Why the upper bound on CHSH is 2\/2 and not 4 ?

[FrediFizzx]

This thread is concerned with explaining what the correct bound should be since we know it isn't 2.

Yep, and experiments consistently indicate that it is greater than 2 and so far the indication is that it is never greater than . Including classical experiments. And the quantum theory prediction is that the maximum is so QM never actually "violates" CHSH since that is most likely the real upper bound for CHSH as Joy has explained in this thread. I think I had the "cheat" backwards in what I said previously. I think Joy is right that the real "cheat" is restricting it to 2 by limiting oneself to scalar numbers.

[jreed]

This thread is concerned with explaining what the correct bound should be since we know it isn't 2.

Yep, and experiments consistently indicate that it is greater than 2 and so far the indication is that it is never greater than . Including classical experiments. And the quantum theory prediction is that the maximum is so QM never actually "violates" CHSH since that is most likely the real upper bound for CHSH as Joy has explained in this thread. I think I had the "cheat" backwards in what I said previously. I think Joy is right that the real "cheat" is restricting it to 2 by limiting oneself to scalar numbers.

Here's an explanation of a Bell experiment. I'll explain where the 2 comes from and why anything greater than that isn't possible with classical physics. That includes 2*Sqrt(2) and 4.

The experiment consists of 2 observers, Alice and Bob each with fair coins and detectors that can be rotated to several angles, along with a source of electrons. There are also 4 summers, labeled AB, AB', A'B and A'B'. The summers are all set to zero initially. Alice's detector is able to be set to two angles, a = 0 or a' = 90 degrees. Bob's detector is able to be set to two angles also, b = 45 or b' = 135 degrees.

Now the experiment starts. Alice flips her coin and sets her detector to a(0 degrees) if it's heads, or a'(90 degrees) if it's tails. Bob does the same, flipping his coin and setting his detector at either 45 or 135 degrees. The source, which is halfway between Alice and Bob, emits two electrons which are detected by Alice's and Bob's detectors. The detectors have indicators on them that show whether the electron's spin is aligned (+1) or anti-aligned (-1) with the detector. Alice records this value and calls it A, if she picked angle a, or A' if she picked angle a'. Similarly Bob records his spin value and calls it B or B'. They multiply these values together and get either +1, if both spins were aligned or anti-aligned or -1 if spins were aligned opposite to each other. This value is summed into one of the counters AB, AB', A'B or A'B', depending on angle selection, as should be clear from the notation.

After many experiments, the summers contain the results of all experiments. The following means can be calculated:



by dividing the sum by the number of counts in that sum, N. Since the sum consists of N +1 or -1 values, the mean must always be less than or equal 1 or greater than or equal -1. CHSH is now calculated by the equation:



Which clearly must be less than or equal 2, or greater than or equal -2.

I hope this explanation shows what a Bell believer thinks, and that it is clear as a bell (pun intended).

[Joy Christian]

Since the sum consists of N +1 or -1 values, the mean must always be less than or equal 1 or greater than or equal -1.

Very good. You are catching up. So we have

CHSH is now calculated by the equation:

Indeed. So with the above mean we get the two extreme values,



and



Thus we have



which is indeed as clear as bell.

[Mikko]

Here's an explanation of a Bell experiment. I'll explain where the 2 comes from and why anything greater than that isn't possible with classical physics. That includes 2*Sqrt(2) and 4.

The experiment consists of 2 observers, Alice and Bob each with fair coins and detectors that can be rotated to several angles, along with a source of electrons. There are also 4 summers, labeled AB, AB', A'B and A'B'. The summers are all set to zero initially. Alice's detector is able to be set to two angles, a = 0 or a' = 90 degrees. Bob's detector is able to be set to two angles also, b = 45 or b' = 135 degrees.

Now the experiment starts. Alice flips her coin and sets her detector to a(0 degrees) if it's heads, or a'(90 degrees) if it's tails. Bob does the same, flipping his coin and setting his detector at either 45 or 135 degrees. The source, which is halfway between Alice and Bob, emits two electrons which are detected by Alice's and Bob's detectors. The detectors have indicators on them that show whether the electron's spin is aligned (+1) or anti-aligned (-1) with the detector. Alice records this value and calls it A, if she picked angle a, or A' if she picked angle a'. Similarly Bob records his spin value and calls it B or B'. They multiply these values together and get either +1, if both spins were aligned or anti-aligned or -1 if spins were aligned opposite to each other. This value is summed into one of the counters AB, AB', A'B or A'B', depending on angle selection, as should be clear from the notation.

