Scholarpedia article on Bell's Theorem

[billschnieder]

Bill, I think we have to agree to disagree about the reasonableness of the "no conspiracy" assumption. Your most recent (long) message does help me (to some extent) to understand what you are and aren't bothered by -- namely that it's the experiment "not living up to" the math as opposed to the math itself. But I still think you are profoundly wrong in how you are thinking about what the experiments are and/or should be. You are thinking of them as attempts to somehow recapitulate the steps in the derivation of the inequality.

No! I'm simply pointing out to you that you can not derive an inequality from ONLY 4 unique terms and reasonably expect an experiment which gives you 8 unique terms to satisfy the inequality! I'm simply pointing out that 2 inches + 2cm ≠ 4 inches, violates 2inches + 2inches = 4inches due to a simple violation of the mathematical definition of terms implicit in the equation, not due non-locality or any other spooky business. It is unreasonable to conclude that locality is ruled out without first demonstrating that the terms from QM or Experiment, correspond to the same terms you have in the inequallity. I believe I have have explained convincingly that they aren't.

Of course you are free to continue believing that they are just because of "ordinary scientific reason", whatever that means. But you have not provided any justification, let alone proof that they are.

The truth is that the experiments simply measure the correlations, while the theorem is a proof that (under certain assumptions) the correlations are constrained in a certain way. There is simply no reason these two things should "look like each other" in anything like the way you seem to be demanding.

There is no reason why "2inches" should look like "2cm" either, nor is there any reason why apples should look like oranges.

Thanks both for the stimulating discussion!

Thank you too. I hoped you will not bow out so soon.

 

[ttn]

No! I'm simply pointing out to you that you can not derive an inequality from ONLY 4 unique terms and reasonably expect an experiment which gives you 8 unique terms to satisfy the inequality!

Yes, I now understand that that's your worry. But I think it's just completely wrong headed and baseless. You somehow think that the experiments must sort of perfectly recapitulate all the steps in the derivation, but there is simply no reason at all it should work like that. Instead, the experiment should reflect the *assumptions* that go into the derivation -- in particular, the settings on each side should be made "at the last possible second" so that the kind of locality assumed in the derivation will apply if locality is true, and the ways those settings are made should be sufficiently independent of stuff going on at the source that one can accept that the "no conspiracies" assumption is reasonable.

It is unreasonable to conclude that locality is ruled out without first demonstrating that the terms from QM or Experiment, correspond to the same terms you have in the inequallity. I believe I have have explained convincingly that they aren't.

Well *of course* there's a sense in which "they aren't" -- the QM predictions, and also the experimental results, *don't respect the inequality*. That is the whole point! But I know it's not what you meant exactly. But I think you are coming at this all backwards. The goal is not to make the derivation somehow "reflect" what is happening in the experiments and/or in QM. The goal rather is to make the derivation respect the assumptions of "locality" and "no conspiracies" (and with no other assumptions). Then, when we do the experiments and find that the inequality is violated, we have to conclude that one of those assumptions is in fact false, i.e., does not apply to the actual experiment!

I hoped you will not bow out so soon.

More "winding down" than "bowing out". But it was becoming apparent that further intense discussion would not be likely to be fruitful.

 

[billschnieder]

Yes, I now understand that that's your worry. But I think it's just completely wrong headed and baseless. You somehow think that the experiments must sort of perfectly recapitulate all the steps in the derivation, but there is simply no reason at all it should work like that. Instead, the experiment should reflect the *assumptions* that go into the derivation -- in particular, the settings on each side should be made "at the last possible second" so that the kind of locality assumed in the derivation will apply if locality is true, and the ways those settings are made should be sufficiently independent of stuff going on at the source that one can accept that the "no conspiracies" assumption is reasonable.

I think I have explained myself clearly enough and I think you have understood, although it appears you are still pre-disposed to rejecting the argument without having a genuine rebuttal to it. So I will wind this down as well with the following questions:

1. Are the terms in the CHSH independent terms or are they cyclically dependent on each other?
2. Are the terms calculated from QM and used to compare with the CHSH independent terms or cyclically dependent on each other.
3. Are the terms calculated from experimental results independent terms or are they cyclically dependent.

If you are reasoning correctly, and being honest with yourself, your answers will be

(1) Cyclically dependent
(2) Independent
(3) Independent

Now you claim that the reason the CHSH is violated is because QM is non-local and the experiments are non-local and the CHSH is local. But your answers to those questions will show that you have an additional assumption in the CHSH ie "cyclic dependency between terms" which is violated by both QM and the experiments. You have provided no argument why this is not a more reasonable explanation of the violation than non-locality.

Well *of course* there's a sense in which "they aren't" -- the QM predictions, and also the experimental results, *don't respect the inequality*.

QM and the experiments *don't respect the assumption of cyclic dependency between term* which is required to derive the inequality. You don't need to take my word for it. I have given two simple examples in which violation of cyclic dependency led to violation of the inequalities even though the situations were demonstrably locally causal. This should be enough for anyone who is interested in the truth. At the very least, it should give you pause the next time you proclaim the demise of locality.

But I think you are coming at this all backwards. The goal is not to make the derivation somehow "reflect" what is happening in the experiments and/or in QM.

But I just explained to you why the derivation does not "reflect" what is happening in the experiments and/or in QM! You may not like it, you may call it baseless and wrong but you have not provided any rebuttal that has stood up. You are the one who is clearly wrong.

The goal rather is to make the derivation respect the assumptions of "locality" and "no conspiracies" (and with no other assumptions). Then, when we do the experiments and find that the inequality is violated, we have to conclude that one of those assumptions is in fact false, i.e., does not apply to the actual experiment!

This is a cop-out. If that is what your goal was, you wouldl have started out with 8 unique functions and derived your inequality using those. Using 4 unique functions when you know fully well that experiments can only measure 8 unique functions is cheating not science. Unfortunately, many are continuously being misled by this.

In fact, cyclic dependency is the ONLY assumption required to derive the inequality as Boole showed, not locality or anything else. I encourage you to look up Booles conditions of possible experience, or Vorob'evs cyclicities.

Here is how to derive the inequalities without any physical assumption. This is how Boole did it:

Define a boolean variable v such that v = 0,1 and \overline{v} = 1 - v
Now consider three such boolean variables x, y, z

It therefore follows that:

1 = \overline{xyz} + x\overline{yz} + x\overline{y}z + \overline{x}y\overline{z} + xy\overline{z} + \overline{xy}z + \overline{x}yz + xyz
We can then group the terms as follows so that each group in parentheses can be reduced to products of only two variables.
1 = \overline{xyz} + (x\overline{yz} + x\overline{y}z) + (\overline{x}y\overline{z} + xy\overline{z}) + (\overline{xy}z + \overline{x}yz) + xyz
Performing the reduction, we obtain:
1 = \overline{xyz} + (x\overline{y}) + (y\overline{z}) + (\overline{x}z) + xyz
Which can be rearranged as:
x\overline{y} + y\overline{z} + \overline{x}z = 1 - (\overline{xyz} + xyz)
But since the last two terms on the RHS are either 0 or 1, you can write the following inequality:
x\overline{y} + y\overline{z} + \overline{x}z \leq 1
This is Boole's inequality. In Bell-type situations, we are interested not in boolean variables of possible values (0,1) but in variables with values (+1, -1) so we can define three such variables a, b, c where a = 2x - 1 , b = 2y - 1 and c = 2z -1, and remembering that
\overline{x} = 1 - x
and substituting in the above inequality maintaining on the LHS only terms involving products of pairs, you obtain the following inequality
-ab - ac - bc \leq 1
from which you can obtain the following inequality by replacing a with -a.
ab + ac - bc \leq 1

and then you can combine the above two inequalities into
|ab + ac| \leq 1 + bc
which is a Bell-type inequality.

