## Understanding Bell's mathematics

 Quote by zonde [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.

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 Quote by JenniT 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?

 Quote by JenniT 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.

 Quote by zonde 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?:)

 Quote by zonde 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?

 Quote by JenniT 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?

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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).

 Quote by zonde 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.

Question.

 Quote by ThomasT 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.

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 Quote by ThomasT -- 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.

 Quote by DrChinese 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 [b]". Yes?

Question 2. Is any more simplifying permitted?

Question 3. Did BELL simplify more?

Thank you.

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 Quote by JenniT 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.

 Quote by JenniT 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?

Here's how I'm thinking about it. For |a-b| /= 0o or 90o 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| = 0o and 90o), 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).

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 Quote by ThomasT 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.
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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.

 Quote by DrChinese 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.

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 Quote by ThomasT 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:

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 Quote by ThomasT 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.

 Quote by JesseM 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?

 Quote by DrChinese 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.

 Quote by zonde Sounds like PBS based analyzer with two detectors. So it's fine.
OK.

 Quote by zonde 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.

 Quote by zonde 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.

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?

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