Understanding Bell's mathematics

[Gordon Watson]

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.

ThomasT, I am using this reply by you just as a vehicle for me to say something that I think might help you. Because your understanding of EPR-Bell seems to be confused to me.

I think if we agree on Bell's fundamental mathematics then we will all understand the basis of Bell's theorem better.

Your comments about |a-b| and certain values that are special seems (to me) to be not relevant. I am thinking this is part of your confusion.

For you to be trying another approach while carrying this confusion will lead to more confusion IMHO.

I can see that DrC and JesseM may be unintentionally confusing you more. So if you sort out the mathematics and your need to refer to certain values of |a-b| all our replies might make better sense and help you.

 

[DrChinese]

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?

Don't misquote me! It is not straight multiplication. It is an integral. You must define it correctly. This is a general purpose statistical statement and has absolutely NOTHING to do with physics. It is a straightforward way of expressing a specific type of universe. Look at my car/motorcycle example above.

 

[ThomasT]

Bell's (2) applies just as much for ANY settings of a and b.

You can apply it. But it's only locally viable for EPR settings -- his second illustration. For all other settings, it only works if, in Bell's words, "the results A and B in (2) are allowed to depend on b and a respectively as well as on a and b" -- his third illustration.

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.

Not maybe. It is true for the a=b case. That was the point of his second illustration: nonlocality isn't required for his (2) to agree with the qm (3) for EPR settings.

But I don't think you would find very many people who would agree with that viewpoint.

It doesn't matter whether they agree with it or not. (2) agrees with the qm prediction for EPR settings.

Maybe we should have these peripheral discussions in another thread. JenniT's trying to work some things out in a systematic way.

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

JenniT, sorry for deviating from your notation in some recent posts here. It won't happen again.

 

[Gordon Watson]

Don't misquote me! It is not straight multiplication. It is an integral. You must define it correctly. This is a general purpose statistical statement and has absolutely NOTHING to do with physics. It is a straightforward way of expressing a specific type of universe. Look at my car/motorcycle example above.

DrC, I do not understand misquoting you?

Bell's (2) is an integral. We agree.

You say -- Read it as: F(AB|abH) = F(A|aH) F(B|bH).

I say No, that would be an error.

You say it is a general purpose statistical statement.

I say it is a specific purpose statistical statement. Limited in scope and not applicable here.

Where is misquote please?

Edit: Using notation that is coming to be used here, I read your "Read it as" as being equivalent to --

P(GG'|Habz) = P(G|Haz) P(G'|Hbz) = (1/2) (1/2) = 1/4.

Are you saying that we should read the "contents in the integral" your way and not the integral? That's different.

 

[ThomasT]

ThomasT, I am using this reply by you just as a vehicle for me to say something that I think might help you. Because your understanding of EPR-Bell seems to be confused to me.

I think if we agree on Bell's fundamental mathematics then we will all understand the basis of Bell's theorem better.

That makes sense to me. That's why I'm going to stay tuned to your thread here -- and not respond to any more posts by anyone except you (in this thread).

Your comments about |a-b| and certain values that are special seems (to me) to be not relevant. I am thinking this is part of your confusion.

They seem to me to be not relevant also. That's why I decided to concentrate on another approach which I find interesting.

For you to be trying another approach while carrying this confusion will lead to more confusion IMHO.

What, exactly, is my confusion?

I can see that DrC and JesseM may be unintentionally confusing you more.

I just recently came to understand (at least I think I do) how DrC thinks about Bell's theorem and (at least I think I do) why he thinks that way.

So if you sort out the mathematics and your need to refer to certain values of |a-b| all our replies might make better sense and help you.

Sounds good to me. I'll just stay in the background and observe.

By the way, everyone's replies are making sense (of one sort or another) to me, even if I might happen to disagree. I don't feel a need to refer to certain values of |a-b|, it's just a fact that for certain values of |a-b| Bell's (2) is a viable LR model. The problem of course is that for most values of |a-b| it isn't.

 

[Gordon Watson]

...
(2) agrees with the qm prediction for EPR settings.

...

ThomasT

Beware confusion. This can only be true if YOU are defining EPR settings in some unique way.

Are you referring to the detector settings a and b as EPR settings?

We are using H as the EPR-Bell context. Are you saying --

Bell's (2) agrees with the qm predictions under condition H?

I think it does not?

 

[Gordon Watson]

What, exactly, is my confusion?

Maybe it is our confusion? See my last post?

 

[Gordon Watson]

...

Sounds good to me. I'll just stay in the background and observe.

... I implied [for you to help us] sort out the mathematics. Not stay out.

Edit: What is your view on my question to DrC (and everyone else) about simplifying the P(GG'|Habz) equation? I think it relates to issues that you raise.

From Post#24 --

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?

 

[billschnieder]

Bell's mathematics is anchored on the principle of common cause (PCC) first formulated in mathematical form by Hans Reichenbach (Reichenbach 1956)
* Reichenbach, H. (1956): The Direction of Time, Berkeley, University of Los Angeles Press.
(see http://plato.stanford.edu/entries/physics-Rpcc/)

Bell's reason for going from the generally applicable mathematical equation P(GG'|Habz) = P(G|Habz) P(G'|GHabz) to the non-general equation P(GG'|Habz) = P(G|Haz) P(G'|Hbz)
can be found in PCC. As I discussed in a recent thread, PCC is not universally valid.

In Bell's own words, in his Bertlmann's Socks Paper(BERTLMANN'S SOCKS AND THE NATURE OF REALITY. J. Bell. (1981)),

While discussing the contradiction and referring to equation (11) P(AB|abH) = P1(A|aH)P2(B|bH), where lambda has been replaced by H, Bell said the following:

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 ...