Note that the only assumption required here has been to suppose that we have three two-valued variables x,y,z. No locality, or other physical assumption is required to obtain the inequalities. It is obvious now why Bell or CHSH arrived at the same inequalities like Bell. They happened to be dealing with 3 bi-valued variables (4 in the case of CHSH) and by pushing some completely unneccessary math they fool themselves into thinking locality or no-conspiracy, or realism or any other physical assumption is required.

So then what do we make of violations of this inequality when obviously there is no other assumptions required to derive it, than "trival algebra of 3 two valued variables"? Violation simply means violation of trivial algebra of 3 two valued variables. As I have explained convincingly, the experiments violate it because:

1 - They are not dealing with 3 (or 4 for CHSH) two valued variables, they are dealing with 6 (or 8 for CHSH).
2 - Because of (1), they do not have 3 ( or 4 for CHSH) cyclically dependent terms. They have 4 independent terms.

And then they say, "Oh but experiments confirm QM". Of course, QM predictions are for independent terms and experiments produce independent terms so there is no surprise that they agree with each other and disagree with the inequality which requires cyclically dependent terms.

More "winding down" than "bowing out". But it was becoming apparent that further intense discussion would not be likely to be fruitful.

I suspect if you had a genuine rebuttal, you would present it.

 

[ttn]

1. Are the terms in the CHSH independent terms or are they cyclically dependent on each other?

I don't understand what you mean by "cyclically dependent".

(I don't understand exactly what you think you mean by "independent" either for that matter. Of course there are senses in which the 4 terms, as calculated say in QM, are independent, and senses in which they aren't.)

you have an additional assumption in the CHSH ie "cyclic dependency between terms"

As I said, I don't understand what you even mean by this "cyclic dependency", but -- assuming you mean to be referring to some property that is actually there -- it is *not* an *additional assumption* but rather something that *follows* from the assumptions that are *actually made*. Otherwise you'd be able to tell me where the mistake in the mathematical derivation is.

QM and the experiments *don't respect the assumption of cyclic dependency between term* which is required to derive the inequality.

There's no such assumption. We are extremely clear and explicit about things that are being assumed. You're objecting to something (I don't fully understand what) "downstream". But all the math that gets you down that stream is trivial. Tell me what's wrong with the actual premises, or with the reasoning.

In fact, cyclic dependency is the ONLY assumption required to derive the inequality as Boole showed, not locality or anything else.

It is undoubtedly true that other assumptions (than the ones we use) can lead to the same conclusion, Bell's inequality. For example, any physics textbook will show how to derive the inequality from the assumption of "local deterministic non-contextual hidden variables". (See our section 8 for some discussion.) Perhaps it's also true that the inequality can be derived from "cyclic dependency". Who cares? None of those alternative starting points have anything like the status of "locality" -- that, I take it, is your point. But if

A --> C

and

B --> C

and you find out C is false, it's not like you get to *choose* which derivation of C you like best, and hence which of A or B you would prefer to reject.

 

[billschnieder]

I don't understand what you mean by "cyclically dependent".

Seriously!?

|ab + ac| - bc <= 1, Bell's 3-term inequality for example
Cyclic Dependency means every product shares one term with another product (ie ab, ac, bc)
for ONLY three distinc terms a,b,c.
NEVER violated unless due to mathematical or logical error!
Proof:
a,b,c = (+1,+1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a,b,c = (+1,+1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a,b,c = (+1,-1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a,b,c = (+1,-1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a,b,c = (-1,+1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a,b,c = (-1,+1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a,b,c = (-1,-1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a,b,c = (-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
(Note that only 8 distinct possibilities exist for combinations of the values of a,b,c)
(Note also that the inequality is NEVER violated, NEVER!
It is a logical/mathematical error to expect the inequality to be satisfied by 6 distinct terms
|a1b1 + a2c2| - b3c3 <= 1, is WRONG! There is no cyclicity present, UNLESS a1=a2 and b1=b3 and c2=c3
Proof:
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,+1,+1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,+1,+1,-1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,-1,+1,+1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,-1,+1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,+1,+1,+1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,+1,+1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,-1,+1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,-1,+1,-1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,+1,-1,+1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,+1,-1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,-1,-1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,+1,-1,-1,-1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,+1,-1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,+1,-1,-1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,-1,-1,+1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,-1,+1,-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,+1,+1,+1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,+1,+1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,-1,+1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,-1,+1,-1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,+1,+1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,+1,+1,-1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,-1,+1,+1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,-1,+1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,+1,-1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,+1,-1,-1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,-1,-1,+1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (+1,+1,-1,-1,-1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,+1,-1,+1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,+1,-1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,-1,-1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (+1,-1,-1,-1,-1,-1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,+1,+1,+1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,+1,+1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,-1,+1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,-1,+1,-1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,+1,+1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,+1,+1,-1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,-1,+1,+1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,-1,+1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,+1,-1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,+1,-1,-1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,-1,-1,+1): |(-1) + (-1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,+1,+1,-1,-1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,+1,-1,+1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,+1,-1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,-1,-1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,+1,-1,-1,-1): |(-1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,+1,+1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,+1,+1,-1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,-1,+1,+1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,-1,+1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,+1,+1,+1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,+1,+1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,-1,+1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,-1,+1,-1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,+1,-1,+1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,+1,-1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,-1,-1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,+1,-1,-1,-1,-1): |(+1) + (-1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,+1,-1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,+1,-1,-1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,-1,-1,+1): |(+1) + (+1)| - (-1) <= 1, obeyed=False
a1,a2,b1,b3,c2,c3 = (-1,-1,-1,-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed=True
(Note that 64 distinct possibilities exist for combinations of the values of a,b,c)
(Note that when a1=a2 and b1=b3 and c2=c3, the inequality is NEVER violated)
(Note that the inequality is violated when the above equalities are not obeyed, ie we can not reduce the 6 terms to 3 unque terms)

 

[billschnieder]

As I said, I don't understand what you even mean by this "cyclic dependency", but -- assuming you mean to be referring to some property that is actually there -- it is *not* an *additional assumption* but rather something that *follows* from the assumptions that are *actually made*. Otherwise you'd be able to tell me where the mistake in the mathematical derivation is.

Now explain to me what about locality or no-consipriacy or any other physical assumption youlike led you to say the following in the article:

Bell's inequality theorem. Consider random variables Ziα , i=1,2 , α=a,b,c , taking only the values ±1 . If these random variables are perfectly anti-correlated, i.e., if Z1α=−Z2α , for all α , then:

(1)P(Z1a≠Z2b)+P(Z1b≠Z2c)+P(Z1c≠Z2a)≥1.

...

Theorem. Suppose that the possible values for A1 and A2 are ±1 . Under the mathematical setup described above, assuming the factorizability condition (4), the following inequality holds:

|C(a,b)−C(a,c)|+|C(a′,b)+C(a′,c)|≤2,

- For the CHSH case, why a,b,c,a', and not a1,b1,a2,c2,a'3,b3,a'4,c4 so that the first run measures "a1,b1", the second measures "a2,c2" the third measures "a'3,b3" and the fourth measures "a'4,c4". Why don't you start with the 8 terms and prove the inequality using those?
- For the Bell Case |ab + ac| - bc <= 1, why start with "a,b,c" and not "a1,b1,a2,bc2,b3,c3?

