for nonlocality is due to the, apparently, prevailing opinion regarding the physical meaning of Bell's theorem and violation of BIs. So, it's become the status quo because Bell's ansatz is only generally applicable if some sort of nonlocal 'communication' between the two sides of the experiment is included -- otherwise it's just an unnecessarily restrictive formulation of the joint situation. But as your comments indicated, it would seem to make more sense to look a bit more closely at Bell's implementation of the EPR definition of reality before we trash locality. My current opinion is that the problem isn't with the EPRs elements of reality, but with Bell's too narrow interpretation of just what sort of form a local realistic model might be rendered in.

Sorry for the aside(s), but I just wanted to mention this while I was here -- and anyway, everything eventually comes back to the precursors to Bell's mathematical implementation(s).

 

[Gordon Watson]

I am not sure I understand how your analyzer works. Can you describe it a bit more? Where is photon when your analyzer gives G and where is photon when your analyzer gives R?


If you mean that you introduced H when you wrote
P(AB|H)=P(A|H)P(B|H)
then my objections still hold. It's because you use the same H for both Alice and Bob. But they are spatially separated so they can't be described with the same conditions.
If you write something like P(AB|HH')=P(A|H)P(B|H') then yes you can introduce H and H' as you wish.

P(AB|H) = P(A|H).P(B|HA) reduces to P(AB|H)=P(A|H)P(B|H) for all settings except |a-b| = 0 and 90 degrees. Anyway, sorry for the delay in replying, but I've been thinking about EPR-Bell from a different perspective. Also, rereading lots of threads and papers. So, I'll just be an occasional observer of this thread.

My little excursion into (simplified) probability notation was just to make a point that I thought might be important at the time, but which I currently don't think is the crux of the problem with interpretations of Bell's theorem. He made an assumption about the meaning of the realism (EPR) part of local realism that's even subtler than what the parsing of his locality condition revealed about that -- and it renders BIs physically insignificant except as possible 'entanglement' measures.

If you have some specific questions re the math in Bell's paper, why not just reproduce (either here or in the math forum) the stuff that you're not sure about and one of the advisors or mentors (or Zonde, or me if I happen to be around) can give you a straightforward answer?

--------------------------

By the way, I noticed your question in the nonlocality thread re why should nonlocality be invoked when entanglement setups produce slightly different, but still similar, correlations compared to nonentanglement setups. It's a good question. Indeed, it would seem more logical to look at the similarities between the two and conclude that the two situations are evolving according to the same physical principles and that the former is simply a special case of the latter. The reason that people opt for nonlocality is due to the, apparently, prevailing opinion regarding the physical meaning of Bell's theorem and violation of BIs. So, it's become the status quo because Bell's ansatz is only generally applicable if some sort of nonlocal 'communication' between the two sides of the experiment is included -- otherwise it's just an unnecessarily restrictive formulation of the joint situation. But as your comments indicated, it would seem to make more sense to look a bit more closely at Bell's implementation of the EPR definition of reality before we trash locality. My current opinion is that the problem isn't with the EPRs elements of reality, but with Bell's too narrow interpretation of just what sort of form a local realistic model might be rendered in.

Sorry for the aside(s), but I just wanted to mention this while I was here -- and anyway, everything eventually comes back to the precursors to Bell's mathematical implementation(s).

Thank you, ThomasT. Your aside(s) are very welcome to me. And your direct comments. Please stay in touch. And active.

Question.

P(AB|H) = P(A|H).P(B|HA) reduces to P(AB|H)=P(A|H)P(B|H) for all settings except |a-b| = 0 and 90 degrees.

Using zonde's photon example, are you sure about this statement?

 

[DrChinese]

-- otherwise it's just an unnecessarily restrictive formulation of the joint situation. But as your comments indicated, it would seem to make more sense to look a bit more closely at Bell's implementation of the EPR definition of reality before we trash locality. My current opinion is that the problem isn't with the EPRs elements of reality, but with Bell's too narrow interpretation of just what sort of form a local realistic model might be rendered in.

And since this thread is about the mathematical side of Bell, perhaps you could point out a) exactly what that might be; and b) your idea of a more "reasonable" interpretation to replace it.

If you think it is too narrow to require that the Alice outcome is not affected by the Bob setting, then say so.

 

[Gordon Watson]

And since this thread is about the mathematical side of Bell, perhaps you could point out a) exactly what that might be; and b) your idea of a more "reasonable" interpretation to replace it.

If you think it is too narrow to require that the Alice outcome is not affected by the Bob setting, then say so.

DrC, I wish personally not to get ahead too far of zonde and ThomasT in this thread.