But the part emphasized in bold is clearly misguided if not wrong. In probability theory, it very often is the case that P1 can depend on b and P2 on a even if a has no causal influence on the physical situation at station 2 and b has no causal influence on the physical situation at station 1 since logical dependence does not imply physical causation.
To Bell's thinking, it appears natural to him (albeit due to his misunderstanding) that if no physical influence exists between stations 1 and 2, then he can justifiably reduce the generic equation P(GG'|Habz) = P(G|Habz) P(G'|GHabz) to the specific equation P(GG'|Habz) = P(G|Haz) P(G'|Hbz). Being oblivious to the fact that this equation not only eliminates physical causation, but also all logical dependence, he proceeds to use it even in situations such as the EPR case in which logical dependence exists -- it is obvious that for certain settings, the outcome at station 1 MUST be opposite that at station 2. The latter statement in italics, is a statement of logical dependence between stations 1 and 2, which can not, and must not be ignored in analysing the EPR situation.

 

[DrChinese]

Bell's mathematics is anchored on the principle of common cause (PCC) first formulated in mathematical form by Hans Reichenbach (Reichenbach 1956)
* Reichenbach, H. (1956): The Direction of Time, Berkeley, University of Los Angeles Press.
(see http://plato.stanford.edu/entries/physics-Rpcc/)

Bell's reason for going from the generally applicable mathematical equation P(GG'|Habz) = P(G|Habz) P(G'|GHabz) to the non-general equation P(GG'|Habz) = P(G|Haz) P(G'|Hbz)
can be found in PCC. As I discussed in a recent thread, PCC is not universally valid.

Who cares if it is "universally valid"? The question is simply: is it a reasonable assumption for Bell's purposes? The answer is YES.

Now, suppose you don't agree. You'd be in the minority. But beyond that, it is already well known that this is not required anyway. There are plenty of other alternatives, see any of my Bell derivation pages for example. I don't bother with this because some people don't follow it for one reason or another. And it is unnecessarily complex.

 

[ThomasT]

I implied (for you to help us) sort out the mathematics. Not stay out.

Ok, thanks.

(2) agrees with the qm prediction for EPR settings.

Beware confusion. This can only be true if YOU are defining EPR settings in some unique way. Are you referring to the detector settings a and b as EPR settings?

EPR settings are |a-b| = 0^o and 90^o.

We are using H as the EPR-Bell context. Are you saying --

Bell's (2) agrees with the qm predictions under condition H?

I think it does not?

I think you and DrC are right. Bell's (2) doesn't agree with qm for any settings. This simplifies things a bit.

 

[ThomasT]

... I implied [for you to help us] sort out the mathematics. Not stay out.

Edit: What is your view on my question to DrC (and everyone else) about simplifying the P(GG'|Habz) equation? I think it relates to issues that you raise.

From Post#24 --

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?

1. Yes
2. (drop the z's ?)
3. No z's ?

I'm not sure what you're trying to illustrate.

Bell's (2) is P(a,b) = int dH rho(H) A(a,H) B(b,H)

How are you translating that, and for what purpose?

 

[Gordon Watson]

EPR settings are |a-b| = 0^o and 90^o.

Thank you ThomasT, I see some progress.

Suggest you drop the term EPR settings here because we are on about Bell's mathematics. I suspect the term is not much use anywhere.

I think that when most of us talk here about Bell's mathematics we are interested in EPR-Bell settings. By which we mean that Alice sets a (any she freely chooses) and Bob sets b (any he freely chooses). So here the mathematics must handle any (a,b) combination.

 

[Gordon Watson]

1. Yes
2. (drop the z's ?)
3. No z's ?

I'm not sure what you're trying to illustrate.

Bell's (2) is P(a,b) = int dH rho(H) A(a,H) B(b,H)

How are you translating that, and for what purpose?

Thank you again Thomas. I see more progress but must go to long meeting. While I am away could you attempt to answer my questions #2 and #3 without questions. And maybe more answers?

Purpose = I am trying to arrive at the heart of Bell's mathematics.

Your final question is good and gets us close to that heart IMO. IMO real progress.

 

[Gordon Watson]

...

Bell's (2) is P(a,b) = int dH rho(H) A(a,H) B(b,H)

How are you translating that, and for what purpose?

In my translation H specifies an EPR-Bell experiment and --

Bell's (2) is P(a,b) = int dz rho(z) A(a,z) B(b,z).

Purpose = to expose IMHO limitation in validity of Bell's mathematics.

See next post.

 

[Gordon Watson]

Who cares if it is "universally valid"? The question is simply: is it a reasonable assumption for Bell's purposes? The answer is YES.

Now, suppose you don't agree. You'd be in the minority. But beyond that, it is already well known that this is not required anyway. There are plenty of other alternatives, see any of my Bell derivation pages for example. I don't bother with this because some people don't follow it for one reason or another. And it is unnecessarily complex.

I can agree with your YES. It is a reasonable assumption for Bell if he is studying EPR elements of reality.

But if Bell is restricting himself to them, his theory may refute just them.

billschneider

points to Bell writing the equivalent of --

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

For generality Bell could have written

(11a) P(GG'|Habz) = P(G|Haz).P(G'|HbzaG).

To get to Bell's (11) from (11a) he neglects the conditioning aG.

But what if z is a variable transformed during the measurement interaction? It is then the case that aG is a condition on the nature of that transformation.

Bell properly has a and b as ordinary vectors representing the detector settings. For generality, let z be a higher-order vector. Then aG conditions all the z-s in P(G'|HbzaG) so that (11a) is more highly correlated than Bell's (11); just as QM is more highly correlated than classical mechanics, QM using the collapse of the wave-function to carry out the aG conditioning remotely.

Does this leave Bell's mathematics neutral on nonlocality and negative on the z-s (lambdas) that he uses? Lambdas (z-s) that are independent of conditioning by aG?

 

[zonde]

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

But I don't know what to make about this H you introduce.
Maybe we can try it this way. I will give the values without H and you can show how H changes things.

P(G) = 0.5, P(G|H) = ?
P(G|a) = 0.5, P(G|Ha) = ?
P(G|az) = cos^2(a-z), P(G|Haz) = ?
P(G|azb) = cos^2(a-z), P(G|Hazb) = ?
P(G|azbG') = cos^2(a-z)cos^2(b-z), P(G|HazbG') = ?