This is crystal clear, you are just stubbornly standing your ground even though you understand what I'm asking very well and you have no answer for it.

There's no such assumption. We are extremely clear and explicit about things that are being assumed.

Then you will have no difficulty answering the above questions. What in the world made you to start with 4 terms instead of 8 for CHSH, and 3 terms instead of 6 for Bell.

You're objecting to something (I don't fully understand what) "downstream". But all the math that gets you down that stream is trivial.

I think you understand very well but don't like it. You are making a trivial mistake which I've pointed out from many different angles in this thread and yet you continue to dodge without providing any counter-argument, claiming not to understand it.

It is undoubtedly true that other assumptions (than the ones we use) can lead to the same conclusion, Bell's inequality.
...
Perhaps it's also true that the inequality can be derived from "cyclic dependency". Who cares? None of those alternative starting points have anything like the status of "locality" -- that, I take it, is your point.

Bah, "cyclic dependenty" is present in EVERY proof. EVERY proof can be reduced to:

(1) Blah, Blah, ... Blah Blah,
(2) Therefore XY + XZ + YZ >= 1 *** <- Cyclic dependency! Required by ALL PROOFS.
(3) ...

Step 1 "Blah blah blah" is just a smokescreen it doesn't matter at all what (1) is so long as you do (2) you will arrive at the inequalities. Violation of the inequalities is not violation of (1), it is violation of (2). This is crystal clear. Cyclic dependency is not an alternate assumption. It is THE most important assumption present in ALL proofs.

But if

A --> C

and

B --> C

and you find out C is false, it's not like you get to *choose* which derivation of C you like best, and hence which of A or B you would prefer to reject.

But that is exactly what you have done. You have picked "locality" to reject even though you know fully well that other assumptions give you the same conclusion. However, as I've clearly explained this is not my point.

My point is more like:

A --> X --> C
and
B --> X --> C
and
X --> C

If C is false, it is X that has been violated not A or B. X is a necessary and sufficient condition to obtain C. A and B are not. The fact that X is trivial algebra does not mean it is not being violated.

 

[rlduncan]

May be the best way to resolve this debate is for someone to analyze the actual experimental data and report their findings. Or, if possible, post the data in a convenient form on this forum (or link) for others to examine. Once sorted, Bell's inequality may be tested using the 3 or 6 data sequences suggested by Bill in this thread. Hopefully, the finished analysis will show if Bell’s inequality can tell us anything about the locality or non-locality of nature.

 

[DrChinese]

... I suspect if you had a genuine rebuttal, you would present it.

Gee Bill, I think you are close to bringing the entire scientific establishment to its knees.

 

[DrChinese]

May be the best way to resolve this debate is for someone to analyze the actual experimental data and report their findings.

This thread is actually about the Scholarpedia article. It is actually not a debate about Bell itself. Not supposed to be, anyway.

 

[rlduncan]

I agree. Except that tnn stated that "Anyway, hopefully people will at some point get around to actually reading the thing and then raising questions about the proofs, arguments, definitions, etc." The factoring step for the derivation of the CHSH inequality has been questioned as to the applicability to the EPR experiments. This may also be resolved using the Bell inequality.

 

[ttn]

Now explain to me what about locality or no-consipriacy or any other physical assumption youlike led you to say the following in the article:

I don't know how to answer that except to say: try reading the article and trying to follow the arguments presented.

- For the CHSH case, why a,b,c,a', and not a1,b1,a2,c2,a'3,b3,a'4,c4 so that the first run measures "a1,b1", the second measures "a2,c2" the third measures "a'3,b3" and the fourth measures "a'4,c4". Why don't you start with the 8 terms and prove the inequality using those?
- For the Bell Case |ab + ac| - bc <= 1, why start with "a,b,c" and not "a1,b1,a2,bc2,b3,c3?

Um, "a" and "b" and "c" are three possible angles along which the polarizations might be made. These 3 possibilities simply aren't "indexed" to a particular run in the way you're describing.

This is crystal clear, you are just stubbornly standing your ground even though you understand what I'm asking very well and you have no answer for it.

Yes, I know you think it's clear, but what you're demanding actually makes no sense at all.

But that is exactly what you have done. You have picked "locality" to reject even though you know fully well that other assumptions give you the same conclusion. However, as I've clearly explained this is not my point.

My point is more like:

A --> X --> C
and
B --> X --> C
and
X --> C

If C is false, it is X that has been violated not A or B. X is a necessary and sufficient condition to obtain C. A and B are not. The fact that X is trivial algebra does not mean it is not being violated.

Now I'm starting to think you don't understand elementary logic. If A --> X --> C, and B --> X --> C, and X --> C, and C is false, then all three of A, B and X are false.

I think you'd be hard pressed to show that X --> C, though (i.e., that this phantom notion you call, dubiously, "cyclic dependency", implies Bell's inequality). I'll wait for the paper where you explain that.

The real point, though, is that you can assume locality ("A" above) and then *do math*, and you get the inequality. The thing you're calling "X" is actually just some step along the way in the math -- *not* anything like an *assumption*. What you're saying is basically equivalent to this: you can't derive a Bell inequality without using a plus sign, so maybe we should blame the violation of the inequalities on the use of plus signs, instead of saying that they prove nonlocality.

 

[billschnieder]

I don't know how to answer that except to say: try reading the article and trying to follow the arguments presented.

You present no argument why ALL BELL/CHSH proofs all of a sudden decide to postulate cyclic dependency between terms. Now you have no excuse for not understanding what I mean by cyclic dependency. Here again I present the evidence from your article:

Bell: P(Z1a≠Z2b)+P(Z1b≠Z2c)+P(Z1c≠Z2a)≥1. Why three angles, why make the terms cyclically dependent? WHY? What about locality or no-conspiracy makes it necessary for you to introduce cyclic dependency into the terms? You have no answer.

CHSH: |C(a,b)−C(a,c)|+|C(a′,b)+C(a′,c)|≤2,. Why 4 angles, why make the terms cyclically dependent? WHY? What about locality or no-conspiracy makes it necessary for you to introduce cyclic dependency between the terms? You have no answer.

All you gave is:

(1) Blah, Blah, ... Blah Blah,
(2) Therefore XY + XZ + YZ >= 1 *** <- Cyclic dependency! Required by ALL PROOFS.
(3) ...

You can not even prove that any physical assumption is required. In fact, (1) can be any thing whatsoever, so long as it involves 3 variables X,Y, Z of value ±1. I have enumerated all the possibilities in a recent post. To which you had absolutely nothing to say in response.

Um, "a" and "b" and "c" are three possible angles along which the polarizations might be made. These 3 possibilities simply aren't "indexed" to a particular run in the way you're describing.

Why not? So you set the instrument angle to "a" and then you collect a series of outcomes ±1. Except you forget that what is actually measured is not the list of outcomes when the instrument was set at angle "a". For the Bell case here is what you are measuring (considering the angle "a", only):

Run 1:
- outcome when instrument is set at "a" given that a corresponding outcome was measured at the other station with that instrument set at "b". ie (a|b)

Run 2:
- outcome when instrument is set at "a" given that a corresponding outcome was measured at the other station with that instrument set at "c". ie (a|c)

Now you don't need to be a rocket scientist so realize how naive it is to assume that (a|b) = (a|c). So clearly there is every justification for indexing the angles appropriately. From what you know from classical physics, it is in fact stupid to assume that the two are equivalent.

I think you'd be hard pressed to show that X --> C, though (i.e., that this phantom notion you call, dubiously, "cyclic dependency", implies Bell's inequality). I'll wait for the paper where you explain that.