While I wait for their answers, would you comment on this please (from notation proposed by me above) --

P(GG'|Hazb) = P(G|Hazb).P(G'|HazbG) = P(G|Haz).P(G'|HazbG).

Question 1. MY simplifying permitted because Bell [.. and me also ..] requires as you say "that the Alice outcome [G] is not affected by the Bob setting ". Yes?

Question 2. Is any more simplifying permitted?

Question 3. Did BELL simplify more?

Thank you.

 

[zonde]

For G (or +1), photon is absorbed in the ordinary ray detector. For R (or -1), photon is absorbed in extraordinary ray detector. (Easy to make. I supply DrC, Mermin, Clauser, Aspect, Zeilinger. You want some

Sounds like PBS based analyzer with two detectors. So it's fine.

No problem to make you happy?

1. With EPR-Bell common condition H, Alice controls orientation a, sees R or G, assumes z has arrived. Bob controls orientation b, sees R' or G', assumes z has arrived.

2. z is Bell's lambda for your photon example.

3. I write formula. You give answer:

I didn't quite understood what answers I was supposed to write but I guess I am happy with a, b and z where a and b are local to Alice and Bob but z is shared between them.
As I understand in general case H is supposed to be non-local so it requires caution when we talk about local and non-local contexts.

So I will write that: P(GG'|abz)=P(G|az)P(G'|bz)
Is it ok?

 

[ThomasT]

Using zonde's photon example, are you sure about this statement?

Here's how I'm thinking about it. For |a-b| /= 0^o or 90^o then when a detection is registered at either end, then that doesn't alter the prediction at the other end. So for all |a-b| except the EPR settings (the EPR settings are |a-b| = 0^o and 90^o), then P(AB|H) = P(A|H)P(B|HA) reduces to P(AB|H)=P(A|H)P(B|H).

Wrt the EPR settings it doesn't reduce because when a detection is registered at one end, then that alters the prediction at the other end.

One might say that it should reduce even for EPR settings because the probabilities are conditioned on H which represents everything in the past light cones of A and B -- or else ftl is implied.

But this reduction doesn't imply ftl because the contingencies that alter the prediction at one end given a detection at the other are facts of the experimental setup in the past light cones of both A and B.

However, whether A or B will detect isn't known at the outset (this knowledge isn't in the past light cones of A and B). So, at the outset of any given trial, the probability of detection at A and the probability of detection at B is always just .5 (even for EPR settings).

 

[DrChinese]

Wrt the EPR settings it doesn't reduce because when a detection is registered at one end, then that alters the prediction at the other end.

That cannot ("alters") happen unless there is an ftl influence. So you are arguing in reverse. Bell's entire point here is that the Alice setting (or result) does not affect Bob's result in a local realistic world. So the idea that something different happens according to Theta (A-B) being the "EPR" setting - or not - makes no sense.
--------------------------------------------------------------
I think you may find it beneficial to read the separability statement - Bell's (2) - a little differently. Read it as:

F(AB|abH) = F(A|aH) F(B|bH)

Which is the equivalent to how both zonde and JenniT have it... with AB are a specific outcome for settings a and b with hidden variables H. And remember that we are integrating so that we are not trying to get a simple product. So here is an example:

We have a dataset of 5 cars and 5 motorcycles (these are the hidden variables H). All of the cars have automatic transmissions and none of the motorcycles do. 1 car and 4 motorcycles are black, the rest are white.

The P(automatic,black) is .1 but that is not equal to P(automatic) * P(black) which is .5 *.5, i.e. .25 and the formula does not work. So don't just multiply or you will get the wrong relationships. Instead, what we want for each individual case is:

F(automatic, black) = F(automatic) * F(black), yielding either a one or a zero.

We get 9 zeros and 1 one. That averages to .1 (over 10 trials) which is correct. That is separability. There can be any bias or correlation you like in the universe. In fact, you might easily expect that there is such bias. The only thing Bell is saying here is that the result of a 2 part question must be a product state of the individual questions. Keep in mind that in our example, we are essentially having Alice ask if the transmission is automatic", and Bob asks if the color is black. Then they match their results on a case by case basis.

I would not call this a severe restriction. It is about as basic as you can get for what might be called a locality condition.

 

[ThomasT]

That cannot ("alters") happen unless there is an ftl influence.

A detection at one end doesn't alter what does happen at the other end. It alters the prediction of what will happen at the other end. This is the case wrt EPR settings where perfect correlation and perfect anticorrelation are observed.

So, F(AB|abH) = F(A|aH) F(B|AbH) doesn't reduce to F(AB|abH) = F(A|aH) F(B|bH) for EPR settings, because for those settings F(B|AbH) /= F(B|bH).

But this doesn't imply ftl because the contingencies that alter the prediction at B given a detection at A are facts of the experimental setup in the past light cones of both A and B.