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 would this P(GG'|abz)=P(G|az)P(G'|bz) be fine if we would be discussing non-entangled case?

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?

This is not universally valid (1/2)(1/2) = 1/4.
Take for example |a-z|=|b-z|=90deg.
In that case P(G|az)=0 and P(G'|bz)=0 so adding H condition can't possibly change it to P(G|Haz)=1/2 and P(G'|Hbz)=1/2.

 

[Eye_in_the_Sky]

Purpose = I am trying to arrive at the heart of Bell's mathematics.

__________________________

BELL LOCALITY

Section II ("Formulation") of Bell's original paper begins with a summary of the EPR argument in terms of the example of Bohm and Aharonov (spin-½ particle-pair in "singlet" state).

In that section, without providing any mathematical details, Bell promptly arrives at the following conclusion:

... it follows that the result of any such measurement must actually be predetermined.

It is in connection with this (EPR-type) argument that the "Bell Locality" condition becomes relevant. That is, the "Bell Locality" condition is intended to provide a mathematical basis by which one is able arrive at the above conclusion of "predetermined outcomes" – not just for QM ... but for any theory.

[In another thread, I have referred to this part of the Bell argument as "stage 1".]
__________________________

DEFINITION

"Bell Locality" (for the "Alice-and-Bob, spin-½" scenario) is defined by the following two symmetrical probability conditions:

(a) P(A|a,b,B,λ) = P(A|a,λ)

(b) P(B|a,b,A,λ) = P(B|b,λ) .

In the above:

A ≡ Alice's outcome (±1) ,
B ≡ Bob's outcome (±1) ,
a ≡ Alice's setting (some unit vector) ,
b ≡ Bob's setting (some unit vector) ,

and λ denotes a complete specification of the "state" of the particle pair with respect to a spacelike hypersurface S having the following characteristic (see diagram):

S intersects the (two) backward light-cones (associated with the measurements of Alice and Bob) in their regions of non-overlap.
__________________________

APPLICATION to "stage 1"

From conditions (a) and (b) in conjunction with the usual rule for conditional probabilities, it follows that

P(A,B|a,b,λ) = P(A|a,λ) P(B|b,λ) .

To this relation, we then apply the condition of "perfect anti-correlation" for equal settings, namely,

[1] P(A=s,B=s|a=n,b=n,λ) = 0 (arbitrary s and n) ,

and arrive at

[2a] P(A|a,λ) = 0 or 1 (never in-between)

and

[2b] P(B|b,λ) = 0 or 1 (never in-between)

[for details see section III, page 6, of the following paper:

http://arxiv.org/PS_cache/quant-ph/pdf/0601/0601205v2.pdf ] .

From [2a] and [2b], it is seen quite clearly that we are dealing with "predetermined outcomes".
________

CONCLUSION of "stage 1"

Any theory which satisfies both

(i) "Bell Locality"

and

(ii) "perfect anti-correlation" for equal settings

will necessarily have "predetermined outcomes" with respect to each (and every) complete specification λ.
____

In that case, we may as well just write (in place of [2a] and [2b]):

A(a,λ) = ±1 , B(b,λ) = ±1 .

This is where the mathematics begins in Bell's original paper (equation (1) therein).

[In another thread, I have referred to this part of the Bell argument (i.e. from equation (1) and onward) as "stage 2".]
__________________________

JOINING "stage 1" to "stage 2"

Upon joining together the arguments of these two "stages", we arrive at the following:

Any theory which satisfies both

(i) "Bell Locality"

and

(ii) "perfect anti-correlation" for equal settings

will necessarily satisfy "Bell's inequality".
__________________________

EVALUATING a specific CANDIDATE theory

Suppose we are given a specific candidate theory for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob, and that (upon calculating) we find for some λ, A, B, a, and b that

P(A|a,λ) P(B|b,λ) ≠ P(A,B|a,b,λ) .

Then, the candidate theory in question does NOT satisfy "Bell Locality".

For example, this is the case for QM:

P(A|a,λ) P(B|b,λ) = ½ ∙ ½ = ¼ ≠ P(A,B|a,b,λ) .
__________________________

MEANING of "Bell Locality" VIOLATION

... I am sorry, I have not yet fully sorted this matter out! ...

____

 

[DrChinese]

I can agree with your YES. It is a reasonable assumption for Bell if he is studying EPR elements of reality.

But if Bell is restricting himself to them, his theory may refute just them.

As Eye_in_the_Sky points out, this addresses the case where you accept that the results of observations must be predetermined. That is a fairly wide case, certainly nothing to sneeze at.

 

[billschnieder]

EVALUATING a specific CANDIDATE theory

Suppose we are given a specific candidate theory for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob, and that (upon calculating) we find for some λ, A, B, a, and b that

P(A|a,λ) P(B|b,λ) ≠ P(A,B|a,b,λ) .

Then, the candidate theory in question does NOT satisfy "Bell Locality".

A machine produces pairs of balls at a time. Each ball contains two switches, the red switch and the blue switch . Each pair is created such that on one ball, pressing the switch causes the ball to light-up the color of the switch but on the other ball pressing the switch causes the ball to light up the opposite color (blue or red). The latter statement is the hidden variable which governs all the outcomes (z) and is determined when the balls are created at the machine. One of pair is randomly sent to Alice and the other to Bob.

P(A|az) = Probability that Alice will see a Red color after pressing a switch (a)
P(B|bz) = probability that Bob will see a Blue color after pressing a switch (b)

P(AB|abz) = Probability that both Alice and Bob will see opposite colors after Alice presses switch (a) and Bob presses switch (b).