After many experiments, the summers contain the results of all experiments. The following means can be calculated:



by dividing the sum by the number of counts in that sum, N. Since the sum consists of N +1 or -1 values, the mean must always be less than or equal 1 or greater than or equal -1. CHSH is now calculated by the equation:



Which clearly must be less than or equal 2, or greater than or equal -2.

No, it needn't be. You defined CHSH in terms of observations, which are random observations, and therefore the CHSH itself is a random value. For comparison, toss a coin. The expected number of heads is 1/2 but an observed value more or less. A theory does not predict the result of a random process. Same way, in certain theories the expected value of CHSH is no more than 2, but individual observations can be.

[Joy Christian]

The bottom line is that there is no way to get the bound on CHSH to be 2 without cheating.

[jreed]

Since the sum consists of N +1 or -1 values, the mean must always be less than or equal 1 or greater than or equal -1.

Very good. You are catching up. So we have

CHSH is now calculated by the equation:

Indeed. So with the above mean we get the two extreme values,



and



Thus we have



which is indeed as clear as bell.

Sorry, but I must apologize for missing one important step. Remember I said that Alice and Bob each flip a coin to randomize their choice of detector angles. This means that in the mean calculations the term <AB>, for example, can be written as <A><B>, since the mean of the product of two uncorrelated random variables can be written as the product of the means and similarly for the other terms. This allows CHSH to be written as:



or, factoring this:



Now since all these terms are still bounded by -1 and 1, it's clear as a bell that CHSH must be greater than -2 and less than 2.

How did I cheat? I don't like cheaters! Please let me know so I can correct this.

[Joy Christian]

Sorry, but I must apologize for missing one important step. Remember I said that Alice and Bob each flip a coin to randomize their choice of detector angles. This means that in the mean calculations the term <AB>, for example, can be written as <A><B>, since the mean of the product of two uncorrelated random variables can be written as the product of the means and similarly for the other terms. This allows CHSH to be written as:



or, factoring this:



Now since all these terms are still bounded by -1 and 1, it's clear as a bell that CHSH must be greater than -2 and less than 2.

How did I cheat? I don't like cheaters! Please let me know so I can correct this.

This time you did not cheat. You did far worse.

You are supposed to know the elementary fact that in the EPR-B experiments we always observe



and

.

I will let you figure out now what these values give you for your revised expressions for CHSH.

[FrediFizzx]

You are supposed to know the elementary fact that in the EPR-B experiments we always observe



and

.

And I believe those are actually Bell conditions also.

[Joy Christian]

You are supposed to know the elementary fact that in the EPR-B experiments we always observe



and

.

And I believe those are actually Bell conditions also.

And predictions of quantum mechanics, as well as of my local model -- cf. Eqs. (3) and (18) of this paper: http://arxiv.org/pdf/quant-ph/0703179.pdf.

[FrediFizzx]

Here's an explanation of a Bell experiment. I'll explain where the 2 comes from and why anything greater than that isn't possible with classical physics. That includes 2*Sqrt(2) and 4.

We are well versed in the usual story about CHSH so you don't have to explain it. What you (and others) need to explain is why Joy's classical physical model gets since you think it is not possible.

Code: Select all

//Adaptation of Albert Jan Wonnink's original code
//http://challengingbell.blogspot.com/2015/03/numerical-validation-of-vanishing-of.html
function getRandomLambda()
{
     if( rand()>0.5) {return 1;} else {return -1;}
}

     batch test()
{
     set_window_title("Test of Joy Christian's CHSH derivation");
     N=20000; //number of iterations (trials)
     I=e1^e2^e3;
     s=0;
     a1=1.00*e1 +0.0001*e2 + 0.0001*e3;
     b1=0.7071068*e1 + 0.7071068*e2 + 0.0001*e3;
     a2=0.0001*e1 + 1.00*e2 + 0.0001*e3;
     b2=0.7071068*e1 + -0.7071068*e2 + 0.0001*e3;


     for(nn=0;nn<N;nn=nn+1) //perform the experiment N times
     {
          lambda=getRandomLambda(); //lambda is a fair coin,
         //resulting in +1 or -1
          mu=lambda * I;  //calculate the lambda dependent mu

          A1=-mu.a1;
          A2=-mu.a2;
          B1=mu.b1;
          B2=mu.b2;
          q=0;
          if(lambda==1) {q=(A1 B1)+(A1 B2)+(A2 B1)-(A2 B2);}
          else {q=(B1 A1)+(B2 A1)+(B1 A2)-(B2 A2);}
          s=s+q;
      }
      mean_F_A_B=s/N;
      print(mean_F_A_B, "f");
      prompt();
}

Result is,

mean_F_A_B = 2.828427 + 0.000000*e2^e3 + 0.000000*e3^e1

[minkwe]

Which clearly must be less than or equal 2, or greater than or equal -2.

