Understanding bell's theorem: why hidden variables imply a linear relationship?

[lugita15]

billschnieder and I continued this discussion in another thread, but on his suggestion I'm bringing it back here.

Your no-conspiracy condition is essentially that Scenario X and Scenario Y (from above) are exactly the same,

Yes, the no-conspiracy condition says that if a statement is meaningful in both scenarios, then the probability is equal for both scenarios.

in other words, your no-conspiracy condition is equivalent to saying, the QM result from a single wavefunction must be the same as the QM result from three different wavefunctions.

Where in the world did you get that from? We're talking about different possible measurements we could perform on a system with the same wavefunction. We're not talking about different wavefunctions.

And I showed you that step (3) was incomplete, Step (3) What does no-conspiracy say about P(AB|w). According to your logic, no-conspiracy also implies that P(AB|w)=P(AB|x,y).

No-conspiracy states that if a statement S is meaningful in both x and w, then P(S|w)=P(S|x) (and similarly for y and z). But A & B is not meaningful in x, so no-conspiracy doesn't tell you anything in this case.

Also, what do you mean by P(A & B|x,y)? Do you mean a combined space which is the union of x and y? Well, my reasoning doesn't talk about combined spaces like that. It only discusses x, y, z, and w.

But x and y are two different sets of photons, which means P(AB|x,y) is undefined/meaningless.

Again, I didn't say anything about P(A & B|x,y).

All you have proven is the triviality that the joint probablity distribution P(ABC|x,y,z) for outcomes from three different sets of photons (x,y,z) is undefined, although the joint probability distribution P(ABC|w) from the single set of photons (w) is well defined.

I didn't say anything about P(A & B & C|x,y,z). And again, since A & B & C is meaningless in x, y, or z, the no-conspiracy condition says nothing in this case either.

 

[billschnieder]

Yes, the no-conspiracy condition says that if a statement is meaningful in both scenarios, then the probability is equal for both scenarios.

But then, such a condition is seriously flawed.

Where in the world did you get that from? We're talking about different possible measurements we could perform on a system with the same wavefunction. We're not talking about different wavefunctions.

Please review post #240 above:

Le us represent the sets of photons as one set "w" for scenario (c), and three different sets ("x", "y","z") for scenario (b). Then the probabilities for (c) are more accurately represented as P(A|w), P(B|w), P(C|w) ... the probabilities for scenario (b) are more accurately represented as P(A|x), P(B|y), P(C|z).

So clearly in the first scenario, we are talking about different measurements that could be done on the same real system, but since we can only measure the system once, the other two terms are counter-factual AND clearly, in the second scenario we are talking about three different measurements on three different systems "x", "y", "z" (hence 3 wavefunctions) and there is nothing counterfactual in it.

Your argument, at least as I understand it, is that:
- the probabilities from the first scenario should be the same as the probabilities from the second scenario, because according to what you call the "no-conspiracy" condition, if we can talk meaningfully of P(A|w), P(B|w), P(C|w) in the first scenario, and we can talk meaningfully of P(A|x), P(B|y), P(C|z) in the second scenario, then those probabilities must be equal, ie P(A|x) = P(A|w) and P(B|y) = P(B|w) and P(C|z) = P(C|w), and if P(C|w) <= P(A|w) + P(B|w), then it must also be true that under "no-conspiracy", P(C|z) <= P(A|x) + P(B|y). From which you argue that since P(A|x) = P(B|y) = 0.25 and P(C|z) = 0.75, the inequalty P(C|z) <= P(A|x) + P(B|y) will be violated. .75≤.25+.25.

Now, is the above not exactly your argument. Please, let me know if I've misunderstood your argument in any way. Once we are clear on what your argument is, I will proceed to show why such a "no-conspiracy" condition is seriously flawed.

 

[lugita15]

So clearly in the first scenario, we are talking about different measurements that could be done on the same real system, but since we can only measure the system once, the other two terms are counter-factual AND clearly, in the second scenario we are talking about three different measurements on three different systems "x", "y", "z" (hence 3 wavefunctions) and there is nothing counterfactual in it.

Well, before measurement the system is described by the same wavefunction in each of the three cases x, y, and z.

Your argument, at least as I understand it, is that:
- the probabilities from the first scenario should be the same as the probabilities from the second scenario, because according to what you call the "no-conspiracy" condition, if we can talk meaningfully of P(A|w), P(B|w), P(C|w) in the first scenario, and we can talk meaningfully of P(A|x), P(B|y), P(C|z) in the second scenario, then those probabilities must be equal, ie P(A|x) = P(A|w) and P(B|y) = P(B|w) and P(C|z) = P(C|w), and if P(C|w) <= P(A|w) + P(B|w), then it must also be true that under "no-conspiracy", P(C|z) <= P(A|x) + P(B|y). From which you argue that since P(A|x) = P(B|y) = 0.25 and P(C|z) = 0.75, the inequalty P(C|z) <= P(A|x) + P(B|y) will be violated. .75≤.25+.25.

Now, is the above not exactly your argument. Please, let me know if I've misunderstood your argument in any way. Once we are clear on what your argument is, I will proceed to show why such a "no-conspiracy" condition is seriously flawed.

Yes, that is exactly my argument.

EDIT: In fact, your statement of my argument is unnecessarily cumbersome. We just need to say that P(C|w) <= P(A|w) + P(B|w), and by no-conspiracy P(A|x) = P(A|w), P(B|y) = P(B|w), P(C|z) = P(C|w), and according to QM we have P(A|x)=P(B|y)=.25 and P(C|z)=.75, so we have P(A|w)=P(B|w)=.25 and P(C|w)=.75, so we have .75 <= .25 + .25.

So we don't even need the intermediate step P(C|z) <= P(A|x) + P(B|y).

 

[billschnieder]

Well, before measurement the system is described by the same wavefunction in each of the three cases x, y, and z.

I guess this is a key point of disagreement. There are three different systems of photons "x", "y" and "z". They can not be described by the same wavefunction. It is only in "w" that we have a single system.

Yes, that is exactly my argument.

Let me explain then what your problem is:

1) Theoretically: An inequality such as P(C|w) <= P(A|w) + P(B|w), is not an independent mathematical truth, but just a simplification of the equality P(C|w)= P(A|w)+P(B|w)-2P(AB|w) from which it is derived. Then, you claim that because of "no-conspiracy", P(A|x) = P(A|w) and P(B|y) = P(B|w) and P(C|z) = P(C|w). Now, if that is the case, then it must also be the case that P(C|z) = P(A|x) + P(B|y) - ?. What in your opinion is the missing term, there must be one, otherwise your no-conspiracy condition is inconsistent. I would guess that it should be 2P(AB|xy) = 2P(AB|w) because of "no-conspiracy". If that is the case, let us remind ourselves at this point what A,B all mean:

Where A denotes statement "The result at -30° differs from the result at 0°", B denotes the statement "The result at 0° differs from the result at 30°", and C denotes the statement "The result at -30° differs from the result at 30°".

P(AB|w) is therefore the probability that the set of photon pairs "w" were destined to mismatch at both (-30°, 0°) AND (30°, 0°). Note here, that it is the same set that must satisfy both conditions.

P(AB|xy) is the probability that the set of photon pairs "x" was destined to mismatch at (-30°, 0°) and a completely different set of pairs "y" was also destined to mismatch at (30°, 0°). Note here that each set has just a single condition to fulfill.

 

[billschnieder]

Then, your no conspiracy condition boils down to the suggestion that a requirement for a single system to fulfill two conditions simultaneously, is the same as a requirement for two different systems to each fulfill only one of the two conditions. In simpler terms. The suggestion is akin to the suggestion that the probability of finding a tall girl in a class, is the same as the probability of finding a tall person in one class and a girl of any height in a different class.

2) Practically: When talking of measurements, since P(C|w), P(A|w), and P(B|w) are all derived from a single set of photon pairs, on which we can only ever measure one, two of the three must be counter-factual and unmeasurable. Since P(C|z), P(A|x), and P(B|y) are each derived from a different set of photon pairs, they are each directly measurable without any counter-factuals. Your no-conspiracy condition then becomes the suggestion that it is okay to substitute the results measured from three different systems into an inequality based on unmeasurable counterfactuals describing a single system.

Here is what Bell says about this idea:

"It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made."

P(C|z), P(A|x), and P(B|y) are each objectively measurable.
P(C|w) <= P(A|w) + P(B|w), is an impossible relation between results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.

So it is the mixing of "w", with "x,y,z" that results in the violation. In otherwords, it is the so-called "no-conspiracy assumption" itself that has to be abandoned.

 

[billschnieder]

P(C|z) = P(A|x) + P(B|y) - ?

On the other hand, if you refuse to provide the last term, let us leave it as ? and follow the calculation through.

0.75 = 0.25 + 0.25 - ?

and

? = - 0.25

Where is the violation? Why do you have a problem with ? = -0.25.

 

[lugita15]

I guess this is a key point of disagreement. There are three different systems of photons "x", "y" and "z". They can not be described by the same wavefunction. It is only in "w" that we have a single system.

I think this is just a minor point. x is the set of photon pairs for which we're going to measure polarization at -30 and 0, y is the set of photon pairs for which we're going to measure 0 and 30, and z is the set of photon pairs for which we're going to measure -30 and 30. All I was saying is that according to quantum mechanics, the wave function of a photon pair in any of these three sets is exactly the same. Do you really disagree with that?

Let me explain then what your problem is:

1) Theoretically: An inequality such as P(C|w) <= P(A|w) + P(B|w), is not an independent mathematical truth, but just a simplification of the equality P(C|w)= P(A|w)+P(B|w)-2P(AB|w) from which it is derived.

That's why I rewrote things in terms of equations in post #249.

Then, you claim that because of "no-conspiracy", P(A|x) = P(A|w) and P(B|y) = P(B|w) and P(C|z) = P(C|w). Now, if that is the case, then it must also be the case that P(C|z) = P(A|x) + P(B|y) - ?. What in your opinion is the missing term, there must be one, otherwise your no-conspiracy condition is inconsistent.

Just refer to my post #249: .75 = .25 + .25 - 2P(A & B|w). Remember, the no-conspiracy condition only applies to statements S which are meaningful in both x and w (and similarly for y and z). It does not apply to A & B, which is not meaningful for x, y, or z.