Here is the so-called 'full universe' of possibilities defined by z: where abs means Alice presses blue switch and brs means Bob presses red switch. The color after the hyphen, is what Alice and Bob see as a result of their pressing their switches.
Alice, Bob
1: abs-blue, brs-blue
2: abs-blue, bbs-red
3: abs-red, bbs-blue
4: abs-red, brs-red
5: ars-blue, bbs-blue
6: ars-blue, brs-red
7: ars-red, brs-blue
8: ars-red, bbs-red

Let us calculate the probability P(AB|abz) for the situation for which:
a = abs (ie, Alice presses blue switch),
b = bbs (ie, Bob presses blue switch)

first using the generic chain rule. Note that the chain rule is universally valid.
(x) P(AB|abs, bbs, z) = P(A|abs, bbs, z) P(B|abs, bbs, z, A)

and then compare with Bell's specific choice
(y) P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z)

Presumably, as Eye_in_the_Sky pointed out, if these values are different, then the situation violates Bell's locality.

The procedure for calculating the following probabilities is according to standard practice. Count the case instances in the full universe in which everything to the right of '|' in the expression occurs, this is the denominator. Within that subset, count the number of case instances in which everything to the left of '|' in the expression occurs, this is the numerator.

It follows therefore that:

P(A|abs, z) = 2/4 = 1/2 (based on cases 1,2,3,4 where abs occurs)
P(B|bbs, z) = 2/4 = 1/2 (based on cases 2,3,5,8 were bbs occurs)
P(A|abs, bbs, z) = 1/2 (based on cases 2,3 where both abs and bbs occur)
P(B|abs, bbs, z, A) = 1 (based on case 3 where both abs and bbs occur and Alice got a red light)

Putting everything together,

P(AB|abz) from (x) = 1/2 * 1 = 1/2

P(AB|abz) from (y) = 1/2 * 1/2 = 1/4

since P(A|abz) P(B|Aabz) /= P(A|az) P(B|bz) in this case, does that mean this case is non-local? What does this say about Bell's locality condition?
Note that in this example, looking only at the colors at Alice, she appears to get random results, and similar for Bob. But when looking at coincidences, they show perfect anti-correlation when they press the same colored switch -- a Microcosm of the EPR experiment.

 

[JesseM]

A machine produces pairs of balls at a time. Each ball contains two switches, the red switch and the blue switch . Each pair is created such that on one ball, pressing the switch causes the ball to light-up the color of the switch but on the other ball pressing the switch causes the ball to light up the opposite color (blue or red). The latter statement is the hidden variable which governs all the outcomes (z) and is determined when the balls are created at the machine. One of pair is randomly sent to Alice and the other to Bob.

P(A|az) = Probability that Alice will see a Red color after pressing a switch (a)
P(B|bz) = probability that Bob will see a Blue color after pressing a switch (b)

So A represents the event of Alice seeing a Red color, and B represents the event of Bob seeing a blue color? If so:

P(AB|abz) = Probability that both Alice and Bob will see opposite colors after Alice presses switch (a) and Bob presses switch (b).

This should actually be the probability Alice sees Red and Bob sees blue given the switches they pressed, not general probability that they "see opposite colors". The total probability they see opposite colors would be P(AB|abz) + P(A'B'|abz), where A' represents Alice seeing Blue and B' represents Bob seeing Red.

Here is the so-called 'full universe' of possibilities defined by z: where abs means Alice presses blue switch and brs means Bob presses red switch. The color after the hyphen, is what Alice and Bob see as a result of their pressing their switches.
Alice, Bob
1: abs-blue, brs-blue
2: abs-blue, bbs-red
3: abs-red, bbs-blue
4: abs-red, brs-red
5: ars-blue, bbs-blue
6: ars-blue, brs-red
7: ars-red, brs-blue
8: ars-red, bbs-red

But your "full universe" does not actually include the hidden variables information about whether Alice has the "normal" ball where pressing a given color switch causes the ball to light up that color, or the "opposite" ball where pressing a given color switch causes the ball to light up the opposite color. We could include that information as two possible hidden variable-states aNbO (meaning Alice got the Normal ball and Bob got the Opposite ball) or aObN (meaning Alice got the Opposite ball and Bob got the Normal one).

Let us calculate the probability P(AB|abz) for the situation for which:
a = abs (ie, Alice presses blue switch),
b = bbs (ie, Bob presses blue switch)

first using the generic chain rule. Note that the chain rule is universally valid.
(x) P(AB|abs, bbs, z) = P(A|abs, bbs, z) P(B|abs, bbs, z, A)

and then compare with Bell's specific choice
(y) P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z)

Bell's specific choice is only meant to apply in cases where you include the hidden-variable information. So, your "z" must include the information about whether the hidden state was aNbO or aObN. If it does, equation (y) will indeed apply; for example, P(AB|abs, bbs, aObN) = 1 (because if Alice presses the blue switch and she had the Opposite ball, it's guaranteed with probability 1 that Alice will see Red, which is what event A stood for, and similarly if Bob presses the blue switch and he had the Normal ball, it's guaranteed with probability 1 that he'll see Blue which is what B stood for), and it's also true that P(A|abs, aObN) = 1 and P(B|bbs, aObN) = 1.

The procedure for calculating the following probabilities is according to standard practice. Count the case instances in the full universe in which everything to the right of '|' in the expression occurs, this is the denominator. Within that subset, count the number of case instances in which everything to the left of '|' in the expression occurs, this is the numerator.

It follows therefore that:

P(A|abs, z) = 2/4 = 1/2 (based on cases 1,2,3,4 where abs occurs)

If your "z" does not include the relevant hidden-variable information, but only states that this must be one of four possible cases where Alice pushes the blue switch, then there's no reason to expect the equation P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z) will be valid, you are badly misunderstanding Bell's reasoning if you think he'd expect the equation to apply under such assumptions.

Now, if you want to consider a sum over different possible hidden variables states, it would be true that P(AB|abs, bbs) would be a sum over all possible values of z of P(AB|abs, bbs, z)*P(z), and by the argument I mentioned above, a sum over all values of z of P(AB|abs, bbs, z)*P(z) is equal to a sum over all values of z of P(A|abs, z)*P(B|bbs, z)*P(z).