No need to wait. See post #125 above for the proof, or any of the published articles I cited earlier in the thread.

The thing you're calling "X" is actually just some step along the way in the math -- *not* anything like an *assumption*.

Call it whatever you like. The point is that you can not obtain the inequality without it, and it is violated by the data gathering and manipulation procedues of experiments.

What you're saying is basically equivalent to this: you can't derive a Bell inequality without using a plus sign, so maybe we should blame the violation of the inequalities on the use of plus signs, instead of saying that they prove nonlocality.

No. Despite your caricature attempt, your position is that since it is OK to use a plus sign, it must be okay to say 2 inches + 2 cm ≠ 4 inches violates the equality 2 + 2 = 4. My position is that you are not adding the same type of thing as implied by the equation. So what you are violating is the mathematical equivalence of the terms in the equation. What the experiments are violating is the cyclicity required to derive the inequality as demonstrated in post #125. You have no answer for that.

 

[billschnieder]

I encourage anyone interested in this discussion to read the following article

ITAMAR PITOWSKY
George Boole's 'Conditions of Possible Experience' and the Quantum Puzzle
Brit. J. Phil. Sci. 45 (1994). 95-125

Excerpts:

In the mid-nineteenth century George Boole formulated his 'conditions of possible
experience'. These are equations and inequalities 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
Home inequalities, and many others.
...
CAN BOOLE'S CONDITIONS BE VIOLATED?
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.
...
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. There are
various possible reasons for that, and they are listed below in an increasing
order of abstractness:
(a) Failure of randomnes...
(b) Measurement biases...
(c) No distribution...
(d) Mathematical oddities...
...
These are the cases, of which I am aware, where Boole's conditions might be
violated. Another possibility, which has been neglected, is the case where we
erroneously believe that some logical relation among the events obtains, and
thus, wrongly expect some condition to be satisfied. Strictly speaking, this case
does not represent a violation of Boole's conditions, but rather an error of
judgement.

 

[harrylin]

Could you say exactly what you thought was inaccurate? I couldn't understand, from what you wrote, what you had in mind exactly.

I could, but that doesn't belong in the QM group. In a nutshell, your definitions imply or suggest that people such as Lorentz and Bell did not teach special relativity when they claimed to teach a theory which they called special relativity and which Einstein and others clearly labeled as such. If you like, we could discuss it further with private messages, or start a post in the relativity group about it.
My point here was that, regretfully, this gave me the impression that your article is based on rather superficial (and thus potentially inaccurate) information that was directly taken from other books and papers, without digging sufficiently deeper - like one would expect of Wikipedia, but of course not what you intended for "Scholarpedia".

 

[ttn]

You present no argument why ALL BELL/CHSH proofs all of a sudden decide to postulate cyclic dependency between terms.

But, for the hundredth time, they *don't* POSTULATE any such thing. Whatever exactly it is that you are objecting to is a CONSEQUENCE of the assumptions, not an additional assumption.

Bell: P(Z1a≠Z2b)+P(Z1b≠Z2c)+P(Z1c≠Z2a)≥1. Why three angles, why make the terms cyclically dependent? WHY? What about locality or no-conspiracy makes it necessary for you to introduce cyclic dependency into the terms? You have no answer.

One is of course free to consider whatever experimental setups one chooses. This happens to be the combination of setups that Bell's theorem is *about*. That's why this particular set of setups is considered. But "Here are some assumptions which, if true, should apply to any setup; let's consider these 3 setups to get an interesting result" is hardly the same as introducing some controversial new postulate.

All you gave is:

(1) Blah, Blah, ... Blah Blah,
(2) Therefore XY + XZ + YZ >= 1 *** <- Cyclic dependency! Required by ALL PROOFS.
(3) ...

You can not even prove that any physical assumption is required. In fact, (1) can be any thing whatsoever, so long as it involves 3 variables X,Y, Z of value ±1. I have enumerated all the possibilities in a recent post. To which you had absolutely nothing to say in response.

The 1/2/3 above is, I'm sure, an accurate description (i.e., confession) of the state of your consciousness in trying to follow the argument. That is, you haven't grasped it as an argument at all, but you instead just zone out during the whole explication of premises. But, it's a logical argument whether you sleep through the first part or not. And if you object to something in the middle, you should have the integrity -- the curiosity -- to try to trace that thing you object to back to the premises, to find out exactly where it came from and how the premises do or don't support it, instead of just dismissing the whole thing as "blah blah blah".

Now you don't need to be a rocket scientist so realize how naive it is to assume that (a|b) = (a|c).

But THIS IS NOT AN ASSUMPTION. It is rather something that FOLLOWS from the assumptions that are actually, explicitly made -- locality here in particular. The "blah blah blah" part that you slept through includes an explicit acknowledgment of the assumption that the outcome on one side might be allowed to depend on the setting on that side and on the pre-measurement state of the particles, but should not depend on the distant setting. (Your notation is also vague in the sense that the "no conspiracy" assumption is also partly responsible for what you evidently mean here: if the "pre-measurement state of the particles" depended on the distant setting, that would be another way that in principle the distant setting could affect the nearby outcome. This possibility however is excluded by the assumption that the "pre-measurement state of the particles" does not depend on the settings, i.e., the "no conspiracy" assumption.)

So clearly there is every justification for indexing the angles appropriately. From what you know from classical physics, it is in fact stupid to assume that the two are equivalent.

Obviously I don't agree with the last sentence. But logically the main point is that any "justification for indexing the angles appropriately" will constitute a denial of one or both of "locality" and "no conspiracies".


We have both made our positions clear, and I have to say that arguing with you just frankly isn't all that enjoyable. So feel free to have the last word if you want it, but I won't continue arguing with you any further. It's clear that nothing will come of it. People still watching the discussion will then have to make up their own minds.

 

[ttn]

I could, but that doesn't belong in the QM group. In a nutshell, your definitions imply or suggest that people such as Lorentz and Bell did not teach special relativity when they claimed to teach a theory which they called special relativity and which Einstein and others clearly labeled as such. If you like, we could discuss it further with private messages, or start a post in the relativity group about it.
My point here was that, regretfully, this gave me the impression that your article is based on rather superficial (and thus potentially inaccurate) information that was directly taken from other books and papers, without digging sufficiently deeper - like one would expect of Wikipedia, but of course not what you intended for "Scholarpedia".

I don't think any huge discussion is needed here, and this thread seems like a perfectly good place to have a short one, since after all your comments are about the scholarpedia article.

You suggest that our understanding/presentation of what constitutes "relativity" is superficial/thin. I (perhaps not surprisingly) think that's just backwards, and actually it's the way most people talk that is inappropriately superficial. "Special relativity" does not refer merely to a certain set of equations that one finds in relativity textbooks; it refers to a certain *physical theory* which involves various, um, ontological commitments. In particular, what normal physicists mean by "special relativity" includes the *denial of the idea that there is a dynamically privileged reference frame aka ether*. This is partly a historical issue. Lorentz had proposed a theory in which there was a dynamically privileged reference frame or "ether" -- the same thing that everybody just assumed existed in the context of Maxwell's electromagnetic theory -- but in which a certain rather strange mathematical symmetry in effect conspired to make it impossible for us to empirically detect our motion through this ether. Let's call this view "Lorentzian relativity" -- it is of course the view that Bell was describing in his lovely "how to teach" article.

The point is: then Einstein came along and proposed what is now usually called "special relativity" -- though let's call it "Einsteinian relativity" here for the purpose of extra clarity. According to "Einsteinian relativity", there *is no ether*; instead, all reference frames are fundamentally, dynamically equivalent.