I would guess that it should be 2P(AB|xy) = 2P(AB|w) because of "no-conspiracy".

No, I'm not claiming anything about P(A & B|x,y).

 

[lugita15]

Then, your no conspiracy condition boils down to the suggestion that a requirement for a single system to fulfill two conditions simultaneously, is the same as a requirement for two different systems to each fulfill only one of the two conditions. In simpler terms. The suggestion is akin to the suggestion that the probability of finding a tall girl in a class, is the same as the probability of finding a tall person in one class and a girl of any height in a different class.

I don't see how the no-conspiracy condition boils down to that at all, because I see P(A & B|x,y) as irrelevant.

2) Practically: When talking of measurements, since P(C|w), P(A|w), and P(B|w) are all derived from a single set of photon pairs, on which we can only ever measure one, two of the three must be counter-factual and unmeasurable. Since P(C|z), P(A|x), and P(B|y) are each derived from a different set of photon pairs, they are each directly measurable without any counter-factuals.

Yes, that's a fair summary,

Your no-conspiracy condition then becomes the suggestion that it is okay to substitute the results measured from three different systems into an inequality based on unmeasurable counterfactuals describing a single system.

Yes, and I think that it is okay in this case.

Here is what Bell says about this idea:

"It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made."

That quote isn't about the assumptions of Bell's theorem, it's about Von Neumann's earlier hidden variable theorem that Bell critiqued as being too demanding in its assumptions. I don't think Bell thought of his own theorem the same way.

P(C|z), P(A|x), and P(B|y) are each objectively measurable.
P(C|w) <= P(A|w) + P(B|w), is an impossible relation between results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.

I don't know why you call it an impossible relation. It's a relationship between unmeasurable things, but the no-conspiracy condition allows us to say that certain unmeasurable quantities are equal to certain measurable quantities.

So it is the mixing of "w", with "x,y,z" that results in the violation.

What is wrong with an equation that contains x, y, z, and w? What is wrong with putting any numbers you want in an equation? That's fundamental to mathematics. Even if two numbers were taken from vastly different sources, if they have the same value, then they can be freely substituted into any statement without changing the truth value. That's Leibniz's famous Identity of Indiscernibles: if F(a) is true, and a=b, then F(b) must always be true. Is that what you're disputing?

In otherwords, it is the so-called "no-conspiracy assumption" itself that has to be abandoned.

OK, if you want to abandon it, you can, but at least address the argument I gave earlier in the thread for why it should be true:

I am talking about the result I would get if I perform a particular experiment on a particle, regardless of whether I actually perform that experiment. Now how do I connect this to real experiments, which are obviously concerned with possibility b)? I make the crucial assumption, which I expect that you disagree with or think is misleading, that the following two probabilities are always equal:
1. The probability that this photon would go through a polarizer if it is oriented at angle x, given that the polarizer is actually oriented at angle x.
2. The probability that this photon would go through a polarizer if it is oriented at angle x, given that the polarizer is NOT actually oriented at angle x, but instead some different angle y.

Now why do I assume that these two probabilities are equal? Because I am assuming that the answers to the following two questions are always the same:
1. What result would you get if you send this photon through a polarizer oriented at angle x, given that the polarizer is actually oriented at angle x?
2. What result would you get if you send the photon through a polarizer oriented at angle x, given that the polarizer is NOT oriented at angle x, but instead some different angle y?

And why do I assume that these questions have the same answer? That seems to me to be a consequence the no-conspiracy condition: the properties (or answers to questions) that are predetermined cannot depend on the specific measurement decisions that are going to be made later, since by assumption those decisions are free and independent.

What do you disagree with here? Do you disagree that questions 1 and 2 always have the same answer?

 

[lugita15]

P(C|z) = P(A|x) + P(B|y) - ?

On the other hand, if you refuse to provide the last term, let us leave it as ? and follow the calculation through.

0.75 = 0.25 + 0.25 - ?

and

? = - 0.25

Where is the violation? Why do you have a problem with ? = -0.25.

I have a problem with .75 = .25 + .25 - 2P(A & B|w) as shown in post #249.

 

[billschnieder]

All I was saying is that according to quantum mechanics, the wave function of a photon pair in any of these three sets is exactly the same. Do you really disagree with that?

Of course, they can not be exactly the same, there are 3 different systems with different space-time coordinates. You probably mean that the QM predictions for the three different systems
"x", "y", "z" as concerns this particular experiment are the same, which I will agree with.

.75 = .25 + .25 - 2P(A & B|w). Remember, the no-conspiracy condition only applies to statements S which are meaningful in both x and w (and similarly for y and z). It does not apply to A & B, which is not meaningful for x, y, or z.No, I'm not claiming anything about P(A & B|x,y).

But that is a nonsensical equation .75 = .25 + .25 - 2P(A & B|w). By your own argument, if your no-conspiracy condition can not say anything about P(A & B|x,y) then it can not say anything about 2P(A & B|w) either. Or do you want to explain how A&B is meaningful in both x and w (and similarly for y and z). By writing .75 = .25 + .25 - 2P(A & B|w), you are backhandedly trying to use it to say something about 2P(A & B|w). You say you are not making any claim about P(A & B|x,y) but I'm telling you that you are, indirectly.

How come you do not want to say anything about the equality for the second scenario only. In other words, why did you need w, why didn't you just write down the inequality using ONLY x, y, and z? What I'm asking you is, is there a genuine expression of the form

P(C|z) = P(A|x) + P(B|y) - ?

involving ONLY x,y,z and no w? If not, why not, and if so what is it?

 

[billschnieder]

That quote isn't about the assumptions of Bell's theorem, it's about Von Neumann's earlier hidden variable theorem that Bell critiqued as being too demanding in its assumptions.

Of course Bell did not intend to argue against his own theorem in a paper written before his theorem. But the point is that his argument is sound, and obviously applies just as well to his theorem whether he thought about it or not.

I don't know why you call it an impossible relation. It's a relationship between unmeasurable things

You've answered your own question. As far as measurements are concerned, it is an impossible relation.

What is wrong with an equation that contains x, y, z, and w? What is wrong with putting any numbers you want in an equation?

Nothing. Until you start drawing physical inferences from the equation. As I explained previously, if the time to put on just shoes is Ta, and the time to put on just socks is Tb, then you can write any equation you like which includes Ta and Tb. But you can not conclude that the Ta + Tb = Time to put on shoes and then socks.

Even if two numbers were taken from vastly different sources, if they have the same value, then they can be freely substituted into any statement without changing the truth value.

That's just not correct. Two numbers do not have the same value just because you claim they do. It is the physical situation producing the two values that determines whether they have the same value or not, not lugita. And I've already explained to you that the probability of finding a tall girl in a class is not the same as the probability of finding a tall person in one class and finding a girl of any height in another class.

That's Leibniz's famous Identity of Indiscernibles: if F(a) is true, and a=b, then F(b) must always be true. Is that what you're disputing?OK, if you want to abandon it, you can

What are you talking about?

Continuing in the next post ...

 

[billschnieder]

I make the crucial assumption, ... that the following two probabilities are always equal:
1. The probability that this photon would go through a polarizer if it is oriented at angle x, given that the polarizer is actually oriented at angle x.
2. The probability that this photon would go through a polarizer if it is oriented at angle x, given that the polarizer is NOT actually oriented at angle x, but instead some different angle y.

To illustrate the error, I will rephrase it as follows:
*) The expected result R1, if the photon is measured at angle x
*) The expected result R3, if the photon were instead measured at angle x, given that we already measured it at angle y and obtained result R2.

Your crucial point is that R1 must be equal to R3 because of no-conspiracy. Of course that is false. I will give you a clear counterexample example using Bernouli's urn.

We have an urn with 2 balls, 1 white and 1 red, we pick two balls in sequence without replacement.
*) The probability of the first ball being Red, R1
*) The probability of the first ball being Red, R3, given that the second ball was revealed to be Red R2.
According to your no-conspiracy condition, R1 = R3 which is false. R1 = 1/2 and R3 = 0. As I hope you appreciate, the underlined statements (R2) place restrictions on what we are allowed to say about R3. By ignoring this part of your argument, it has obscured your ability to see the error.

Not only do we have results for the actually measured angle y, we are also using it jointly in the same expression as the results at angle x.

the properties (or answers to questions) that are predetermined cannot depend on the specific measurement decisions that are going to be made later, since by assumption those decisions are free and independent.

Obviously false or naive. In the Bernouli example above, the result of the first pick was dependent on the result of the second pick which was made later.

So I will appreciate answers to my own questions too:

What I'm asking you is, is there a genuine expression of the form

P(C|z) = P(A|x) + P(B|y) - ?

involving ONLY x,y,z and no w? If not, why not, and if so what is it?

 

[lugita15]

Of course, they can not be exactly the same, there are 3 different systems with different space-time coordinates. You probably mean that the QM predictions for the three different systems
"x", "y", "z" as concerns this particular experiment are the same, which I will agree with.

The wavefunction of a system can be written as a product of a positional part and a spin part. All I meant is that the spin state of each photon pair is the same.

But that is a nonsensical equation .75 = .25 + .25 - 2P(A & B|w).

Why is it nonsensical?

By your own argument, if your no-conspiracy condition can not say anything about P(A & B|x,y) then it can not say anything about 2P(A & B|w) either.

Whether the no-conspiracy condition says anything about P(A & B|x,y) is not really something that concerns me, because it's completely irrelevant to my argument. But yes, the no-conspiracy condition does not allow you to conclude that P(A & B|w) = P(A & B|x), for instance, because A & B is not meaningful in x. To repeat, the no-conspiracy condition states that if S is meaningful in both x and w, then P(S|w)=P(S[x).

Or do you want to explain how A&B is meaningful in both x and w (and similarly for y and z).

No, it's definitely not meaningful in x, y, or z.

By writing .75 = .25 + .25 - 2P(A & B|w), you are backhandedly trying to use it to say something about 2P(A & B|w).

All I'm saying is that you can't directly apply the no-conspiracy condition to P(A & B|w).

You say you are not making any claim about P(A & B|x,y) but I'm telling you that you are, indirectly.

How am I doing that?

How come you do not want to say anything about the equality for the second scenario only. In other words, why did you need w, why didn't you just write down the inequality using ONLY x, y, and z?

Well, you can write the inequality in terms of only x, y, and z, but you can't write the equation in terms of only x, y, and z. How is that a problem?