Note that this equation:

P(AB|abs, bbs) = (sum over all values of z) P(A|abs, z)*P(B|bbs, z)*P(z)

...is directly analogous to equation (2) in Bell's original paper at http://www.drchinese.com/David/Bell_Compact.pdf

In this case we have a particularly simple version of z that can only take two values, aObN or aNbO. So the above would become:

P(AB|abs, bbs) = P(A|abs, aObN)*P(B|bbs, aObN)*P(aObN) + (A|abs, aNbO)*P(B|bbs, aNbO)*P(aNbO)

Since A means Alice got Red, A is guaranteed to occur if she pressed the blue switch and had the Opposite ball (abs, aObN) and guaranteed not to occur if she pressed the blue switch and had the Normal ball (abs, aNbO). So, P(A|abs, aObN)=1 and P(A|abs, aNbO)=0.
Likewise P(B|bbs, aObN)=1 and P(B|bbs, aNbO)=0. So, the above reduces to:

P(AB|abs, bbs) = 1*1*P(aObN) + 0*0*P(aNbO) = P(aObN)

Which is exactly what you'd expect, since given that they both pressed the blue switch, A (Alice getting Red) and B (Bob getting Blue) is guaranteed to happen of Alice got the Opposite ball and Bob got the Normal ball, and guaranteed not to happen if the balls were reversed. Whatever the probability that Alice got the Opposite ball and Bob got the Normal ball, that should be the same as the probability of P(AB|abs, bbs).

 

[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).

It isn't known by the experimenters themselves, but for an imaginary being who knows H, the full set of local variables at every point in the past light cone of the measurements at some time t, the results might be predictable (in a local realist universe with perfectly deterministic rules, it would be predictable with probability 1). And any equation featuring H, like F(AB|abH), can be viewed as the frequency or probability as seen by that imaginary observer who knows H, not the frequency/probability as seen by the experimenters.

So again, do you disagree that for such an imaginary observer with that extra knowledge, living in a universe obeying local realist laws, it should be true that F(AB|abH) = F(A|aH) F(B|AbH) under the definition of H I gave in post #29? If you do disagree, can you address the argument about how assuming otherwise would imply FTL information transmission?

 

[Gordon Watson]

But I don't know what to make about this H you introduce.
Maybe we can try it this way. I will give the values without H and you can show how H changes things.

P(G) = 0.5, P(G|H) = ?
P(G|a) = 0.5, P(G|Ha) = ?
P(G|az) = cos^2(a-z), P(G|Haz) = ?
P(G|azb) = cos^2(a-z), P(G|Hazb) = ?
P(G|azbG') = cos^2(a-z)cos^2(b-z), P(G|HazbG') = ?
...

zonde, I am sorry and apologize if my preoccupation-with-precision or my mistakes have led me to give you grief.

IMO probability concerns the study of a function P(X|Y), read as "the probability of X conditional on Y". So a function P(X) is not part of probability theory IMO.

IMO, in our discussion H is the condition that defines your EPR-Bell experiment with photons in identical states. H = the implied condition in your notation.

Your expressions above have this H implied, otherwise you would not know how to give values for each expression. In mathematics, implied conditions can lead to trouble ...

Above you have written --

<< P(G|azbG') = cos^2(a-z)cos^2(b-z), P(G|HazbG') = ? >>

IMO P(G|HazbG') = cos^2(a-b).

Is this a mistake?

Cheers, JenniT

 

[Gordon Watson]

__________________________

...

EVALUATING a specific CANDIDATE theory

Suppose we are given a specific candidate theory for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob, and that (upon calculating) we find for some λ, A, B, a, and b that

P(A|a,λ) P(B|b,λ) ≠ P(A,B|a,b,λ) .

Then, the candidate theory in question does NOT satisfy "Bell Locality".

For example, this is the case for QM:

P(A|a,λ) P(B|b,λ) = ½ ∙ ½ = ¼ ≠ P(A,B|a,b,λ) .
__________________________

MEANING of "Bell Locality" VIOLATION

... I am sorry, I have not yet fully sorted this matter out! ...
____

Dear Eye_in_the_Sky, thank you. I have not studied your total response in depth but I love your candidate theory to bits.

Suppose we are given candidate theory X for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob. Let the possible outcomes for Alice be A = {G, R}, for Bob be B = {G', R'}. Upon calculating with X we find for some λ, A, B, a, and b that

P(G|X,a,λ) P(G'|X,b,λ) ≠ P(G,G'|X,a,b,λ) .

Then X [in your view] does NOT satisfy "Bell Locality".

For example [in your view], this is the case for QM:

P(G|X,a,λ) P(G'|X,b,λ) = ½ ∙ ½ = ¼ ≠ P(G,G'|X,a,b,λ) .
__________________________

MEANING of "Bell Locality" VIOLATION and the error in X.

If G and G' are correlated, probability theory teaches [equation (1)] that

(1) P(G,G'|X,a,b,λ) = P(G|X,a,b,λ).P(G'|X,a,b,λ,G) =

(2) P(G|X,a,λ).P(G'|X,a,b,λ,G);

(2) following from (1) because (with Einstein, Bell, and many others) we agree that setting b can have no relevance for outcome G. A realistic locality condition.

But Bell goes further. Bell supposes [Bertlmann's Socks, page 13] that

(3) P(G'|X,a,b,λ,G) = P(G'|X,b,λ),

a result [you say] known as Bell locality.

Bell supposes that the condition aG has no relevance for λ. So Bell locality is a restraint on the λ-s under consideration in candidate theory X. This explains why theory X fails to be realistic.

"Bell locality" might be less confusing if known as "Bell's supposition"?

The condition aG has relevance for λ because it indicates how the λ-s in P(G'|X,a,b,λ,G) respond when subject to a measurement interaction with a measuring device oriented a -- they yield outcome G -- that is the relevance and physical significance of condition aG = |a,G.

[See the one emphasized phrase in Bohr's response to EPR; a response which Bell did not understand.]

Thus condition aG eliminates, from consideration in P(G'|X,a,b,λ,G), just those λ-s that are irrelevant.