Note: as I mentioned to JenniT, I'm thinking about Bell from a different perspective for the time being. Maybe there's something in the probability stuff, maybe not.

 

[JesseM]

A detection at one end doesn't alter what does happen at the other end. It alters the prediction of what will happen at the other end. This is the case wrt EPR settings where perfect correlation and perfect anticorrelation are observed.

So, F(AB|abH) = F(A|aH) F(B|AbH) doesn't reduce to F(AB|abH) = F(A|aH) F(B|bH) for EPR settings, because for those settings F(B|AbH) /= F(B|bH).

But this doesn't imply ftl because the contingencies that alter the prediction at B given a detection at A are facts of the experimental setup in the past light cones of both A and B.

And what if H represents all local physical facts in the past light cones of the regions where measurement results A and B occurred, at some moment after the time when the two past light cones stopped overlapping (as depicted in Fig. 4 here)? In this case, if you want to know the probability that setting b will give measurement result B over here, and meanwhile another measurement is being made far away with setting a, then if you already know H, the full information about all local physical variables in the past light cone of the measurement b at some time after the last moment when the past light cones of a and b overlapped (so that nothing in H can have a causal effect on the outcome at a), then learning that measurement a resulted in outcome A should tell you nothing further about the probability that measurement b will result in outcome B.

If this isn't apparent to you even after some reflection, consider the argument I made in post #41 of this thread:

If we can learn something about the probability an event A with spacelike separation from us (say, an event happening on Alpha Centauri right now in our frame) by observing some event B over here, and that's some new information beyond what we already could have known from all the prior events L in our past light cone (including past events which might also be in the past light cone of A and thus could have had a causal influence on it), then this is a form of FTL information transfer. Say A was the event of a particular alien horse on Alpha Centauri winning a race, and B was the event of a buzzer going off in my room; then I know that if I hear the buzzer go off, I should place a bet that when reports of the race reach Earth by radio transmission 4 years later, that particular horse will be the winner, and that will be a piece of information that no one who didn't have access to the buzzer could deduce by examining events in my past light cone. If you think this type of scenario is consistent with relativistic causality in a local realist universe, then I don't know what else to tell you, the idea that you can't gain any new information about an event A by observing an event B at a spacelike separation from it, if you already know all possible information about events in the past light cone of B (or just in a cross-section of the past light cone taken at some time after the last moment when the past light cones of A and B intersected, as I imagined in my analysis in posts 61/62 on the other thread, and is also the assumption used in this paper which discusses relativistic causality as it applies to Bell's analysis, which you should probably look through if my own arguments don't convince you) can basically be taken as the definition of relativistic causality. If you disagree, can you propose an alternate one that's stated in terms of what kind of information you can gain about distant events based only on local observations? Or do you think relativity and local realism place absolutely no limits on information you can gain about events outside your past light cone, allowing arbitrary forms of FTL communication?

 

[DrChinese]

So, F(AB|abH) = F(A|aH) F(B|AbH) doesn't reduce to F(AB|abH) = F(A|aH) F(B|bH) for EPR settings, because for those settings F(B|AbH) /= F(B|bH).

This is false. Just change my example above so that all cars are white and all motorcycles black and you will see that that even for classical perfect correlations, the separability requirement applies and works just fine.

You keep multiplying the wrong things, as I already mentioned. So you see Bell's (2) as not working for perfect symmetric/antisymmetric settings, which is 180 degrees backwards. There is no evidence for or against that per se. It is not until you get to the realism requirement, in which other relationships must also exist (unit vector c) that the problems arise with the local realistic requirements.

 

[ThomasT]

And what if H represents all local physical facts in the past light cones of the regions where measurement results A and B occurred, at some moment after the time when the two past light cones stopped overlapping (as depicted in Fig. 4 here)? In this case, if you want to know the probability that setting b will give measurement result B over here, and meanwhile another measurement is being made far away with setting a, then if you already know H, the full information about all local physical variables in the past light cone of the measurement b at some time after the last moment when the past light cones of a and b overlapped (so that nothing in H can have a causal effect on the outcome at a), then learning that measurement a resulted in outcome A should tell you nothing further about the probability that measurement b will result in outcome B.

Here's how I'm thinking about it:

The information regarding whether A or B will detect isn't known at the outset (this knowledge isn't in the past light cones of A and B). So, at the outset of any given trial, the probability of detection at A and the probability of detection at B is always just .5 (even for EPR settings).

On the other hand, what is in the past light cones of A and B is the experimental preparation and setup, which allows that if we've agreed to use the EPR setting, |a-b| = 0, then if A registers a detection, then the probability of detection at B (which was .5) at the outset of the trial, is thereby altered to 1.