In a deeper sense, if you're asking why we need w at all in this proof, that's because one of the assumptions of Bell's theorem is counterfactual definiteness.

What I'm asking you is, is there a genuine expression of the form

P(C|z) = P(A|x) + P(B|y) - ?

involving ONLY x,y,z and no w? If not, why not, and if so what is it?

No, there isn't. (I mean, of course there are equations of that form, but nothing relevant.) And the reason there isn't is that the no-conspiracy condition applies to P(A|w), P(B|w), and P(C|w), but it doesn't apply to P(A & B|w), because A & B is only meaningful in w.

 

[lugita15]

Of course Bell did not intend to argue against his own theorem in a paper written before his theorem. But the point is that his argument is sound, and obviously applies just as well to his theorem whether he thought about it or not.

OK, fair enough, it's just that the way you've been using the quote makes it seem like Bell was acknowledging a shortcoming in his own theorem.

You've answered your own question. As far as measurements are concerned, it is an impossible relation.

Well, if all you mean is that P(C|w) <= P(A|w) + P(B|w) is a relationship between quantities that are not directly measurable, then yes I agree with that.

Nothing. Until you start drawing physical inferences from the equation.

I think this is where we fundamentally disagree. If the mass of a stick and the mass of a stone are equal, and the mass of the stick is great than the mass of an ant, then the mass of the stone is also greater than the mass of the ant. Numbers which are equal can be freely substituted for each other in any statement without changing the truth value of that statement, whether that statement is about the physical world or not. Do you really disagree with that? That's Leibniz's law.

As I explained previously, if the time to put on just shoes is Ta, and the time to put on just socks is Tb, then you can write any equation you like which includes Ta and Tb. But you can not conclude that the Ta + Tb = Time to put on shoes and then socks.

But all your example shows is that the time to do a then b is not equal to the time to do a + the time to do b, or in symbols, T_(a then b) does not equal T_a + T_b. In other words, the "then" operation of actions does not correspond to the addition operation. But we're not dealing with that kind of issue at all here.

That's just not correct. Two numbers do not have the same value just because you claim they do. It is the physical situation producing the two values that determines whether they have the same value or not, not lugita.

I didn't say that any two numbers are equal if I say so. All I said is IF they'e equal, then they can be freely substituted. So if you want to disagree with me, you have to disagree with P(A|w)=P(A|x). But if I'm right about that, then after that you can't disagree, can you, with me substituting P(A|x) into the equation P(C|w)= P(A|w)+P(B|w)-2P(AB|w), and drawing valid physical inferences from a correct equation?

And I've already explained to you that the probability of finding a tall girl in a class is not the same as the probability of finding a tall person in one class and finding a girl of any height in another class.

I agree with that, but I don't see how I'm doing anything analogous to that.

What are you talking about?

I'm talking about Leibniz's Law, known as the Identity of Indiscernibles. It says that if F(a) is a true statement (like "a is an apple"), and a=b is a true statement, then F(b) ("b is an apple") must always be a true statement. Do you disagree with that?

 

[lugita15]

To illustrate the error, I will rephrase it as follows:
*) The expected result R1, if the photon is measured at angle x
*) The expected result R3, if the photon were instead measured at angle x, given that we already measured it at angle y and obtained result R2.

Your crucial point is that R1 must be equal to R3 because of no-conspiracy.

I am NOT asserting that the probability that a given photon would have yielded a certain result R1 at angle x, given that it was measured at y and yielded R2, is equal to the probability that it that will yield R1 if we measure it at x. Instead I'm saying that the probability that a given photon have yielded R1 at x, given that it was measured at y, but NOT given the result of the measurement, is equal to the probability that it will yield R1 if we measure it at x.

That was a long string of words, so let me write it in fraction form. This is what I am NOT saying, but what you seem to think I'm saying:
(number of photons yielded R2 at y but would have yielded R1 at x)/(number of photons which yielded R2 at y) = (number of photons which yielded R1 at x)/(number of photons measured at x).

This is what I AM saying:
(number of photons which were measured at y but would have yielded R1 at x)/(number of photons which were measured at y) = (number of photons which yielded R1 at x)/(number of photons measured at x).

If you disagree with this statement, you can, but please tell me your thoughts on the argument I presented for it:

"Now why do I assume that these two probabilities are equal? Because I am assuming that the answers to the following two questions are always the same:
1. What result would you get if you send this photon through a polarizer oriented at angle x, given that the polarizer is actually oriented at angle x?
2. What result would you get if you send the photon through a polarizer oriented at angle x, given that the polarizer is NOT oriented at angle x, but instead some different angle y?

And why do I assume that these questions have the same answer? That seems to me to be a consequence the no-conspiracy condition: the properties (or answers to questions) that are predetermined cannot depend on the specific measurement decisions that are going to be made later, since by assumption those decisions are free and independent."
Do you agree or disagree that these two questions have the same answer?

Of course that is false. I will give you a clear counterexample example using Bernouli's urn.

We have an urn with 2 balls, 1 white and 1 red, we pick two balls in sequence without replacement.
*) The probability of the first ball being Red, R1
*) The probability of the first ball being Red, R3, given that the second ball was revealed to be Red R2.
According to your no-conspiracy condition, R1 = R3 which is false. R1 = 1/2 and R3 = 0. As I hope you appreciate, the underlined statements (R2) place restrictions on what we are allowed to say about R3. By ignoring this part of your argument, it has obscured your ability to see the error.

This is not at all what I'm doing.

 

[billschnieder]

I am NOT asserting that the probability that a given photon would have yielded a certain result R1 at angle x, given that it was measured at y and yielded R2, is equal to the probability that it that will yield R1 if we measure it at x. Instead I'm saying that the probability that a given photon have yielded R1 at x, given that it was measured at y, but NOT given the result of the measurement, is equal to the probability that it will yield R1 if we measure it at x.

I can only assume then that you do not understand what you are saying. Do you at least agree now, that the two probabilities would not be the same if R2 were given? If you can agree to this, then you have understood the argument and all I have to do is to convince you that in your argument, R2 is indeed given.

If you disagree with this statement, you can, but please tell me your thoughts on the argument I presented for it

What do you think I've been doing? You are just repeating the same post and ignoring my argument against it. Do you even understand the argument at all, let-alone agreeing with it?

Now why do I say R2 must be given. Because you have both results in the same expression from which you are trying to draw physical conclusions. You are arguing as if you only had a single probability written on one sheet of paper, and another probability written on another sheet of paper. Rather you have an expression P(A|x) + P(B|y) which includes BOTH the results at x and the results at y, so how can the results at y not be given?! You have "given" it, by having it in the expression. You can't have your cake and not have it.

This is not at all what I'm doing.

It is indeed, as I've explained now, maybe 5 different ways. Each time you simply say it is not. Just because you phrase your argument in a way that obscures the argument does not mean I can not see through it and go right through to the core issue. That is what I have done, many times already.

 

[billschnieder]

Numbers which are equal can be freely substituted for each other in any statement without changing the truth value of that statement ... Do you really disagree with that?

But the numbers are not equal, isn't that the argument? Otherwise why are you asking me to explain why I claim the numbers are not equal.

But all your example shows is that the time to do a then b is not equal to the time to do a + the time to do b

You completely missed the point which was to show you that the value of T_a, T_b in the expression T_a + T_b, will be different from the value of just T_a or T_b separately. The point being that the physical situation of in the expression "T_a + T_b" (putting on shoes and then Socks) determines which value T_a or T_b will have in the expression and you can not simply take values from considering T_a separately and plug them in expecting things to be meaningful. Of course you are free to do it, but do not be surprised when nonsensical paradoxes result.

All I said is IF they'e equal, then they can be freely substituted.

And I'm arguing that the free standing T_b is not the same as the T_b when considering it jointly with T_a. I'm arguing that they are not equal.

So if you want to disagree with me, you have to disagree with P(A|w)=P(A|x).

No I don't. I disagree with you going from

P(C|w)= P(A|w)+P(B|w)-2P(AB|w)
to P(C|z)= P(A|x)+P(B|y)-2P(AB|w)

And apparently, you too disagree with it when if agree that there is no genuine expression of the form P(C|z) = P(A|x) + P(B|y) - ? involving only x, y, z.

I disagree with "P(A|w)=P(A|x) AND P(B|w)=P(B|y) AND P(C|w)=P(C|z) simultaneously"

I mean, of course there are equations of that form, but nothing relevant.

Please, please it is very relevant. This is at the core of the disagreement, how can it not be relevant. Please provide the equation. Unless you just want to wiggle out of admitting that there is none. The question did not include any w, so why even bring no-conspiracy into it. Do you have a complete expression including just x, y, z for of the form P(C|z) = P(A|x) + P(B|y) - ? or not? If you do please provide it. If none exists, please explain why none exists. This is a crucial question.

 

[lugita15]

I can only assume then that you do not understand what you are saying. Do you at least agree now, that the two probabilities would not be the same if R2 were given?

I agree that the percentage of photons yielding R2 for y, for which x would have yielded R1, is different from the percentage of photons measured at x, for which x yielded R1. But I also believe that the percentage of photons measured at y, for which x would have yielded R1, is the SAME as the percentage of photons measured at x, for which x yielded R1. Do you agree or disagree with this second statement?

What do you think I've been doing? You are just repeating the same post and ignoring my argument against it. Do you even understand the argument at all, let-alone agreeing with it?

I think I understand my argument, but I don't know what your counterargument is. At least so far as I can tell, you haven't told me whether my 2 questions always have the same answer. In post #262, you responded to that post by giving the example of Bernoulli's urn, which was about probabilities. But my two questions have absolutely nothing to do with probabilities.

Now why do I say R2 must be given. Because you have both results in the same expression from which you are trying to draw physical conclusions. You are arguing as if you only had a single probability written on one sheet of paper, and another probability written on another sheet of paper. Rather you have an expression P(A|x) + P(B|y) which includes BOTH the results at x and the results at y, so how can the results at y not be given?!

First of all, we have some ambiguity as to what x and y mean. The x and y in my two questions are angle settings of a polarizer. The x and y in P(A|x) + P(B|y) are probability spaces, x being the set of photons for which you measure at -30 and 0, and y being the set of photons for which you measure at 0 and 30. To minimize confusion, in future why don't we use θ1 and θ2 to refer to angle settings, and continue to use x, y, z, and w to refer to probability spaces?