Or, better, clearer:

Condition aG identifies, for consideration in P(G'|X,a,b,λ,G), just those λ-s that are relevant. That is, the λ-s that would respond, if subject to a measurement interaction with a measurement device oriented a, to yield outcome G -- that being the relevance and physical significance of aG.

Bell's supposition is a restriction on candidate λ-s.

Bell's supposition should not be associated with locality, nor with realistic constraints on locality.

Note 1:
A realistic constraint on locality was exercised in reducing (1) to (2).

Note 2:
Parameter independence is allowed --
(4) P(G'|X,a,b,λ,G) = P(G'|X,b,λ,G).

Note 3:
Outcome independence is allowed --
(5) P(G'|X,a,b,λ,G) = P(G'|X,a,b,λ).

Note 4:
The one thing not allowed, when X relates to EPR-Bell settings, is Bell's supposition
(3) P(G'|X,a,b,λ,G) = P(G'|X,b,λ).

IMHO.

 

[ThomasT]

At first glance, this looks ok. By that I mean it makes sense to me. Hopefully some more sophisticated observers than I will comment.

 

[ThomasT]

It isn't known by the experimenters themselves, but for an imaginary being who knows H, the full set of local variables at every point in the past light cone of the measurements at some time t, the results might be predictable (in a local realist universe with perfectly deterministic rules, it would be predictable with probability 1). And any equation featuring H, like F(AB|abH), can be viewed as the frequency or probability as seen by that imaginary observer who knows H, not the frequency/probability as seen by the experimenters.

So again, do you disagree that for such an imaginary observer with that extra knowledge, living in a universe obeying local realist laws, it should be true that F(AB|abH) = F(A|aH) F(B|AbH) under the definition of H I gave in post #29? If you do disagree, can you address the argument about how assuming otherwise would imply FTL information transmission?

More importantly for this thread, JesseM, it would be helpful if you would give your assessment of JenniT's treatment in her post #59.

 

[zonde]

zonde, I am sorry and apologize if my preoccupation-with-precision or my mistakes have led me to give you grief.

Apologies accepted.

IMO probability concerns the study of a function P(X|Y), read as "the probability of X conditional on Y". So a function P(X) is not part of probability theory IMO.

From wikipedia http://en.wikipedia.org/wiki/Probability" :
"In mathematics, a probability of an event A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A).
...
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B"."

IMO, in our discussion H is the condition that defines your EPR-Bell experiment with photons in identical states. H = the implied condition in your notation.

Your expressions above have this H implied, otherwise you would not know how to give values for each expression. In mathematics, implied conditions can lead to trouble ...

These following expressions do not describe EPR-Bell experiments but simple experiments with single photon beam and single analyzer and are covered by Malus law:
P(G) = 0.5
P(G|a) = 0.5
P(G|az) = cos^2(a-z)
You can read about it here http://en.wikipedia.org/wiki/Malus%27_law"
There are expressions I=I0cos^2(theta) and I/I0=1/2.

So I do not see any reason to introduce any additional conditions for those particular expressions.

Above you have written --

<< P(G|azbG') = cos^2(a-z)cos^2(b-z), P(G|HazbG') = ? >>

IMO P(G|HazbG') = cos^2(a-b).

Is this a mistake?

Cheers, JenniT

Yes, this is a mistake. You completely ignore z in your expression yet you include it in conditions that way saying that expression holds for any value of z.
You can test this easily if you specify such values z=90°, a=0°, b=45°.
In that case cos^2(a-z)=0 (given horizontal polarization for photons in question and vertical orientation of analyzer at Alice there are no G clicks for Alice) so independently from any other conditions probability is 0, but
cos^(a-b)=0.5 i.e. it is not zero as should be.

 

[JesseM]

Dear Eye_in_the_Sky, thank you. I have not studied your total response in depth but I love your candidate theory to bits.

Suppose we are given candidate theory X for which we can calculate the various probabilities associated with the possible outcomes of Alice and Bob. Let the possible outcomes for Alice be A = {G, R}, for Bob be B = {G', R'}. Upon calculating with X we find for some λ, A, B, a, and b that

P(G|X,a,λ) P(G'|X,b,λ) ≠ P(G,G'|X,a,b,λ) .

Then X [in your view] does NOT satisfy "Bell Locality".

Bell locality just says it should be possible to find a set of information about hidden variables λ that's sufficiently complete such that the two sides of the above equation will be equal. It doesn't say the opposite, that you can't find a more restricted λ such that the two sides are unequal--of course you can! For example, λ could be defined in such a way that it contains no information about anything in the past light cone of either measurement, in which case it should be completely irrelevant and P(G|X,a,λ) = P(G|X,a) and so forth. In that case the above inequality may be correct, despite the fact that X is a local realistic theory. Again, Bell is just saying that under local realism it should be possible to define some λ so the two sides of the equation become equal.

But Bell goes further. Bell supposes [Bertlmann's Socks, page 13] that

(3) P(G'|X,a,b,λ,G) = P(G'|X,b,λ),

a result [you say] known as Bell locality.

Bell supposes that the condition aG has no relevance for λ. So Bell locality is a restraint on the λ-s under consideration in candidate theory X. This explains why theory X fails to be realistic.

"Bell locality" might be less confusing if known as "Bell's supposition"?

It's not a supposition, any physicist will agree that under a local realist theory, as long as enough information about local variables (hidden or otherwise) in the past light cones of G and G' is contained in λ, your equation (3) should be satisfied. If this doesn't make sense to you, perhaps you could address my post #29?

Also, consider the following:

--In a local realist theory, all physical facts--including macro-facts about "events" spread out over a finite swatch of space-time--ultimately reduce to some collection of local physical facts defined at individual points in spacetime (or individual 'bits' if spacetime is not infinitely divisible). See my first few comments in this post from another thread. So, any fact of the matter about the result of a measurement can be reduced to a set of local facts about events associated with the smallest possible units of spacetime. Without loss of generality, then, let G and G' be two possibilities for what happens at some single point in spacetime P.

--In a deterministic local realist theory, if λ represents the complete set of local physical facts that lie in the past light cone of P at some time t prior to P, then this allows us to determine whether G or G' occurs with probability one, so any additional information would not cause us to alter our probability estimate. Thus in this case it should be clear that your equation (3) above is satisfied.