Now it sounds like you want to say that both "Einsteinian relativity" and "Lorentzian relativity" are perfectly well subsumed under "special relativity" -- they are merely different interpretations of "special relativity", or something like that. I actually think that is right, and I certainly think that "Lorentzian relativity" is a going option, i.e., that no experiment has refuted it as a possibility. But here I openly acknowledge being in a great minority among regular physicists. If you ask any normal physicist whether "special relativity" is compatible with the possible existence of an ether, a dynamically privileged reference frame, they will say "no bleeping way!" and never talk to you again! =)

So, for purposes of communication, we tend to use the phrase "special relativity" to mean basically what most other people use that phrase to mean. But of course, in the discussion, we explicitly distinguish the "Einsteinian" and "Lorentzian" views, distinguish the idea of "relativity at the level of what can be observed" (which is compatible with both Einsteinian and Lorentzian approaches) and "fundamental relativity" (which the Lorentzian approach violates), etc. This is all more or less exactly following Bell, who for example notes repeatedly that Bohmian Mechanics (in so far as it requires a dynamically privileged reference frame, or some equivalent) fails to respect fundamental relativity, etc. It's true that in his "how to teach" article he described himself as teaching "special relativity", but I think it's clear that here he was doing a kind of propaganda, i.e., trying to "soft sell" an idea that, if presented more bluntly, causes normal physicists to simply shut down and stop listening (because they have been dogmatized against the Lorentzian view to the point that they believe, erroneously, that it was somehow experimentally refuted by the MM exp or whatever).

Basically the main point is just that it's incredibly easy to reconcile non-locality (which remember we know is there because of Bell's theorem and the experiments) with "lorentzian relativity" -- Bell always called this "the simplest solution", etc., and we agree. Indeed, since we all like Bohmian Mechanics, we are quite happy to agree! But reconciling quantum non-locality with *Einsteinian relativity* -- i.e., what most normal physicists think of as just plain "relativity" -- is much harder. It can in principle be done, sort of, probably. (See for example Tumulka's relativistic GRW model.) But it's very very difficult, and it's basically an open question whether non-locality can be reconciled with "fundamental relativity" in the context of a "fully serious" theory (e.g., something that makes all the predictions of ordinary QFT).

Do you disagree with any of this? Or do you merely prefer using the words "special relativity" in a less narrow way, like the way Bell evidently uses them in "how to teach"?

 

[mattt]

ttn, seriously I don't know how can you be so patient.

To the rest: there is a mathematical threorem "The CHSH-Bell Inequality: Bell's Theorem without perfect correlations". As far as I could check, the mathematical proof (of the precise mathematical statement) is correct.

I don't even care the names we put to the two premises ("mathematical setup" + "factorization condition", or "mathematical general structure capturing the possible ways a theory produces numerical predictions" + "locality", or "mathematical general structure capturing the possible ways a theory produces numerical predictions" + "separability", whatever...), or the name we put to the thesis of that mathematical theorem, because no words can be as precise as a mathematical statement itself.

So, is there anyone else here (apart from me) that actually has tried to check if the mathematical proof is correct or not?

 

[billschnieder]

Maybe you think by shouting the contrary, you are actually refuting my argument, but you are not. And since you continue to misrepresent my argument, I will continue to clarify it.

But, for the hundredth time, they *don't* POSTULATE any such thing. Whatever exactly it is that you are objecting to is a CONSEQUENCE of the assumptions, not an additional assumption.

Look, I can grant to you that the factorizability condition is a consequence of an assumption. However it is a lie that the cyclicity implied in P(Z1a≠Z2b)+P(Z1b≠Z2c)+P(Z1c≠Z2a)≥1 is due to any physical assumption. You could simply have said Bell picked those ones because they worked. There is no reason in your paper or any other Bell-type proof why Bell picked P(Z1a≠Z2b)+P(Z1b≠Z2c)+P(Z1c≠Z2a), instead of say P(Z1a≠Z2b)+P(Z1c≠Z2d)+P(Z1e≠Z2f) and you know it. Nothing about Locality or no-conspiracy forces you to pick the former and not the latter. The only reason that can be infered from it is "because it works". In other words, the latter would have given an inequality which is NEVER violated by QM or Experiments.

Now I have carefully explained to you that the reason why it works is BECAUSE of the cyclicity, that is why NO Bell-type proofs use the non-cyclical type P(Z1a≠Z2b)+P(Z1c≠Z2d)+P(Z1e≠Z2f). I even posted a simulation using all possible values, clearly demonstrating the importance of the cyclicity. To which you had absolutely nothing so say. So contrary to what you like, the cyclicity is the most important component of the proof. Call it "trival agebra" or whatever you like, it still doesn't change this fact.

One is of course free to consider whatever experimental setups one chooses.

Exactly, and there is nothing about locality or no-conspiracy that forces ALL Bell proofs to use ONLY the ones that involve cyclicity but not the ones without cyclicity. I suspect you will respond that you do not understand what cyclicity means, or completely ignore it as you have been doing.

This happens to be the combination of setups that Bell's theorem is *about*. That's why this particular set of setups is considered.

But you do not deny that for three separate runs, the experiments are measuring a1,b1,a2,c2,b3,c3 (six separate terms). All you argue is that a1=a2, b1=b3, c2=c3, naively thinking that the the angles are the only parameters relevant for the outcomes and making the logical error of equating the outcomes. Your (and Bell's) only justification is the assumption that the distribution of hidden parameters is identical for each term. You then make the leap to argue that the only way this assumption can be violated is if there is nonlocality or no-conspiracy. But as I've explained, this is just plain naive.

Bell is assuming that in such an experiment "When the instrument is set to the angle 'a', the distribution of all hidden parameters which affect the outcome is exactly identical from one run of the experiment to the next". Note that failure of this assumption implies that the cyclicity is broken and the inequality can be violated as demonstrated by Boole 100 years before Bell, and as proven in my simulation in post #125.

I have already explained in this thread that the way experiments are performed using coincidence circuitary, the experimenters are not measuring simply "Outcomes when angle is set at 'a'" but they are measuring "Outcomes when angle is set at 'a' given that an outcome is also measured at the other station with it's angle set at 'b'". If you think the two are the same or that the latter violates no-conspiracy or locality, then you lack basic understanding of logic and probability theory.

I have demonstrated convincingly that experiments violate the cyclicity by measuring and using 6 different terms rather than the 3 used by Bell. I have also demonstrated that it is unreasonable to expect the cyclicity to be maintained given what we know about light from classical physics. Therefore violation of the inequality says absolutely nothing about locality or no-conspiray. Violation simply implies violation of the cyclicity. Boole recognized this 100 years before Bell and he did not question locality/reality the way those with a penchant for mysticism are prone to doing these days.

Obviously I don't agree with the last sentence. But logically the main point is that any "justification for indexing the angles appropriately" will constitute a denial of one or both of "locality" and "no conspiracies".

Now let us break down what this implies about what you believe:

- You believe contrary to Malus law that the angular difference between the two sides, does not affect the rate of coincidence detection.
- You believe that every property of the complete system "instrument + particle" relevant for the outcomes actually observed are identical when "a" is measured coincidentally with "b" and when "a" is measured with "c", despite the fact that the angle between "a" and "b" is different from the angle between "a" and "c".
- You believe only conspiracy or non-locality can explain why all relevant properties of the complete system of "instrument + particle" for two separate runs, performed at different times, and filtered using coincidence circuitary governed by a different angular differences might be different, so long as they used the same macroscopic angle setting.

What a naive view of physics. Funding agencies may be fooled by this kind of "snake-oil". Not me.