You have "given" it, by having it in the expression. You can't have your cake and not have it.

Instead of arguing about what's been "given" or not, why don't you simply tell me whether you agree or disagree that the following two fractions are equal?
1. (Number of photons pairs which yielded different results when measured at -30 and 0)/((Number of photon pairs which were measured at -30 and 0).
2. (Number of photon pairs which were measured at 0 and 30 but would have yielded different results at -30 and 0)/(Number of photon pairs which were measured at 0 and 30)

It is indeed, as I've explained now, maybe 5 different ways. Each time you simply say it is not.

Your example of Bernoulli's urn dealt with different probabilities than the ones I'm dealing with. I wrote the probabilities that I'm dealing with in fraction form above.

 

[lugita15]

But the numbers are not equal, isn't that the argument? Otherwise why are you asking me to explain why I claim the numbers are not equal.

If you'r only claim is that P(A|w)=P(A|x) is wrong for some reason, that's one thing that we can discuss. But you also seem to claim that "it is the mixing of "w", with "x,y,z" that results in the violation". What is the reason why equations involving x, y, z, and w are not allowed, or physically meaningless, or whatever your contention is?

You completely missed the point which was to show you that the value of T_a, T_b in the expression T_a + T_b, will be different from the value of just T_a or T_b separately.

I have no idea what this sentence means. Can you explain it different words?

The point being that the physical situation of in the expression "T_a + T_b" (putting on shoes and then Socks) determines which value T_a or T_b will have in the expression and you can not simply take values from considering T_a separately and plug them in expecting things to be meaningful. Of course you are free to do it, but do not be surprised when nonsensical paradoxes result.

Again, I don't quite understand what you're saying here, and I don't know why you're associating T_a + T_b with putting on shoes then socks, rather than just the time to put on shoes plus the time to put on socks. Can you use a different example to illustrate your point, whatever it is?

And I'm arguing that the free standing T_b is not the same as the T_b when considering it jointly with T_a. I'm arguing that they are not equal.

T_b is just a fixed number, just like 5. Whether you put it on its own in a statement (with physical significance or not), or whether you put it in a statement (with physical significance or not) which also has T_a in it, doesn't it still retain its value?

No I don't. I disagree with you going from

P(C|w)= P(A|w)+P(B|w)-2P(AB|w)
to P(C|z)= P(A|x)+P(B|y)-2P(AB|w)

And apparently, you too disagree with it when if agree that there is no genuine expression of the form P(C|z) = P(A|x) + P(B|y) - ? involving only x, y, z.

I definitely don't disagree with going from P(C|w)= P(A|w)+P(B|w)-2P(AB|w) to P(C|z)= P(A|x)+P(B|y)-2P(AB|w). What do you see wrong with that step?

I disagree with "P(A|w)=P(A|x) AND P(B|w)=P(B|y) AND P(C|w)=P(C|z) simultaneously"

This is a rather strange statement. What does it mean for three statements to be true, versus three statement being true simultaneously? Are you operating in some non-classical logic or something?

Please, please it is very relevant. This is at the core of the disagreement, how can it not be relevant. Please provide the equation. Unless you just want to wiggle out of admitting that there is none. The question did not include any w, so why even bring no-conspiracy into it. Do you have a complete expression including just x, y, z for of the form P(C|z) = P(A|x) + P(B|y) - ? or not? If you do please provide it. If none exists, please explain why none exists. This is a crucial question.

There aren't any useful equations of that form which involve just x, y, and z. There's lots of useless and unimportant equations of that form, but they're of zero relevance to what we're talking about. (For instance P(C|z) = P(A|x) + P(B|y) - (1 - P(not A|x) - P(not B|y) + P(not C|z)) or something silly like that.) Anyway, the point is, why do we need to express the Bell inequality in terms of an equation that only involves x, y, and z? What is wrong with an equation that involves x, y, z, and w? And why can't we derive physical inferences from such an equation?

And also, why can't we just sidestep this whole issue, like I did in post #249, and just directly get to .75 = .25 + .25 - 2P(A & B|w) from P(C|w)= P(A|w)+P(B|w)-2P(AB|w), P(A|w)=P(A|x)=.25, P(B|w)=P(B|y)=.25, and P(C|w)=P(C|z)=.75?

 

[billschnieder]

What is wrong with an equation that involves x, y, z, and w? And why can't we derive physical inferences from such an equation?

Because the expression is incorrect to start with. It is incorrect because you have assumed that terms are equal, which are not. The fact that a genuine expression exists for w, and none exists for xyz, confirms that. In other words, due to the physical situation "w", there is a single joint probability distribution p(ABC|w) from which you can extract P(A|w), P(B|w), and P(C|w) but there is no joint probability distribution p(ABC|xyz) in the physical situation "xyz" from which you can extract P(A|x),P(B|y), and P(C|z). Because of the physical situation, "w" the probabilities P(A|w), P(B|w), and P(C|w) are mutually dependent having originated from the same system, but the physical situation "xyz" mandates that the probabilities P(A|x), P(B|y), and P(C|z) are independent, having originated from three separate systems.

After substituting the probabilities from the physical situation "xyz" into an expression derived for the physical situation "w", the violation you obtain tells you clearly that the probabilities were not the same as you naively assumed. And the impossibility of deriving a similar expression directly in physical situation "xyz" without first going through "w" is a big hint that your assumptions are wrong.

To summarize:
1) There are two scenarios/physical sittuations involved in this discussion:

Scenario "xyz", involving the three probabilities:

P(A|x) = what we would get if we measure system "x" at angles (a,b)
P(B|y) = what we would get if we measure system "y" at angles (b,c)
P(C|z) = what we would get if we measure system "z" at angles (a,c)​

Scenario "w", involving the three probabilities:

P(A|w) = what we would get if we measure system "w" at angles (a,b)
P(B|w) = what we would have gotten had we measured system "w" at angles (b,c) instead of (a,b)
P(C|w) = what we would have gotten had we measured system "w" at angles (a,c) instead of (a,b)​

2) Scenario "xyz" is different from Scenario "w". The three probabilities in "w" are not equal to the three probabilities from "xyz".
3) Bell's inequalities are derived for scenario "w" ONLY. A similar inequality cannot be derived for scenario "xyz".
4) The predictions of QM are for scenario "xyz" only, not "w"
5) Experimental results are for scenario "xyz" only
6) Using probabilities from "xyz" in an expression derived for "w", amounts to introducing an assumption that the probabilities are all equal. It is this assumption that should be rejected when a violation results.

So what part of this argument don't you understand, or do understand but disagree with and why?

 

[billschnieder]

Again, I don't quite understand what you're saying here, and I don't know why you're associating T_a + T_b with putting on shoes then socks, rather than just the time to put on shoes plus the time to put on socks.

Because to calculate the time required to put on shoes and then socks, I have to add the time required to put on shoes to the time required to put on socks.

T_b is just a fixed number, just like 5. Whether you put it on its own in a statement (with physical significance or not), or whether you put it in a statement (with physical significance or not) which also has T_a in it, doesn't it still retain its value?

No. T_b is just a fixed number like 5 in the physical situation in which you are only putting on socks. T_b is just a fixed number like 8, in the physical situation in which we are putting on shoes and then socks. T_b in in the first physical situation, is not the same value as T_b in the second physical situation, even though in both cases T_b is still the time it takes to put on socks.

I definitely don't disagree with going from P(C|w)= P(A|w)+P(B|w)-2P(AB|w) to P(C|z)= P(A|x)+P(B|y)-2P(AB|w). What do you see wrong with that step?

What I have been explaining all this while, that the probabilities from the "xyz" scenario are not the same as those in the "w" scenario, so it is not correct to substitute them.

This is a rather strange statement. What does it mean for three statements to be true, versus three statement being true simultaneously? Are you operating in some non-classical logic or something?

Nothing strange there at all if you understand degrees of freedom, and probability theory. Like I explained to you using the Bernouli's urn example, the probability of Red on the first draw when considered alone, has a different value from the probability of red on the first draw when considered together with the probability of red on the second draw, already given.

And also, why can't we just sidestep this whole issue, like I did in post #249

Because it is assumed that you want to make sound, consistent and valid arguments rather than just sneaking in paradoxes behind the back door.

 

[billschnieder]

This is a rather strange statement. What does it mean for three statements to be true, versus three statement being true simultaneously? Are you operating in some non-classical logic or something?

George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle
ITAMAR PITOWSKY, Br J Philos Sci (1994) 45 (1): 95-125. doi: 10.1093/bjps/45.1.95

Abstract:
In the mid-nineteenth century George Boole formulated his ‘conditions of possible experience’. These are equations and ineqaulities that the relative frequencies of (logically connected) events must satisfy. Some of Boole's conditions have been rediscovered in more recent years by physicists, including Bell inequalities, Clauser Horne inequalities, and many others ...

On page 100:

George Boole is best known as one of the fathers of modern logic. Somewhat less known in his work on the theory of probability, most of it published in his classical book, Boole [1854]. ... Boole's problem is simple: we are given rational numbers which indicate the relative frequencies of certain events. If no logical relations obtain among the events, then the only constraints imposed on these numbers are that they each be non-negative and less than one. If however, the events are logically interconnected, there are further equalities or inequalities that obtain among the numbers. The problem thus is to determine the numerical relations among frequencies, in terms of equalities and inequalities, which are induced by a set of logical relations among the events. The equalities and inequalities are called 'conditions of possible experience'.

 

[billschnieder]

On page 105 it says:

One thing should be clear at the outset: none of Boole's conditions of possible experience can ever be violated when all the relative frequencies involved have been measured in a single sample. The reason is that such a violation entails a logical contradiction. For example, suppose that we sample at random a hundred balls from an urn. Suppose, moreover that 60 of the balls sampled are red, 75 are wooden and 32 are both red and wooden. We have p1=0.6, p2= 0.75, p12 = 0.32. But then p1+p2-p12 > 1. This clearly represents a logical impossibility, for there must be a ball in the sample (in fact three balls) which is 'red', is 'wooden', but not 'red and wooden'; absurd.
Similar logical absurdities can be derived if we assume a violation of any of the relevant conditions, no matter how complex they appear to be. This is the reason for the title 'conditions of possible experience'. In case we deal with relative frequencies in a single sample, a violation of any of the relevant Boole's conditions is a logical impossibility.
But sometimes, for various reasons, we may choose or be forced to measure the relative frequencies of (logically connected) events, in several distinct samples. In this case a violation of Boole's conditions may occur.