So, wrt any settings that allow such contingent alterations in the the probability of an individual detection then F(B|AbH) /= F(B|bH) and F(A|BaH) /= F(A|aH) and F(AB|abH) /= F(A|aH) F(B|bH).

But this doesn't imply ftl because the contingencies that alter the prediction at B given a detection at A, and vice versa, are facts of the experimental setup in the past light cones of both A and B.


Am I missing something?

 

[ThomasT]

This is false. Just change my example above so that all cars are white and all motorcycles black and you will see that that even for classical perfect correlations, the separability requirement applies and works just fine.

You keep multiplying the wrong things, as I already mentioned. So you see Bell's (2) as not working for perfect symmetric/antisymmetric settings, which is 180 degrees backwards. There is no evidence for or against that per se. It is not until you get to the realism requirement, in which other relationships must also exist (unit vector c) that the problems arise with the local realistic requirements.

Yes, I understand that Bell's (2) only works for EPR settings. It works for those settings because F(AB|abH) = F(A|BaH) F(B|AbH) holds for those settings without implying ftl. And, yes, I understand that the locality and realism requirements are intertwined.

Anyway, as I said, I've abandoned the probability considerations temporarily because I don't think that they really illuminate the problem with Bell's LR model.

 

[Gordon Watson]

Sounds like PBS based analyzer with two detectors. So it's fine.

OK.

I didn't quite understood what answers I was supposed to write but I guess I am happy with a, b and z where a and b are local to Alice and Bob but z is shared between them.

The answers I thought you would give would be those that you derive for your photon example.

As I understand in general case H is supposed to be non-local so it requires caution when we talk about local and non-local contexts.

So I will write that: P(GG'|abz)=P(G|az)P(G'|bz)
Is it ok?

Well, No.

As I see it, H is required so that we know that the source and detectors are EPR-Bell compatible; so that we know we are discussing EPR-Bell. Your caution cannot have H just dropped.

So you should be happy if I upgrade your effort to

P(GG'|Habz)=P(G|Haz)P(G'|Hbz)

and unhappy when I say it equals (1/2)(1/2) = 1/4.

Because your photon experiment (defined by H) would not give that result, would it?

 

[DrChinese]

Yes, I understand that Bell's (2) only works for EPR settings. It works for those settings because F(AB|abH) = F(A|BaH) F(B|AbH) holds for those settings without implying ftl.

And again, this is false. Bell's (2) applies just as much for ANY settings of a and b. There is nothing special about the a=b case. Keep in mind that (2) is just a general statement of ANY 2 sets of classical variables & functions. There is nothing complicated about it, and there was nothing particularly controversial about it. You can probably find variations of this in standard statistical texts. That is why Bell chose this, because he knew it would be understood as basic.

I don't care for it myself because for many people it leads to unneeded confusion. That is why I ignore it in my derivations. There are other things that work just as well and don't lead to a debate.

To address the special a=b case (what you call the EPR case) a bit more: Everyone (pre-Bell) thought this case made sense for ALL a and b and never thought much about it. Because in and of itself, Bell (2) is not obviously violated by the QM predictions. What I think you are trying to say is that based on what we know today, maybe (2) is true for the a=b case. But I don't think you would find very many people who would agree with that viewpoint. It is clearly false for many settings of a, b, c. And whether you want to call it "true" for a=b is something of a semantics issue. Kinda like saying "all men are boys" and claiming it is true for the case where you only have only boys.

 

[Gordon Watson]

I think you may find it beneficial to read the separability statement - Bell's (2) - a little differently. Read it as:

F(AB|abH) = F(A|aH) F(B|bH)

Which is the equivalent to how both zonde and JenniT have it... with AB are a specific outcome for settings a and b with hidden variables H. And remember that we are integrating so that we are not trying to get a simple product.

NOT JenniT PLEASE!

IMHO, if Bell's (2) is to be read as you say, then we have found Bell's mistake!

I asked earlier if I could be shown this above formulation in Bell's work. I thought no one showed it to me?

Who agrees that this above reading is correct? That it is the example that I asked for?



A recent question at Post #24 to DrC about P(GG'|Hazb) was aimed at sorting this confusion out.



From Post #24 --

DrC, I wish personally not to get ahead too far of zonde and ThomasT in this thread.

While I wait for their answers, would you comment on this please (from notation proposed by me above) --

P(GG'|Hazb) = P(G|Hazb).P(G'|HazbG) = P(G|Haz).P(G'|HazbG).

Question 1. MY simplifying permitted because Bell [.. and me also ..] requires as you say "that the Alice outcome [G] is not affected by the Bob setting ". Yes?

Question 2. Is any more simplifying permitted?

Question 3. Did BELL simplify more?

Thank you.