 

[billschnieder]

ttn, seriously I don't know how can you be so patient.

To the rest: there is a mathematical threorem "The CHSH-Bell Inequality: Bell's Theorem without perfect correlations". As far as I could check, the mathematical proof (of the precise mathematical statement) is correct.

If you were following carefully you would have understood that I'm not questioning the math. Rather I'm questioning the suggestion and common errorneous practice of using terms from QM and experiments on the LHS of the inequalities.

 

[mattt]

If you were following carefully you would have understood that I'm not questioning the math. Rather I'm questioning the suggestion and common errorneous practice of using terms from QM and experiments on the LHS of the inequalities.

But then you won't agree either with other versions (weaker than this one) of Bell's Theorem.

What I like of this mathematical theorem is that it is the strongest I have seen of this kind (Bell type theorem), the most general mathematical premises.

So you are arguing about how good or bad (in your opinion) are his mathematical premises with respect to capturing the conditions and procedures in the real experiments, aren't you?

 

[ttn]

I said I wasn't going to argue with Bill anymore, but for the sake of anybody else reading, I wanted to be sure the following was clear:

Now let us break down what this implies about what you believe:

- You believe contrary to Malus law that the angular difference between the two sides, does not affect the rate of coincidence detection.

That is just out of the blue. Of course it's true that the angular difference between the two sides does affect the rate of coincidence detection. (Incidentally, this is *not* "Malus law". Malus' law is about the probability for light to pass through two successive polarizers with some relative angle between them. The EPR-Bell setup involves two hunks of light, with each hunk going in opposite directions, and with each hunk being subjected to only a single polarization measurement.) Why Bill thinks something I've said commits me to denying this, is a mystery (but not such a big surprise) to me.

- You believe that every property of the complete system "instrument + particle" relevant for the outcomes actually observed are identical when "a" is measured coincidentally with "b" and when "a" is measured with "c", despite the fact that the angle between "a" and "b" is different from the angle between "a" and "c".

It shouldn't be put that way. The "no conspiracy" assumption does not say that the state of the complete system is identical every time a is measured with b, etc. It says only that, on average, the statistical distribution of the different possible states (whatever the heck those might be) is the same, no matter which angles we measure along. That is, for the bajillion particle pairs that happened to get measured along a and b, the statistical distribution of states is about the same as for the bajillion particle pairs that happened to get measured along b and c, etc. That is, the source just makes particle pairs the same way each time (where "the same way" probably involves some randomness and hence a large set of possible states) regardless of which settings the polarizers will be in. Denying this requires a kind of "pre-established harmony" -- indeed, one might say "a conspiracy" -- between the random/hidden variables determining the settings, and those determining the particle states.

- You believe only conspiracy or non-locality can explain why all relevant properties of the complete system of "instrument + particle" for two separate runs, performed at different times, and filtered using coincidence circuitary governed by a different angular differences might be different, so long as they used the same macroscopic angle setting.

Subject to the clarifications above, that is basically correct. That is, after all, what the theorem shows: "no conspiracies" and "locality" jointly entail something that is found in the experiments to be false.

 

[ttn]

ttn, seriously I don't know how can you be so patient.

Thanks mattt, I appreciate that. I guess it was probably a rhetorical question, but I'll answer anyway for fun. Part of the answer is that this is the kind of thing one expects at a place like this, and it's kind of fun to engage with it every so often (but it does get old fast). A more serious part of the answer is that I feel about Bill more or less the way that most normal physicists no doubt feel about me. So I'm kind of "on the premise" of trying very hard to extend other people -- I mean, specifically, people who have a view that's unfamiliar to me and that sounds wrong to me and that I don't fully understand -- the same courtesy that I wish people would more often extend to me when I talk about Bohm's theory or other such things. Another part of the reason is that, both by profession and "demeanor", I'm a teacher. So I assume, sometimes for too long, that people are actually interested in understanding things better, that they will be open to good arguments if only they hear them put sufficiently clearly, etc. But, as you saw, I've basically given up with Bill. =)

So, is there anyone else here (apart from me) that actually has tried to check if the mathematical proof is correct or not?

I looked at it pretty carefully and I think it looks pretty convincing, too!

 

[billschnieder]

But then you won't agree either with other versions (weaker than this one) of Bell's Theorem.

The fact that the terms in the inequality do not correspond to the terms being measured in the experiment is a non-starter.

What I like of this mathematical theorem is that it is the strongest I have seen of this kind (Bell type theorem), the most general mathematical premises.

Then you will like Boole's proof even better. It is the most general of them all and it can never be violated unless there is a logical error. I encourage you to check out Itamar Pitowsky's exposition of it cited, or the original version.

Here is another one originally suggested by Sica (cited earlier in the thread http://dx.doi.org/10.1016/S0030-4018(99)00417-4):

Assume that we have three lists of numbers, each of length N, with each number restricted to values ±1.The lists are denoted a, b, and c and their respective members by ai, bi, and ci, i = 1...N.

It follows that
(1a) a_ib_i - a_ic_i = a_i(b_i-c_i) = a_ib_i(1-b_ic_i)

By summing this equation over the list, dividing by N and taking absolute values, noting that bi = 1/bi we get:(1b) \left | \frac{\sum_{i=1}^{N} a_ib_i}{N} - \frac{\sum_{i=1}^{N} a_ic_i}{N}\right | \leq \frac{\sum_{i=1}^{N} \left | a_ib_i \right |\left | 1 - b_ic_i \right |}{N} = \frac{\sum_{i=1}^{N} (1 - b_ic_i)}{N}

And finally:
(2) \left | \frac{\sum_{i=1}^{N} a_ib_i}{N} - \frac{\sum_{i=1}^{N} a_ic_i}{N}\right | \leq 1 - \frac{\sum_{i=1}^{N} b_ic_i}{N}<br />

or

(3) | <ab> - <ac> | <= 1 - <bc>

Bells inequality from ONLY ONE assumption: That we have 3 lists of numbers with values ±1. This inequality can NEVER be violated by any 3 lists of numbers with values ±1. Starting with 4 lists, you get the CHSH.

Sica says:

It is perhaps startling to the reader to find that Bell's inequality has been obtained here with no mention of locality or nonlocality, properties of probability functions, factorization assumptions, etc. All these assumptions are peripheral to the central fact: identity (2) and inequality (3) follow from nothing but the arithmetic of ±1's and the assumption of limits.

He goes on to derive the CHSH inequality from similar assumptions.

And concludes:

The present paper has shown that the form of Bell's inequalities appropriate for comparison with experiments is an identity based on minimal assumptions. If these minimal assumptions are not met by the data, the conditions for validity of the identity will be violated, and the inequality may (or may not) be violated. This may happen if it is not noticed that the data for two correlation estimates uniquely determines the third in the three correlation case, and that data for three correlations determine the fourth in the four correlation case. That this has not been generally recognized may stem from the belief, based on early derivations, that Bell's inequality is a fact about statistics. But in fact, as has been shown above, it is a constraint of arithmetic, quite independently of statistics.

 

[billschnieder]

That is just out of the blue. Of course it's true that the angular difference between the two sides does affect the rate of coincidence detection. (Incidentally, this is *not* "Malus law". Malus' law is about the probability for light to pass through two successive polarizers with some relative angle between them.

It's not out of the blue. Rather it is out of the view that since the same angle value "a" was set during the a1,b1 run as during and a2,c2 run, a1 must be identical to a2. The view which believes that (a1|b1, ie outcome at angle "a" given that an outcome was measured at angle "b") = (a2|c2, outcome at angle "a" given that an outcome was measured at "c"). If you are now claiming that (a1|b1) ≠ (a2|c2) in Bell test experiments, then you are admitting that:
- the distribution of λ is not the same in each term.
- you have 6 unique terms not 3 as required by the inequality.