So, No, I'm not using any alternate logic. I'm reasoning the situation as prescribed by one of the fathers of logic, and explaining to you that you are not reasoning logically about the situation.

see for example, https://www.physicsforums.com/showpost.php?p=3856772&postcount=125

 

[DevilsAvocado]

We have an urn with 2 balls, 1 white and 1 red, we pick two balls in sequence without replacement.
*) The probability of the first ball being Red, R1
*) The probability of the first ball being Red, R3, given that the second ball was revealed to be Red R2.
According to your no-conspiracy condition, R1 = R3 which is false. R1 = 1/2 and R3 = 0. As I hope you appreciate, the underlined statements (R2) place restrictions on what we are allowed to say about R3. By ignoring this part of your argument, it has obscured your ability to see the error.

Not only do we have results for the actually measured angle y, we are also using it jointly in the same expression as the results at angle x.

To summarize:
1) There are two scenarios/physical sittuations involved in this discussion:

Scenario "xyz", involving the three probabilities:

P(A|x) = what we would get if we measure system "x" at angles (a,b)
P(B|y) = what we would get if we measure system "y" at angles (b,c)
P(C|z) = what we would get if we measure system "z" at angles (a,c)​

Scenario "w", involving the three probabilities:

P(A|w) = what we would get if we measure system "w" at angles (a,b)
P(B|w) = what we would have gotten had we measured system "w" at angles (b,c) instead of (a,b)
P(C|w) = what we would have gotten had we measured system "w" at angles (a,c) instead of (a,b)​

2) Scenario "xyz" is different from Scenario "w". The three probabilities in "w" are not equal to the three probabilities from "xyz".
3) Bell's inequalities are derived for scenario "w" ONLY. A similar inequality cannot be derived for scenario "xyz".
4) The predictions of QM are for scenario "xyz" only, not "w"
5) Experimental results are for scenario "xyz" only
6) Using probabilities from "xyz" in an expression derived for "w", amounts to introducing an assumption that the probabilities are all equal. It is this assumption that should be rejected when a violation results.

So what part of this argument don't you understand, or do understand but disagree with and why?

Bill, I think you’re lost in the ‘Bayesian jungle’... can’t see the forest for the trees...
You and lugita could probably continue this discussion for 100 years without agreement, because you are discussing two different things.

I have some simple Yes/No -questions that will hopefully reveal the root of bafflement:

1. Do you understand that Bell's theorem is mainly about correlations (not probabilities)? Yes/No:
   
   
2. Do you understand that for a single entangled photon, the measurement probability outcome according to QM is always 100% random (50/50 +1/-1) no matter what you do or what angle you set? Yes/No:
   
   
3. Do you realize that when you write P(A|x) you are talking about statistical correlations for the relative angle (a - b), for hundreds or thousands of entangled pairs? Yes/No:
   
   
4. Do you realize that above is also always true for any other settings, i.e. P(B|y), P(C|z), etc? Yes/No:
   
   
5. Do you understand that it’s completely meaningless to talk about probabilities/correlations for a single photon/pair (except when perfectly aligned)? Yes/No:
   
   
6. Do you understand that P(A|x) says absolutely nothing about the properties of a single photon before measurement? Yes/No:
   
   
7. Do you accept the experimental outcome/statistics of EPR-Bell test experiments? Yes/No:
   
   
8. Do you admit that to set an upper limit for ‘classical’ Local Realism, we must operate according to these ‘classical’ rules only, i.e. not according to the rules of QM? Yes/No:
   
   
9. Do you acknowledge that that the definite property of LHV in Local Realism unconditionally leads to Counterfactual definiteness (CFD)? Yes/No:
   
   
10. Do you acknowledge that LHV & CFD must naturally be included in the test of Local Realism, in from of Bell's inequality? Yes/No:
    
    
11. Do you acknowledge that classical propositional logic do not work satisfactory in QM systems before measurement and the transition to [macroscopic] statistics? Yes/No:
    
    
12. Are you aware of that your “Bayesian/logical investigation” on QM states before measurement is erroneous and that Quantum Logic is still ‘under development’? Yes/No:
    
    
13. Do you understand that to use Bayesian probability theory in QM, you are forced to deal with measurements and [macroscopic] statistics, and that any “pre-measurement probability/states” is [yet] out of the question? Yes/No:
    
    
14. Do you admit that if QM violates Bell's inequality – it violates Bell's inequality – no matter if it’s “Little Green Men” running an “intergalactic-non-local-signaling-system” or what, it is violated, okay? Yes/No:

Appreciate if you could answer these 14 simple questions with Yes/No – it will save us a lot of time.

Thanks
DA

 

[billschnieder]

I have some simple Yes/No -questions that will hopefully reveal the root of bafflement:

Somehow I doubt it.

[*]Do you understand that Bell's theorem is mainly about correlations (not probabilities)? Yes/No

Have you stopped beating your wife? Yes/No. I understand that that Bell's theorem can be about both. In the other thread which was closed, we discussed correlations, in this thread, we are discussing about probabilities. Do you understand that?

[*]Do you understand that for a single entangled photon, the measurement probability outcome according to QM is always 100% random (50/50 +1/-1) no matter what you do or what angle you set?

huh? That is a nonsensical statement. As I have explained to you 100 times previously, each specific single photon, when measured, gives a definite outcome. QM says absolutely nothing and can say absolutely nothing about this single outcome. What appears random is the SERIES of outcomeS from a SERIES of similarly prepared photons. It is for this SERIES that QM makes predictions. Obviously you do not understand this.

[*]Do you realize that when you write P(A|x) you are talking about statistical correlations for the relative angle (a - b), for hundreds or thousands of entangled pairs?

P(A|x) is a probability of mismatch when the set of photon pairs "x" is measured at the angle pair (a,b). P(A|x) is NOT a correlation.
etc ...
etc ...
I encourage you to go back and re-read this thread from page 15, carefully, because you obviously have not, as your "yes/no" wife-beater-type questions indicate. I'm not willing to attempt to correct all your misconceptions here, especially those that have already been corrected in excruciating detail in this very thread since page 15.

Appreciate if you could answer these 14 simple questions with Yes/No – it will save us a lot of time.

Rather it would save us time if you could state clearly what part of the argument in post #270 (which you quoted) you disagree with and why exactly.

 

[billschnieder]

In the other thread, when I asked the questions:

1) Do you agree that there are two scenarios involved in this discussion:

Scenario X, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(b,c) = correlation for what we would get if we measure (b,c)
C(a,c) = correlation for what we would get if we measure (a,c)

Scenario Y, involving the three correlations:

C(a,b) = correlation for what we would get if we measure (a,b)
C(a,c) = correlation for what we would have gotten had we measured (a,c) instead of (a,b)
C(b,c) = correlation for what we would have gotten had we measured (b,c) instead of (a,b)

2) Do you agree that Scenario X is different from Scenario Y?
3) Do you agree that the correlations in Bell's inequalities correspond to Scenario Y NOT Scenario X?
4) Do you agree that correlations calculated from QM correspond to Scenario X not Scenario Y?
5) Do you agree that correlations measured in experiments correspond to Scenario X not Scenario Y?

Your answers there were
1) Yes, of course
2) Yes, of course
3) Yes, of course
4) Yes, of course
5) Yes, of course

Scenario X => QM theory => EPR-Bell experiments => what actually works
Scenario Y => assumptions of Local Realism => definite values all the time => CFD => not working
For clarity we exchange Scenario X = QM theory and Scenario Y = Local Realism, and then it’s easy to see where it goes wrong.

2) Do you agree that Scenario X is different from Scenario Y?

Yes of course, it’s big difference between QM theory and Local Realism.

3) Do you agree that the correlations in Bell's inequalities correspond to Scenario Y NOT Scenario X?

Yes of course, Bell's inequalities sets the upper limit for what any model of Local Realism can achieve.

4) Do you agree that correlations calculated from QM correspond to Scenario X not Scenario Y?

Yes of course, calculated correlations of QM theory naturally will correspond to QM theory.

5) Do you agree that correlations measured in experiments correspond to Scenario X not Scenario Y?

Yes of course, when we perform QM experiments in form of EPR-Bell test experiments, it will naturally correspond to QM theory.

 

[billschnieder]

You completely missed the point which was to show you that the value of T_a, T_b in the expression T_a + T_b, will be different from the value of just T_a or T_b separately.

I have no idea what this sentence means. Can you explain it different words?

I will let Bell answer this in his own words:
Page 449 in his 1966 Paper "On the problem of Hidden variables in Quantum mechanics", Reviews of Modern Physics, Vol 38 (3), 1966.

The essential assumption can be criticized as follows. At first sight the required additivity of expectation values seems very reasonable, and it is rather the non-additivity of allowed values (eigenvalues) which requires explanation. Of course the explanation is well known: A measurement of a sum of noncommuting observables cannot be made by combining trivially the results of separate observations on the two terms.

On page 451 of the same article Bell says

The danger in fact was not in the explicit but in the implicit assumptions. It was tacitly assumed that measurement of an observable must yield the same value independently of what other measurements may be made simultaneously. Thus as well as P(Φ3) say, one might measure either P(Φ2) or P(ψ2), here Φ2 and ψ2 are orthogonal to Φ3 but not to one another. These different possibilities require different experimental arrangements; there is no a priori reason to believe that the results for P(Φ3) should be the same.

 

[DevilsAvocado]

I understand that that Bell's theorem can be about both.

Okay, then you don’t understand Bell's theorem.

[my bolding]

huh? That is a nonsensical statement. As I have explained to you 100 times previously, each specific single photon, when measured, gives a definite outcome. QM says absolutely nothing and can say absolutely nothing about this single outcome. What appears random is the SERIES of outcomeS from a SERIES of similarly prepared photons. It is for this SERIES that QM makes predictions. Obviously you do not understand this.

Okay, now you’re refuting that most fundamental property of the most successful scientific theory ever, the very same theory that makes your computer compute and your internet connection connect. Grave...

P(A|x) is a probability of mismatch when the set of photon pairs "x" is measured at the angle pair (a,b). P(A|x) is NOT a correlation.