--An intrinsically probabilistic local realist theory is a somewhat more subtle case, but for any probabilistic local realist theory it should be possible to break it up into two parts: a deterministic mathematical rule that gives the most precise possible probability of a given event happening at point P based on information in the past light cone of P (if information outside that past light cone of P was required to get the most precise possible probability estimate, the theory would not be a local one), and a random "seed" number whose value is combined with the probability to determine what event actually happened. This "most precise possible probability" does not represent a subjective probability estimate made by any observer, but is the probability function that nature itself is using, the most accurate possible formulation of the "laws of physics" in a universe with intrinsically probabilistic laws.

For example, if the mathematical rule determines the probability of G is 70% and the probability of G' is 30%, then the random seed number could be a randomly-selected real number on the interval from 0 to 1, with a uniform probability distribution on that interval, so that if the number picked was somewhere between 0 and 0.7 that would mean G occurred, and if it was 0.7 or greater than G' occurred. The value of the random seed number associated with each probabilistic choice (like the choice between G and G') can be taken as truly random, uncorrelated with any other event in the universe, while the precise probability of different events could be generated deterministically from a λ which contained information about all local physical facts at all points in spacetime in the past light cone of P. In this case, it would again be true that your equation (3) above would be satisfied.

Do you think it is possible to imagine a "local realist" theory where one of the above would not be true?

 

[JesseM]

More importantly for this thread, JesseM, it would be helpful if you would give your assessment of JenniT's treatment in her post #59.

I've done so now, but I would still like to see your response to my comments in post #57.

 

[ThomasT]

I've done so now, but I would still like to see your response to my comments in post #57.

Ok, thanks, the way I currently think about this is below.

It isn't known by the experimenters themselves, but for an imaginary being who knows H, the full set of local variables at every point in the past light cone of the measurements at some time t, the results might be predictable (in a local realist universe with perfectly deterministic rules, it would be predictable with probability 1). And any equation featuring H, like F(AB|abH), can be viewed as the frequency or probability as seen by that imaginary observer who knows H, not the frequency/probability as seen by the experimenters.

So again, do you disagree that for such an imaginary observer with that extra knowledge, living in a universe obeying local realist laws, it should be true that F(AB|abH) = F(A|aH) F(B|AbH) under the definition of H I gave in post #29? If you do disagree, can you address the argument about how assuming otherwise would imply FTL information transmission?

I agree that F(B|AbH) reduces to F(B|bH) for all settings. I see now that the only additional info we're conditionalizing on with F(B|AbH) is the detection attribute at A. We haven't included the setting, a, associated with A.

So, even without an omniscient imaginary observer, F(AB|abH) = F(A|aH) F(B|AbH) = F(A|aH) F(B|bH).

Is this correct?

 

[JesseM]

I agree that F(B|AbH) reduces to F(B|bH) for all settings. I see now that the only additional info we're conditionalizing on with F(B|AbH) is the detection attribute at A. We haven't included the setting, a, associated with A.

So, even without an omniscient imaginary observer, F(AB|abH) = F(A|aH) F(B|AbH) = F(A|aH) F(B|bH).

Is this correct?

Well, "omniscient imaginary observer" shouldn't be taken too literally, it's just a kind of shorthand for the fact that H can include information not available to any real experimenter, like information about local hidden variables. Does your H still include that information? Are you defining it as I did, so that it includes information about all local physical variables in the past light cones of the measurements (or in cross-sections of the past light cones taken at some time t after the last moment the two light cones overlapped)?

 

[ThomasT]

Well, "omniscient imaginary observer" shouldn't be taken too literally, it's just a kind of shorthand for the fact that H can include information not available to any real experimenter, like information about local hidden variables. Does your H still include that information? Are you defining it as I did, so that it includes information about all local physical variables in the past light cones of the measurements (or in cross-sections of the past light cones taken at some time t after the last moment the two light cones overlapped)?

One problem I have is that I see the H in F(AB|abH) as referring to something different than the H's in F(A|aH) and F(B|bH). The H in F(AB|abH) is irrelevant wrt individual results and the H's in F(A|aH) and F(B|bH) are irrelevant wrt joint detection.

So how would you formulate the joint probability statement if this is the case?

Let's say the H in F(AB|abH) denotes some relationship between H_a and H_b for starters.

 

[Gordon Watson]

Apologies accepted.


From wikipedia http://en.wikipedia.org/wiki/Probability" :
"In mathematics, a probability of an event A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A).
...
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B"."


These following expressions do not describe EPR-Bell experiments but simple experiments with single photon beam and single analyzer and are covered by Malus law:
P(G) = 0.5
P(G|a) = 0.5
P(G|az) = cos^2(a-z)
You can read about it here http://en.wikipedia.org/wiki/Malus%27_law"
There are expressions I=I0cos^2(theta) and I/I0=1/2.

So I do not see any reason to introduce any additional conditions for those particular expressions.


Yes, this is a mistake. You completely ignore z in your expression yet you include it in conditions that way saying that expression holds for any value of z.
You can test this easily if you specify such values z=90°, a=0°, b=45°.
In that case cos^2(a-z)=0 (given horizontal polarization for photons in question and vertical orientation of analyzer at Alice there are no G clicks for Alice) so independently from any other conditions probability is 0, but
cos^(a-b)=0.5 i.e. it is not zero as should be.

Dear zonde, thank you for these additional details.

I think it best to leave my apology stand.

But to withdraw any implication of a mistake on my part.

Instead I confess to gross confusion.

I thought we were discussing Bell's mathematics in the context of entangled photons.

To use P(A) [as you do] and to then specify some conditions B [which you do] is to invoke (P(A|B).

Your B = These following expressions do not describe EPR-Bell experiments but simple experiments with single photon beam and single analyzer and are covered by Malus law. wow.

Does your wiki reference give anywhere a value for P(A) without a B?

I would like to see that.