As concerns Malus Law.
(1)- Single stream of photons with 2 polarizers A & B in sequence. Probability of detection is the probability that the photon passed through B given that it also passed through A.

(2)- Two steams of correlated photon pairs with one polarizer each A & B, with coincidence circutary. Probability of detection for any photon, is the probability that it passed B given that it's counterpart passed A.

It is naive, given what we know classically about Malus Law and coincidence circuitary, to think that outcomes from freely setting the device to "a" will give you results that do not dependen on the "b" in Bell test experiments.

 

[rlduncan]

ttn,

billschnieder has provided three experiments relating to this thread for you to comment on.

1. Post # 102 - coin toss experiment
2. Post # 107 - Lyon, Paris, Lille study
3. Post # 125 – cyclic dependency

Your failure to make a comment is very telling. Why not start with the coin toss experiment and explain how it is possible this simple experiment can violate a Bell-type inequality? Notice the similarity to the EPR experiments. Am I to conclude that nature is nonlocal because the inequality was violated? If yes, then why do we need to use entangled photons?

 

[DrChinese]

Why not start with the coin toss experiment and explain how it is possible this simple experiment can violate a Bell-type inequality? Notice the similarity to the EPR experiments. Am I to conclude that nature is nonlocal because the inequality was violated? If yes, then why do we need to use entangled photons?

Although it has nothing to do with this thread, I will be glad to answer. Obviously, if there is a sufficient deviation from any fair sample, you can get even a classical analogy to give nonsensical results. When was the last time you saw 30 heads in a row? It is ridiculous to talk about "cyclic dependency" when it comes to Bell tests because they have been done with random choices of angle settings. Typical of billschnieder that he manages to get folks to bite on this obvious red herring. I especially love the part about it being "very telling". Do you really think science stands or falls on a comment in PF?

If you are so confident, take the DrChinese challenge! Give me a miniature sample of 30 with realistic results (3 angle settings, 0/120/240 degrees, plus or minus, labeled a/b/c). I will select 2 of the 3 settings for each trial of the 30 randomly. We'll see if you can get a sample with a match rate anywhere near 25%, the quantum prediction.

Good luck! You are going to need it!

a/b/c

1: ++-
2: -+-
3: +-+
...or whatever you want to make up.

By the way, this is basically the same test Richard Gill wrote about recently. But me being me, I choose to name it after me. LOL.

 

[harrylin]

I don't think any huge discussion is needed here, and this thread seems like a perfectly good place to have a short one, since after all your comments are about the scholarpedia article.

You suggest that our understanding/presentation of what constitutes "relativity" is superficial/thin.

Not exactly... I'll try to state it more precisely: my reading of parts of your writing that I can judge, gave me the impression that your writing is based on a rather superficial reading of a few books or articles. Sorry if that wasn't clear. However, happily your new reply changed my impression.

[..]
So, for purposes of communication, we tend to use the phrase "special relativity" to mean basically what most other people use that phrase to mean. But of course, in the discussion, we explicitly distinguish the "Einsteinian" and "Lorentzian" views, distinguish the idea of "relativity at the level of what can be observed" (which is compatible with both Einsteinian and Lorentzian approaches) and "fundamental relativity" (which the Lorentzian approach violates), etc. This is all more or less exactly following Bell, who for example notes repeatedly that Bohmian Mechanics (in so far as it requires a dynamically privileged reference frame, or some equivalent) fails to respect fundamental relativity, etc. It's true that in his "how to teach" article he described himself as teaching "special relativity" [..]

OK - but did you realize that such yielding to mislabeling results in contradictions with the existing literature? Not just with Bell's papers but also with textbooks and articles by Lorentz, Langevin and even some articles by Einstein. I hope that such things didn't occur with other definitions and labels concerning QM and EPR.

Basically the main point is just that it's incredibly easy to reconcile non-locality (which remember we know is there because of Bell's theorem and the experiments) with "lorentzian relativity" -- Bell always called this "the simplest solution", etc., and we agree. Indeed, since we all like Bohmian Mechanics, we are quite happy to agree! But reconciling quantum non-locality with *Einsteinian relativity* -- i.e., what most normal physicists think of as just plain "relativity" -- is much harder.

Einstein's metaphysical opinions flip-flopped so much that, depending on how you look at it, it's extremely easy or impossible.
It may be better to label the different models not with people's names but with the names of the models themselves. You could correct that little issue with a few minor changes, by simply removing some unnecessary statements and labels about relativity; an article on Bell's Theorem doesn't have use for them. Instead you can simply define your own labels when you introduce them, e.g. "fundamental relativity" as referring to the block universe interpretation of relativity.

It can in principle be done, sort of, probably. (See for example Tumulka's relativistic GRW model.) But it's very very difficult, and it's basically an open question whether non-locality can be reconciled with "fundamental relativity" in the context of a "fully serious" theory (e.g., something that makes all the predictions of ordinary QFT).

Do you disagree with any of this? Or do you merely prefer using the words "special relativity" in a less narrow way, like the way Bell evidently uses them in "how to teach"?

My answer is here above. Would you agree with using the label "quantum mechanics" in a narrow way, so that it refers to the Copenhagen interpretation? The objection is similar, and the solution is also similar.

 

[ttn]

OK - but did you realize that such yielding to mislabeling results in contradictions with the existing literature?

Well, contradictions (involving this sort of terminological issue, and much worse as well) are rampant in the literature already, so I don't think there's any set of terminology we could choose that would avoid all contradictions. We just tried to explain what we meant so it would be clear. You don't think it is clear? I find that slightly surprising, since what I wrote in the last post here (and which you evidently understood just fine and even seemingly agreed with for the most part) is just about the same as what we wrote in the article. So I'm really not sure what it was that gave you the original impression.

Not just with Bell's papers but also with textbooks and articles by Lorentz, Langevin and even some articles by Einstein. I hope that such things didn't occur with other definitions and labels concerning QM and EPR.

You mean, you hope there's nothing else in the physics part of the article that might be considered controversial or wrong by some people??

Einstein's metaphysical opinions flip-flopped so much that, depending on how you look at it, it's extremely easy or impossible.

Yes, I agree, it's hard to pin down "what Einstein thought" on this or other issues.

It may be better to label the different models not with people's names but with the names of the models themselves. You could correct that little issue with a few minor changes, by simply removing some unnecessary statements and labels about relativity; an article on Bell's Theorem doesn't have use for them. Instead you can simply define your own labels when you introduce them, e.g. "fundamental relativity" as referring to the block universe interpretation of relativity.

I don't see anything confusing/misleading about labeling the views with the names. I think an article on Bell's Theorem *does* need to go into this stuff. Bell's Theorem, properly understood, is a proof that nature is non-local, and basically the only reason anybody should care about that is because, for about 100 years, relativity has given us all a pretty strong reason to expect locality instead. So the question of how to reconcile non-locality with relativity (and whether it's even possible) is an obvious and important one. And here I think it's essential to get into the question of what "relativity" even means/says/requires, distinguish emergent/superficial/empirical relativity from "fundamental relativity", etc. So again I'm really just not sure what you're meaning to criticize.

My answer is here above. Would you agree with using the label "quantum mechanics" in a narrow way, so that it refers to the Copenhagen interpretation? The objection is similar, and the solution is also similar.