Sorry Bill, you do not understand the basics in Bell's theorem:

P(A|x) = what we would get if we measure system "x" at angles (a,b)

Test angles:
a = 45°
b = 22.5°

RELATIVE angle (a - b) => (45° - 22.5°) => (22.5°)

Correlation STATISTICS for HUNDREDS of entangled pairs at RELATIVE angle 22.5°:
cos^2(22.5°) = 85%

And of course, the type of Bell state (Type I or Type II) decides if we are talking match or mismatch. Period.

I encourage you to go back and re-read this thread from page 15, carefully, because you obviously have not, as your "yes/no" wife-beater-type questions indicate. I'm not willing to attempt to correct all your misconceptions here, especially those that have already been corrected in excruciating detail in this very thread since page 15.

Oh... did I mention that in the “Bayesian probability system”, built just for this occasion, there was also included Yes/No/Absent? According to the “system estimation” – you have written hundreds of posts, including thousands of words, on this subject – and now you refuse to type 14 simple Yes/No...

I think anybody with normal intelligence can see what the probabilities for this suggest.

Thanks Bill, you’ve just saved us a lot of time, and I will of course bookmark this post, and link as soon as you bring up this terribly erroneous logic on PF, for anyone to inspect!

End of story.

Take care!

 

[lugita15]

Because the expression is incorrect to start with. It is incorrect because you have assumed that terms are equal, which are not.

OK, then that's what we need to focus on. Again, I still want to know whether you agree or disagree with the answers to my two questions are always the same. Because if they are always the same, then I can show that P(A|x) must always equal P(A|w).

Let me repeat the basic logic. Consider a photon before it is measured. By counterfactual definiteness, for every angle setting θ1, there exists a definite, predetermined answer to the question "What result would you get if the photon were measured at angle θ1?". Now if the experimenter can freely choose his measurement decision, then the predetermined answer to this question cannot depend on what he is going to do, so it cannot depend on the angle θ2 he is going to measure at, whether θ1=θ2 or θ1≠θ2. Do you agree or disagree with that? (And please answer this question without discussing probabilities, because it has nothing to do with probabilities.)

If you agree, then I can easily show that for any given photon pair, the truth value of A is independent of the angle settings of the two polarizers. And then from there, I can easily show that P(A|w) = P(A|x).

The fact that a genuine expression exists for w, and none exists for xyz, confirms that. In other words, due to the physical situation "w", there is a single joint probability distribution p(ABC|w) from which you can extract P(A|w), P(B|w), and P(C|w) but there is no joint probability distribution p(ABC|xyz) in the physical situation "xyz" from which you can extract P(A|x),P(B|y), and P(C|z).

I'm not sure what you're talking about here. I never use P(A & B & C|w) or P(A & B & C|x,y,z) in my proof at all.

After substituting the probabilities from the physical situation "xyz" into an expression derived for the physical situation "w", the violation you obtain tells you clearly that the probabilities were not the same as you naively assumed.

If we use a bunch of different assumptions to derive a contradiction, then at least one of them is wrong. You think that what's wrong is my equating of terms. But I gave an argument for why I'm equating terms at the beginning of this post, so the question becomes, where do I go wrong in that argument?

And the impossibility of deriving a similar expression directly in physical situation "xyz" without first going through "w" is a big hint that your assumptions are wrong.

Or a hint that counterfactual definiteness is an essential assumption of Bell's theorem.

To summarize:
1) There are two scenarios/physical sittuations involved in this discussion:

Scenario "xyz", involving the three probabilities:

P(A|x) = what we would get if we measure system "x" at angles (a,b)
P(B|y) = what we would get if we measure system "y" at angles (b,c)
P(C|z) = what we would get if we measure system "z" at angles (a,c)​

Scenario "w", involving the three probabilities:

P(A|w) = what we would get if we measure system "w" at angles (a,b)
P(B|w) = what we would have gotten had we measured system "w" at angles (b,c) instead of (a,b)
P(C|w) = what we would have gotten had we measured system "w" at angles (a,c) instead of (a,b)​

Scenarios "xyz" and "w" are just (b) and (c) in the notation you were using earlier, right?

2) Scenario "xyz" is different from Scenario "w". The three probabilities in "w" are not equal to the three probabilities from "xyz".

Of course this is the part I disagree with. See my argument at the top of this post for why.

So what part of this argument don't you understand, or do understand but disagree with and why?

Part 2 is where we part ways.

 

[billschnieder]

OK, then that's what we need to focus on. Again, I still want to know whether you agree or disagree with the answers to my two questions are always the same. Because if they are always the same, then I can show that P(A|x) must always equal P(A|w).

Are you a real person!? Have you read anything I've written in this thread, at all. Do you understand any of it?
I have already shown you that they must not always be equal. I have used several examples, with quotes from Bell himself to explain to you that they are not equal in the two scenarios. Yet you ask me again if I agree that they are always equal.

 

[lugita15]

Are you a real person!? Have you read anything I've written in this thread, at all. Do you understand any of it?
I have already shown you that they must not always be equal. I have used several examples, with quotes from Bell himself to explain to you that they are not equal in the two scenarios. Yet you ask me again if I agree that they are always equal.

As far as I can tell the only response you've given to the argument involving the two questions (which I just restated in my previous post) was in post #262, where you give the example of Bernoulli's urn. But as I said, Bernoulli's urn is an illustration of something involving probabilities, and my two questions do not deal with probability at all.

The two Bell quotes above are also irrelevant to this point: one talks about how the sum of two measurements is not the measurement of the sum, and the other quote talks about probability.

If anything else you've said was a response to my two questions, then I must have missed it, so please tell me if there are any other examples or arguments you have against my two questions having the same answer.

 

[DevilsAvocado]

OK, then that's what we need to focus on. Again, I still want to know whether you agree or disagree with the answers to my two questions are always the same. Because if they are always the same, then I can show that P(A|x) must always equal P(A|w).

Lugita, you aren’t getting anywhere with this. Check out my post in #274 and Bill’s reply in #275 and you’ll see that Bill can’t keep two balls in the air at once. He – for real – thinks that:

• QM is wrong.
• QM measurement outcome is “definite” and “only appears random”.
• QM measurement of a single entangled photon is not 100% random 50/50 +1/-1.
• Bell's theorem is not about statistical correlations, but probabilities for a single photon.
• The cos^2(a - b) rule is not about the relative angle between Alice & Bob.
• Instead we can apply cos^2(θ) for any single photon/angle outcome.
• We can use Bayesian probability theory to investigate single photon properties, before measurement.
• At lot of measurements (hundreds/thousands) is not needed to get the (statistical) correlations.
• Experimental outcome/statistics of EPR-Bell test experiments is wrong.
• The definite property of LHV is not obligatory.
• Bell's inequality is not an upper limit for Local Realism only.
• To test QM against Bell's inequality, QM needs to posses the same basic properties as Local Realism, otherwise the test is groundless, i.e. Local Realism and QM has to be on the same footing, to be rigorously tested against Bell's inequality.

And AFAIK, Bill seems to think that QM and Local Realism is really the same thing, except for some “mysticism buffoons” [i.e. the rest of the world] who got it all wrong...

This discussion will take you nowhere.

 

[billschnieder]

Because if they are always the same, then I can show that P(A|x) must always equal P(A|w)..

Are you for real? You want to argue that two probabilities are always the same, yet you do not want me to mention probabilities when explaining to you that they are not always equal.

 

[billschnieder]

But as I said, Bernoulli's urn is an illustration of something involving probabilities, and my two questions do not deal with probability at all.

Contrary to your claims, they in fact do because you are using them to claim that two probabilities are the same. Isn't that what this is all about. I don't care about your two statements, I go directly to your claim that the probabilities are equal in the two scenarios and I show you that they are not. Besides, I have given you examples that have nothing to do with probability. For example:

T_b is just a fixed number like 5 in the physical situation in which you are only putting on socks. T_b is just a fixed number like 8, in the physical situation in which we are putting on shoes and then socks. T_b in in the first physical situation, is not the same value as T_b in the second physical situation, even though in both cases T_b is still the time it takes to put on socks.

The two Bell quotes above are also irrelevant to this point

That is because you do not understand the argument. They are very relevant. You are making the same tacit assumption Bell refers to in his second quote, that a measurement being made in one scenario will always give the same result in a different scenario, you naively assume that probabilities from scenario "xyz" can be used in scenario "w". You are making the same error, that Bell talks about in the first quote, because you assume that the sum of two noncommuting observables P(A|w) + P(B|w) can be obtained by combining trivially the results of separate observations on the two terms P(A|x) + P(B|y)

 

[billschnieder]

Consider a photon before it is measured. By counterfactual definiteness, for every angle setting θ1, there exists a definite, predetermined answer to the question "What result would you get if the photon were measured at angle θ1?". Now if the experimenter can freely choose his measurement decision, then the predetermined answer to this question cannot depend on what he is going to do, so it cannot depend on the angle θ2 he is going to measure at, whether θ1=θ2 or θ1≠θ2. Do you agree or disagree with that?

Ok for the last time, please pay attention carefully.
1) You do not have a single angle, you have two angles (a,b) for the photon pair, where your θ1 = (a-b). And you have two angles (a,c) where your θ2 = (a-c).
2) You are not considering each result independtly, you are considering them jointly in the same expression, in fact adding them together "answer_at(a,b) + answer_at(a,c)" ...
3) You are claiming that in the expression you have which includes both answers, the answer at (a,c) can not depend on the answer at (a,b).

However, NOTE: we are talking about the same photon pair measured at (a,b) and counterfactually measured at (a,c). Let us call the photons p1 and p2 where p1 is sent to Bob and p2 is sent to Alice. In the factual measurement at (a,b), Bob measures p1 at angle a, and Alice measures p2 at angle b. Now in the counterfactual measurement, Bob is not doing anything differently that what he already did. ONLY Alice's measurement would have been different because she is now doing the measurement at angle "c" instead of "b" at which she measured previously.

Therefore, contrary to what you naively thought, the result you would have obtained counterfactually at (a,c) does depend on the result you obtained for (a,b) for the simple reason that the same photon can not counterfactually give you something different when measured at "a", than what was already obtained at "a". In other words, Bob's result at "a" for the (a,b) measurement, constraints Bob's result at "a" for the counterfactual measurement (a,c). Get it?