Thank you again,

Jenni

 

[JesseM]

One problem I have is that I see the H in F(AB|abH) as referring to something different than the H's in F(A|aH) and F(B|bH). The H in F(AB|abH) is irrelevant wrt individual results and the H's in F(A|aH) and F(B|bH) are irrelevant wrt joint detection.

So how would you formulate the joint probability statement if this is the case?

Let's say the H in F(AB|abH) denotes some relationship between H_a and H_b for starters.

Say H_a represents the full set of information about all local variables in the past light cone of measurement A at some time t prior to A, and H_b represents the full set of information about all local variables in the past light cone of B at the same time t which is also prior to B, with t chosen so that it happens after the last moment the two light cones overlap.

In this case, if we simply define H as the sum of all the information contained in both H_a and H_b, then the equation is equivalent to this:

F(AB|abH_aH_b) = F(A|aH_aH_b)*F(B|bH_aH_b)

Do you still have a problem with this equation? Some of those terms may represent unecessary information--for example, I could give you an argument that F(A|aH_aH_b) = F(A|aH_a), i.e. H_b would give no additional information about the probability/frequency of A given knowledge of a and H_a--but that shouldn't affect their validity.

 

[billschnieder]

This should actually be the probability Alice sees Red and Bob sees blue given the switches they pressed, not general probability that they "see opposite colors". The total probability they see opposite colors would be P(AB|abz) + P(A'B'|abz), where A' represents Alice seeing Blue and B' represents Bob seeing Red.

That is what I meant. Thanks.

But your "full universe" does not actually include the hidden variables information about whether Alice has the "normal" ball where pressing a given color switch causes the ball to light up that color, or the "opposite" ball where pressing a given color switch causes the ball to light up the opposite color. We could include that information as two possible hidden variable-states aNbO (meaning Alice got the Normal ball and Bob got the Opposite ball) or aObN (meaning Alice got the Opposite ball and Bob got the Normal one).

No, I provided the full universe for my hidden "variable" z, which were clearly defined. The term "variable" is actually not appropriate since it carries a connotation of something with multiple values. Maybe that is what is confusing you. That is why t'Hooft prefers to use the terms "beables" as different from "changeables". What you are describing is not relevant for calculating probabilities from the full universe defined by my z.

Bell's specific choice is only meant to apply in cases where you include the hidden-variable information. So, your "z" must include the information about whether the hidden state was aNbO or aObN. If it does, equation (y) will indeed apply; for example, P(AB|abs, bbs, aObN) = 1

No it must not! "z" was clearly defined and my full universe includes all possibilities given that "z" is true. Therefore in my full universe, P(z) = 1. P(B|bbs, aObN) is a nonsensical expression with no meaning. It is similar to saying. What is the probability of Bob getting a blue light if Bob gets a blue light. No surprise that you will never get any result other than 1 or 0 (certainties) because your full universe will consist of exactly one case and no more, in which case it makes no sense to compare such a result with QM where values other than 1 and 0 are obtained.

If your "z" does not include the relevant hidden-variable information, but only states that this must be one of four possible cases where Alice pushes the blue switch, then there's no reason to expect the equation P(AB|abs, bbs, z) = P(A|abs, z) P(B|bbs, z) will be valid, you are badly misunderstanding Bell's reasoning if you think he'd expect the equation to apply under such assumptions.

I'm afraid it is you who is badly misunderstanding probability theory. The conditioning variable must be specific, not just a vague concept of "everything in past light cone". In probability theory, everything after the "|" must be specific enough to enable you to define a hypothesis space. Besides Bell was not dealing with certainties but with probabilities. Your approach MUST NECESSARILY result in a certainties (0, or 1) rather than probabilities. If you disagree, give me an example for which you know and consider everything in the past light cone of an event and yet still resulted in a probability other than 0 or 1.

Now, if you want to consider a sum over different possible hidden variables states, it would be true that P(AB|abs, bbs) would be a sum over all possible values of z of P(AB|abs, bbs, z)*P(z), and by the argument I mentioned above, a sum over all values of z of P(AB|abs, bbs, z)*P(z) is equal to a sum over all values of z of P(A|abs, z)*P(B|bbs, z)*P(z).

Again, "z" does not represent all different possible hidden variable states. You are confusing functional notation with Probability notation. When I write P(A|ab) in a generic equation, a and b are not variables but place-holders for "beables" not variables. Therefore when carrying out a specific calculation such as P(A|bbs, abs), bbs and abs are not variables but specific cases instances ("beables"). Once you change z, you have completely changed the probability space over which a and b are defined and as such you MUST NOT integrate or add up the probabilities anymore.

I don't need to multiply by P(z) because my full universe is already defined by z, ie within my full universe, P(z) is already 1. I don't need to add anything from different hidden variables because I am dealing with a specific hidden variable. I don't need to consider all possible hidden variables because being omniscient about the workings of my machine, I know for sure that it only operates as I described. Yet, according to Bell's choice of equations, my machine is non-local.

Note that this equation:

P(AB|abs, bbs) = (sum over all values of z) P(A|abs, z)*P(B|bbs, z)*P(z)

...is directly analogous to equation (2) in Bell's original paper

I suspect those terms mean something different to you than to me.

P(A|abs, z) means the Probability that A is observed, given that abs is True and z is true. So when you talk about summing over all values of z,what does the term P(A|abs, z) mean to you. If you are thinking of adding up the results from say z1 with those of z2, ..., zn then clearly the results from z1 are due to a completely different context from z2 etc Therefore you can not add them up legitimately.What is the rule of probability theory that permits you to do that addition?

In this case we have a particularly simple version of z that can only take two values, aObN or aNbO.

No! The value of the hidden variable z in my example is the description of the mechanism of the machine I provided. It has no other values other than the description I gave. If you want to provide a different hidden variable that can result in the same probabilities I calculated, go ahead and give your own description of the functioning of the machine. The probabilities you get will be restricted to the space defined by your description. The concept of multiple values for the hidden variable is a misunderstanding carried over from functional notation and probably over-reading into the term "variable".