That's a good and fair question. I guess I think the two cases, (Copenhagen) QM and (Einsteinian) SR, are a little different. As I explained before, I think basically all physicists *actually believe* Einsteinian SR -- it's what they mean when they talk about "SR". And if you probe them by asking such things as whether they think maybe there is an "ether" that we just can't detect, they will answer unamibuously "no, SR doesn't just say we can't detect an ether, it says there isn't one!" And so on. That is, I think they really believe Einsteinian SR in a fully consistent way. On the other hand, although most physicists will claim to endorse Copenhagenish ideas, they don't really believe (or even really understand) what Copenhagen says -- it's just to them about one paragraph of vague words they read in a textbook 30 years ago in grad school, but never studied carefully or took too seriously. Instead, they were trained/dogmatized in the "shut up and calculate" attitude, with its hostility not just to alternatives to Copenhagen, but indeed to Copenhagen itself. And if you probe a normal physicist (don't quote that phrase out of context!) about foundations of QM issues, you find quickly that they don't have any particular view, but instead they have a contradictory hash of only-weakly-held slogans. For example, it is trivially easy to get a normal physicist to take whichever position you want on the question of whether the collapse of the wave function is a real physical process, or is instead merely an updating of our knowledge: just frame the question in the context of "hidden variables", or "non-locality", respectively.

For this reason, I think the only safe way to use the words "quantum mechanics" (with no explicitly qualifier) is to refer only to what's sometimes called "the quantum formalism", i.e., the mathematical algorithms for calculating things. That is, I think "QM" should really be used to refer to the shut-up-and-calculate (non-) interpretation, not Copenhagen. If you want to refer to Copenhagen, call it "Copenhagen QM".

So... that's why I think it's OK/proper to use "SR" to denote the *interpretation* of "the relativistic formalism" that physicists overwhelmingly and deeply endorse, while using "QM" to denote only the formalism itself devoid of any particular interpretation.

On the other hand, I doubt that this or any other choice of terminology causes any real problems of communication/clarity. So, although I'm certainly open to hearing what you found confusing/unclear in the article, I'm probably not interested in having a super long discussion about which terminology is best...

 

[ttn]

In response to rlduncan...

Three fair coins are tossed simultaneously by three individuals. For simplicity, let's them be a, b, and c and each coin is tossed eight times. It follows that the outcomes must obey the following inequality

nab(HH) + nbc(HH) ≥ nac(HH)

That's already not right. Bill gave an example of possible data, evidently to "prove" that the inequality has to be satisfied:

To see this, consider the following outcomes for the three coins
a=HTTTHTHH
b=TTHHTHHH,
c=HTHTTTHH,

2+3 ≥ 3 , the inequality is satisfied.

But suppose the flips turn out instead like this:

a=TTH
b=THT
c=HTH

So nab(HH)=0, nbc(HH)=0, and nac(HH)=1. The inequality is violated. So... why should I believe the inequality in the first place? Did Bill type the wrong thing? Did I understand it wrong?

When a big alleged "knock down concrete example refutation" starts off with an obvious error like this, you maybe shouldn't be surprised that people don't bother to respond, tend to stop listening to your arguments, and don't even bother to look at subsequent "knock down concrete example refutations".

However, if in an experiment you decide to perform three different runs of the experiment such that you obtain

a1=HTTHTHHH
b1=THHTTHTT,

b2=HTHHTHHT
c1=TTTTHHTH,

a2=THHTHTTH
c2=HHHTHTTT,

You now have 1+1 ≥ 3 which violates the inequality. Why is that? The reason is simply because the three terms in the inequality are not independent. They are calculated from only 3 lists of outcomes so that there is a cyclic dependency. However in the latter experiment, we have 6 distinct lists not reduceable to 3!

I am in the happy position of getting to basically agree with Dr Chinese. Of course you can violate a Bell type inequality by just making up lists of how the data might have come out. The point is that such data will imply a violation of one of the assumptions that went into the inequality, or of the relevant QM predictions. So it's not a refutation of Bell's proof; it's a demonstration of it!

The right approach would be instead the following (basically the Dr C challenge): make lists of how you think each particle in each pair will "answer" (H/T) when "asked" any of the (3 or 4) possible "questions". Then I'll go down the list, one pair at a time, and decide randomly for each pair which 2 questions I want to ask. (Here, by "random", I basically just mean that I have to decide which questions to ask before I look at what you've written down for that pair -- also that I won't ask you, who already know what you've written down for each pair, for advice on which questions to ask... I'll instead let which questions I ask be determined by something totally unrelated to you and the lists you made, e.g., I'll roll a die or look at the 5th digit in the current price of porkbellies or ...) We'll keep track of what the outcomes will be and then calculate at the end all 3 or all 4 of the correlation coefficients. (Note that this procedure is in effect a way a implementing the "no conspiracies" assumption.) I assume you understand perfectly well that if we played *that* game, the correlations would respect the inequality. Which of course proves that in *your* game, the way you violate the inequality is because you get to decide what outcomes to assign to each particle pair *after you already know what questions are being asked*.

What I don't understand is why you and Bill don't just openly acknowledge this painfully obvious fact: your beef is with the "no conspiracy" assumption. (Or perhaps also to some extent, and despite your protestations to the contrary, with locality!) All the stuff about "cyclic dependency" is just a red herring shaped hot air balloon.

 

[harrylin]

[..] Yes, I agree, it's hard to pin down "what Einstein thought" on this or other issues. [..] I don't see anything confusing/misleading about labeling the views with the names.

It's exactly this kind of inconsistencies that I noticed in the first place in your article: you recognize that it's hard to pin down "what Einstein thought" on this issue, and still you don't see anything confusing/misleading about labeling a view that he did not consistently have with his name. I find that counter-productive - and as I showed, it's completely unnecessary.

I think an article on Bell's Theorem *does* need to go into this stuff. Bell's Theorem, properly understood, is a proof that nature is non-local, and basically the only reason anybody should care about that is because, for about 100 years, relativity has given us all a pretty strong reason to expect locality instead. So the question of how to reconcile non-locality with relativity (and whether it's even possible) is an obvious and important one. And here I think it's essential to get into the question of what "relativity" even means/says/requires, distinguish emergent/superficial/empirical relativity from "fundamental relativity", etc. So again I'm really just not sure what you're meaning to criticize.

If you read this part of your reply again and then compare it with my earlier suggestions, you will notice that I agree with it; my criticism is, again, that you evidently prefer to include remarks and labels that are controversial to say the least and for which there is absolutely no need. Why would you want to do that in an encyclopedia article?

That's a good and fair question. I guess I think the two cases, (Copenhagen) QM and (Einsteinian) SR, are a little different. As I explained before, I think basically all physicists *actually believe* Einsteinian SR -- it's what they mean when they talk about "SR".
And if you probe them by asking such things as whether they think maybe there is an "ether" that we just can't detect, they will answer unamibuously "no, SR doesn't just say we can't detect an ether, it says there isn't one!" And so on. That is, I think they really believe Einsteinian SR in a fully consistent way. [..] On the other hand, although most physicists will claim to endorse Copenhagenish ideas, they don't really believe (or even really understand) what Copenhagen says [..]

There we go again - Einstein said around 1920 that according GR an ether exists, but earlier he had a "shut up and calculate" attitude! Now, what is your "Einsteinian relativity"? It's completely useless at best, even worse than "Copenhagen". Thus, again, and for a last time (I won't continue about this either): if you want to present a quality article, you scrap this kind of debatable things which you don't need at all, and simple say for example that with "fundamental relativity" the article refers to a block universe model. BTW, I agree with you about QM, that was to illustrate the issue.