Let us assume that the outcomes for each photon can be either + or -. Let us now replace the angles in the expression "answer_at(a,b) + answer_at(a,c)" with the corresponding answers for the counterfactual situation. If the measurement at (a,b) gave the answer (+,-), we now have "answers(+,-) + answers(+,c)". Do you notice that because we already have the answer for the first term, we therefore already have part of the answer for the second term, since we are talking about the same photon? Do you realize that although the result (a,c) is counterfactual to (a,b), it is in fact only Alice's part of the measurement that is counterfactual? Do you realize that to be consistent, Bob's part of the answer can not change from the factual?

Now what about three different photon pairs where the (a,b) measurement is performed on (p1, p2) and the (a,c) measurement is performed on a different pair of photons (p3, p4). In this case, we do not have any counterfactual measurements. Bob's result for p3 is not constrained by Bob's result for p1, because they are different photons which are free to have different results.

In other words, in scenario "xyz", there are no logical relationships between results from the sets of photons "x", "y" and "z" because they are three different sets. However in the scenario "w", we are talking about the same set, and therefore there are logical relationships between measurements on that set.

 

[billschnieder]

Lugita, you aren’t getting anywhere with this. Check out my post in #274 and Bill’s reply in #275 and you’ll see that Bill can’t keep two balls in the air at once. He – for real – thinks that:
...

Obviously, you are wrong and completely off base.

 

[BenjaminTR]

[...]

Therefore, contrary to what you naively thought, the result you would have obtained counterfactually at (a,c) does depend on the result you obtained for (a,b) for the simple reason that the same photon can not counterfactually give you something different when measured at "a", than what was already obtained at "a". In other words, Bob's result at "a" for the (a,b) measurement, constraints Bob's result at "a" for the counterfactual measurement (a,c).

[...]

If I understand, the independence lugita15 is talking about is independence between what properties a particle has when it becomes entangled and what measurement we decide to make. Since the LHV theory is that properties of the particles determine measurement outcomes, we can use measurement outcomes as proxy for properties.
Let A+ be the proposition that the particle would measured + in orientation A, and x be the event that we actually measure in orientation A. Let B+ be the proposition that particle would be measured + in orientation B and y be the event that we actually measure orientation B. Let w be the event that we measure the particle at all. Independence says that P(A+|x) = P(A+|y) = P(A+|w). Same for B+.
In the quoted passage, you are talking about P(B+|A+ & x). No one assumes B+ is independent of A+ & x, so your argument here does not undermine lugita15's or Herbert's proofs.

 

[billschnieder]

If I understand, ...
and x be the event that we actually measure in orientation A. ...
and y be the event that we actually measure orientation B.
Let w be the event that we measure the particle at all.

...

Huh? You have completely misunderstood the argument. Please go back and re-read the thread from page 15, carefully in order to understand what we are discussing.

Le us represent the sets of photons as one set "w" for scenario (c), and three different sets ("x", "y","z") for scenario (b). The the probabilities for (c) are more accurately represented as P(A|w), P(B|w), P(C|w). ...

However the probabilities for scenario (b) are more accurately represented as P(A|x), P(B|y), P(C|z).

 

[lugita15]

1) You do not have a single angle, you have two angles (a,b) for the photon pair, where your θ1 = (a-b). And you have two angles (a,c) where your θ2 = (a-c).

I was talking about individual photons, and then I was going to move to photon pairs. But if you want to start with photon pairs that's fine.

3) You are claiming that in the expression you have which includes both answers, the answer at (a,c) can not depend on the answer at (a,b).

What I'm claiming is that before measurement, the answer you would get at (a,c) is entirely predetermined, and it can't depend on whether you're going to measure at (a,b) or (a,c).

However, NOTE: we are talking about the same photon pair measured at (a,b) and counterfactually measured at (a,c). Let us call the photons p1 and p2 where p1 is sent to Bob and p2 is sent to Alice. In the factual measurement at (a,b), Bob measures p1 at angle a, and Alice measures p2 at angle b. Now in the counterfactual measurement, Bob is not doing anything differently that what he already did. ONLY Alice's measurement would have been different because she is now doing the measurement at angle "c" instead of "b" at which she measured previously.

I agree with this.

Therefore, contrary to what you naively thought, the result you would have obtained counterfactually at (a,c) does depend on the result you obtained for (a,b) for the simple reason that the same photon can not counterfactually give you something different when measured at "a", than what was already obtained at "a". In other words, Bob's result at "a" for the (a,b) measurement, constraints Bob's result at "a" for the counterfactual measurement (a,c). Get it?

The thing is, for any given photon pair, both what you would get at (a,b) and what you would get at (a,c) is entirely predetermined beforehand, before you make the measurement at (a,b). Do you agree or disagree with that with that?

The point is, that the predetermined answer to the question "What would you get at (a,c)" can't depend on what you're going to measure later on, because (by assumption) you have free will and can choose to measure anything you want.

Do you realize that although the result (a,c) is counterfactual to (a,b), it is in fact only Alice's part of the measurement that is counterfactual? Do you realize that to be consistent, Bob's part of the answer can not change from the factual?

Yes, this is all fine.

In other words, in scenario "xyz", there are no logical relationships between results from the sets of photons "x", "y" and "z" because they are three different sets. However in the scenario "w", we are talking about the same set, and therefore there are logical relationships between measurements on that set.

The point is, for any given photon pair, what you would get at (-30,0) is entirely independent of whether you're actually going to measure at (-30,0) or (0,30), because at the time that the answer to the question "What would you get at (-30,0)?" is determined, the experimenter hasn't yet made up his mind about whether he'll measure at (-30,0) or (0,30).

So the question of what you would get at (-30,0) is independent of whether the photon pair is going to be in set x or set y, and thus the percentage of pairs in set x for which you would get R1 at (-30,0) is the same as the percentage of pairs in set y for which you would get R1 at (-30,0). Do you agree or disagree with that?

 

[billschnieder]

What I'm claiming is that before measurement, the answer you would get at (a,c) is entirely predetermined

I agree with this. I also agree that the outcome of any measurement is predetermined no matter the angle.

, and it can't depend on whether you're going to measure at (a,b) or (a,c).

I don't think you fully understand the implication of this claim. It is not correct to say the outcome at (a,b) does not depend on the angle (a,b) or the outcome at (a,c) does not depend on the angle (a,c), since those settings are part of the set-up which produces the outcome. Probably what you are trying to say is that the outcome of the single photon measured at angle "a" in the (a,b) pair must be exactly the same as the outcome of the same photon measured at angle "a" in the (a,c) pair. In other words, the outcome for that particular single photon should not change from (a,b) to (a,c) just because it's sibling was measured at "c" in (a,c) instead of "b" in (a,b). So Alice's choice of setting should not influence Bobs outcome for the same setting "a" and the exact same photon. If this is what you mean, then I agree. But if you mean that the outcome at (a,b) can not depend on the angle pair (a,b) then I disagree with that.

The thing is, for any given photon pair, both what you would get at (a,b) and what you would get at (a,c) is entirely predetermined beforehand, before you make the measurement at (a,b). Do you agree or disagree with that with that?

I agree.

The point is, that the predetermined answer to the question "What would you get at (a,c)" can't depend on what you're going to measure later on, because (by assumption) you have free will and can choose to measure anything you want.

No. I disagree with that. Just because it is predetermined to produce the result at (a,c) doesn't mean it can produce the result without (a,c). Without the setting (a,c) you will not get the result at (a,c) so how can you say the result obtained at (a,c) does not depend on what you actually choose (a,c)? What is predetermined is the condition that "if you set the device to (a,c), you will obtain such and such result". It doesn't make much sense to say the result you obtain at (a,c) does not depend on the setting, which is required to obtain the result! Free will has nothing to do with it. You are free to choose a different setting, you will get a different result. But whenever you freely choose (a,c) you will get the result predetermined for (a,c). Probably what you are trying to say here is that the result for the single photon measured at angle "a", should not depend on whether you measured the pair at (a,c) or you measured (a,b). Again, that means Bob's result for the same photon measured at the same angle should not depend on what angle Alice chose to measure it's sibling photon at. If this is what you mean, I agree.

The point is, for any given photon pair, what you would get at (-30,0) is entirely independent of whether you're actually going to measure at (-30,0) or (0,30), because at the time that the answer to the question "What would you get at (-30,0)?" is determined, the experimenter hasn't yet made up his mind about whether he'll measure at (-30,0) or (0,30).

See explanation above. The results predetermined for a given angle pair can not be independent of the angle pair for which it is predetermined.

 

[billschnieder]

So the question of what you would get at (-30,0) is independent of whether the photon pair is going to be in set x or set y, and thus the percentage of pairs in set x for which you would get R1 at (-30,0) is the same as the percentage of pairs in set y for which you would get R1 at (-30,0). Do you agree or disagree with that?

No! The same photon can not belong to two different sets. If a specific photon is predetermined to give a result at -30, then obviously the result when that specific photon and it's sibling are measured at (-30, 0) respectively CANNOT CONTRADICT the result when that same photon and it's sibling were instead measured at (-30, 30). In other words, the same photon can not be predetermined to have different results at the same angle, however, two different photons can be predetermined to have opposite results at the same angle. Do you agree with this?

Remember I'm explaining why the results from three different sets "x", "y", "z" are not the same as the results from one single set "w".

Scenario "xyz"
A_x: results from measuring photon set x at angles (a,b)
B_y: results from measuring photon set y at angles (a,c)
C_z: results from measuring photon set z at angles (c,b)

Scenario "w"
A_w: results from measuring photon set x at angles (a,b)
B_w: results from measuring photon set y at angles (a,c)
C_w: results from measuring photon set z at angles (c,b)

Your claim is that the results from Scenario "xyz" are the same as the results from scenario "w", ie, A_x = A_w, B_y = B_w and C_y = C_w.

For scenario "w", the results at angle "a" when measured at the angle pair (a,b) must be the same as the results for angle "a" when measured at angle pair (a,c) and the results for angle "b" when measured at angle pair (a,b) must be exactly the same as the results for angle "b" when measured at angle pair (c,b) etc.. for "c" too. This is the case because we are dealing with the same set of photons "w". There is therefore a cyclic logical relationship between the results at (a,b), (a,c) and (c,b).

However, for scenario "xyz", since each result is obtained from a different set of photons, the results at "a" in the (a,b) pair can be different from the results for "a" in the (a,c) pair and the same for "b", and "c". There is therefore no cyclic logical relationship between the results at (a,b), (a,c) and (c,b) in this scenario.

Note. In all of this explanation, the results for a specific photon measured angle "a" does not depend on the angle at which it's sibling was measured at, and all the results are predetermined.

Another way to look at this is to say that results drawn from a single set "w" have additional constraints between them that results drawn from three different sets "x", "y", "z" do not have. These are the constraints that Boole worked on 150 years ago. So knowing the result (a,b) from "w" give's you insight into the properties of "w" which might be important for the results (a,c) and (c,b). However knowing (a,b) from the set "x", gives you no insight into the properties of "y" which are important for the results (a,c) or the results (c,b) from another set z. Because the sets are all different.

This is why I disagree with your claim is that the results from Scenario "xyz" are the same as the results from scenario "w", ie, A_x = A_w, B_y = B_w and C_y = C_w.

 

[DevilsAvocado]

In other words, the same photon can not be predetermined to have different results at the same angle,

Pretty sweet, you’ve summarized Bell’s theorem in a single line.

These are the constraints that Boole worked on 150 years ago.

And many will draw the logical conclusion that Boole was probably not completely familiar with the life of QM photons (that wasn’t ‘invented’ until 1926), hence Boole’s work will almost certainly say very little (if anything?) about the properties of unmeasured QM photons.

This of course, is just a personal speculation.

 

[billschnieder]

Pretty sweet, you’ve summarized Bell’s theorem in a single line.

As usual, you say much about stuff you know little about. This is not Bell's theorem.

And many will draw the logical conclusion that Boole was probably not completely familiar with the life of QM photons (that wasn’t ‘invented’ until 1926)

What you do not know is that Bell's inequalities were originally discovered by Boole. Boole found that one of the constraints that properties from a given set must obey to be logically consistent was Bell's inequalities. Bell re-discovered the same rules without knowing that Boole had already discovered them a century earlier. One thing Boole also found out, was that properties from a single set can never violate the inequalities but properties from different sets can.

Remember this article I mentioned earlier:

George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle
ITAMAR PITOWSKY, Br J Philos Sci (1994) 45 (1): 95-125. doi: 10.1093/bjps/45.1.95

One thing should be clear at the outset: none of Boole's conditions of possible experience can ever be violated when all the relative frequencies involved have been measured in a single sample. The reason is that such a violation entails a logical contradiction. For example, suppose that we sample at random a hundred balls from an urn. Suppose, moreover that 60 of the balls sampled are red, 75 are wooden and 32 are both red and wooden. We have p1=0.6, p2= 0.75, p12 = 0.32. But then p1+p2-p12 > 1. This clearly represents a logical impossibility, for there must be a ball in the sample (in fact three balls) which is 'red', is 'wooden', but not 'red and wooden'; absurd.
Similar logical absurdities can be derived if we assume a violation of any of the relevant conditions, no matter how complex they appear to be. This is the reason for the title 'conditions of possible experience'. In case we deal with relative frequencies in a single sample, a violation of any of the relevant Boole's conditions is a logical impossibility.
But sometimes, for various reasons, we may choose or be forced to measure the relative frequencies of (logically connected) events, in several distinct samples. In this case a violation of Boole's conditions may occur.

Bell's inequality is simply one of "Boole's conditions of possible experience".

 

[billschnieder]

The point is, that the predetermined answer to the question "What would you get at (a,c)" can't depend on what you're going to measure later on, because (by assumption) you have free will and can choose to measure anything you want.

It occurs to me that when you use the word "depend", you may be assuming causal dependence. You are thinking that since what happens later can not change what happens earlier, it means what happens earlier cannot "depend" on what happens later. But this is just wrong, dependence does not have to mean causation, that is why I gave you the Bernouli urn example. Two events can be causally independent and yet logically dependent.

 

[DevilsAvocado]

This is not Bell's theorem.

I didn’t expect you to recognize it.

Bell's inequalities were originally discovered by Boole.

There’s only one “little” glitch, that you constantly refuse to discuss.

We can use classical logic to calculate the properties of unmeasured Local Hidden Variables that acts the classical way, but I sure hope you are not saying that Boole – 150 years ago – developed a “quantum logic” that makes it possible to deal with superposition, entanglement, uncertainty principle, etc??

Besides this erroneous approach – you’re arguing that a single QM photon/pair could give you the correct probabilities/correlations, when we all know that a single QM outcome is always 100% random, no matter what angle. We need hundreds of measured entangled pairs to get the correlations, which you refuse to discuss or admit.

The stochastic nature of QM is fundamental – and absolutely not only “appears random” – to quote your own catastrophic lack of knowledge.

Hence, Boole works excellent for LHV in every instance – before & after measurement – and it even works for a single “LHV photon” outcome, because they must be DEFINITE/PREDETERMINED. However, Boole can’t say anything about QM photons before measurement, and a single QM measurement outcome is always 100% random.

This is what I mean when I say that you can’t keep two balls in the air at once. You refuse to see there’s a BIG difference between unmeasured “LHV photons” and unmeasured QM photons, and not even dear old Boole can help you get them on same footing.

The only thing you can do is to deal with measurement STATISTICS for hundreds/thousands of QM photons, and there goes your “personal theory” down the drain, and this is why you react the way you do.

I’m not expecting any other reaction this time.

 

[billschnieder]

There’s only one “little” glitch, that you constantly refuse to discuss.

We can use classical logic to calculate the properties of unmeasured Local Hidden Variables that acts the classical way, but I sure hope you are not saying that Boole – 150 years ago – developed a “quantum logic” that makes it possible to deal with superposition, entanglement, uncertainty principle, etc??

Besides this erroneous approach – you’re arguing that a single QM photon/pair could give you the correct probabilities/correlations, when we all know that a single QM outcome is always 100% random, no matter what angle. We need hundreds of measured entangled pairs to get the correlations, which you refuse to discuss or admit.

The stochastic nature of QM is fundamental – and absolutely not only “appears random” – to quote your own catastrophic lack of knowledge.

Hence, Boole works excellent for LHV in every instance – before & after measurement – and it even works for a single “LHV photon” outcome, because they must be DEFINITE/PREDETERMINED. However, Boole can’t say anything about QM photons before measurement, and a single QM measurement outcome is always 100% random.

This is what I mean when I say that you can’t keep two balls in the air at once. You refuse to see there’s a BIG difference between unmeasured “LHV photons” and unmeasured QM photons, and not even dear old Boole can help you get them on same footing.

The only thing you can do is to deal with measurement STATISTICS for hundreds/thousands of QM photons, and there goes your “personal theory” down the drain, and this is why you react the way you do.

I’m not expecting any other reaction this time.

There is nothing coherent here to respond to. You probably think you are making a lot of sense but somebody needs to tell you that this is nonsense. It is difficult to discuss with somebody who does not even understand the basics of the topic let alone thinking rationally about any topic.

 

[billschnieder]

Scenario "w"
A_w: results from measuring photon set x at angles (a,b)
B_w: results from measuring photon set y at angles (a,c)
C_w: results from measuring photon set z at angles (c,b)

What I meant to say here is:
A_w: results from measuring photon set w at angles (a,b)
B_w: results from measuring photon set w at angles (a,c)
C_w: results from measuring photon set w at angles (c,b)

 

[DevilsAvocado]

Lugita15, ask Buffalo* Bill exactly what set x and set w represents in terms of number of photons and exactly where in the process – before/after measurement – he perform his magical pet theory.


(and don’t forget to wear raincoat against the flood of personal insults)


*droppings

 

[billschnieder]

Lugita15, ask Buffalo* Bill exactly what set x and set w represents in terms of number of photons and exactly where in the process – before/after measurement – he perform his magical pet theory.

In Bell-test experiments, to measure the probability of mismatches at angle pair (a,b), the Alice sets her polarizer at angle "a", Bob sets his to angle "b", the source is turned on and hundreds of thousands of photons or even millions of photons go through. Everytime a photon goes through Alice's device she records the detector which fired, either a +1 or a -1 for D+ or D- respectively. At the end of the "run", Alice has a long list consisting of +'s and -'s in random order and Bob has a corresponding list. The lists are brought together and you have something that looks like:

ab
-+
++
+-
--
... etc
for as long as there were photon pairs produced, where each row corresponds to results from one single photon pair. It is this list that is used to calculate the C(a,b), or probability of mismatches, or any other statistic you may want to calculate for the angle pair (a,b) in the Bell test experiment. Let us call the set of photons which produced this list "x".

Each photon can only be measured once, being destroyed in the process. Therefore to measure at the angle pair (a,c), a different set of photons has to be used, in a similar manner as above. Let us call this set "y". And similarly, yet a different set "z" is needed to measure at the angle pair (b,c).

Each of the sets "x", "y", "z", is a different set of photons. In other words, no photon belongs to more than one set. No photon is common between those sets because no photon was measured more than once.

An alternative method of doing the experiment involves random switching of the angles by Alice and Bob. However, in the end, to calculate statistics for (a,b), they select only those photons measured when Alice had her device at angle "a" and Bob had his at angle "b". it is this set that we called "x", and the other two are "y" and "z".

Therefore, it should be obvious that for any Bell test experiment the correlations, C(a,b), C(b,c), C(a,c) are each measured on a distinct set of photons. If you prefer, P(A|x), P(B|y), P(C|z) are each measured on a different set of photons. In other words, we have 3 different sets of photons (x, y, z). This is what is referred to in the above discussion as scenario "xyz" and it refers to the way experimental correlations are measured.

The QM predictions are similarly made for separate sets of photons, and not surprisingly, the QM predictions match the experimental results.

 

[DrChinese]

...The QM predictions are similarly made for separate sets of photons, and not surprisingly, the QM predictions match the experimental results.

That is ALL that a Bell test is. Because Bell says that LR theories cannot match QM results, ergo at least one of QM or LR theories are wrong. It isn't QM, as experiment shows.

What I'm claiming is that before measurement, the answer you would get at (a,c) is entirely predetermined, and it can't depend on whether you're going to measure at (a,b) or (a,c).

Glad to see you finally admit that LR requires that for any photon, the results of a measurement must be predetermined. Now all you have left to do is draw the Bell conclusion, and we will let you join the rest of us.