I hope this explanation shows what a Bell believer thinks, and that it is clear as a bell (pun intended).

Did you even bother to read my previous post?

[minkwe]

I will let you figure out now what these values give you for your revised expressions for CHSH.

Not to talk of the fact that his factorization step implies division by zero. Given the level of detail to which the CHSH has been dissected on this forum, I suspect jreed might be joking.

[Mikko]

Remember I said that Alice and Bob each flip a coin to randomize their choice of detector angles. This means that in the mean calculations the term <AB>, for example, can be written as <A><B>, since the mean of the product of two uncorrelated random variables can be written as the product of the means and similarly for the other terms.

This inference is incorrect. <AB> = <A><B> is not generally true. Have you forgotten what A and B mean? There is no reason to assume that A and B are statistically independent. Usually they aren't.

[jreed]

If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

[FrediFizzx]

If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

And here is the link to a PDF of it for those interested.
http://cds.cern.ch/record/142461/files/198009299.pdf
But of course it does not explain why Joy's classical local realistic hidden variable model gets CHSH = . Which is the real physical bound on CHSH. Quantum theory and experiments show that is the actual bound also.

[Joy Christian]

If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

This post reads -- word by word -- as if it were written by Richard Gill. I wouldn't be surprised at all if it were actually written -- word by word -- by Richard Gill, even though he is banned from this forum (as he is banned from many other forums and blogs for misbehaving).

I want to make one other sociological point clear to John Reed, and to anyone else who might be reading this, because I have had enough of the lectures given to me by John Reed about CHSH and Bell inequalities. I wrote my PhD thesis with Prof. Abner Shimony, the "S" in the CHSH. Do you know what that means? That means I am an expert on the foundations of quantum mechanics, and in particular on Bell's theorem. I learned about Bell's theorem not only from Abner Shimony --- the foremost expert on Bell's theorem until he passed away last month --- but from Bell himself, who was a frequent visitor to our research group in the 1980's, until 1990 when he passed away. His famous book came out during that decade, but we were intimately familiar with all the papers in it long before they were republished in that book, from the closed seminars Bell gave to our research group during that decade. I am elaborating on this to stress that I don't need lectures on Bell's theorem or on CHSH from you, John Reed, who has learned about them yesterday, from the dishonest, incompetent, and psychologically disturbed statistician like Richard Gill. And since Gill is lurking on this forum, I am elaborating this also for his benefit. In any case, I learned about "Bertlmann's socks and the nature of reality" from John Bell himself, in a closed seminar to our research group in the 1980's, when that paper had just been published. So don't ever dare to give me a lecture, Gill, about Bell's theorem.

Now about the probabilistic point made in the above post by John Reed -- dictated by Richard Gill. This point has been thoroughly addressed and refuted by Michel Fodje on this forum. See, for example, this thread, and there are plenty more threads like this by Michel on this forum.

But even if we don't accept Michel's argument, all one has to do is to look at my analytical model, as Fred has pointed out above. Just look at the explicit derivation of the bound on CHSH in my "Reply to Gill" [cf. Eq. (26)]. Better still, look at the derivation of the actual correlation E(a, b) = -a.b in Eq. (B10) of that paper. This is explicit analytical derivation, giving correlation between the standard scalar events A(a, s) = +/-1 and B(b, s) = +/-1, where s is the "hidden variable" or initial state.

So get this through your head, Richard Gill. Bell's theorem is dead. It was actually stillborn. And It's corps was put to rest permanently on the 20th of March 2007.

[FrediFizzx]

So get this through your head, Richard Gill. Bell's theorem is dead. It was actually stillborn. And It's corpse was put to rest permanently on the 20th of March 2007.

Amazing that he hasn't figured out or can't figure out yet that he lost the debate. He is still going on about "measurement outcomes". Quantum theory does not use measurement outcomes for its predictions as evidenced by,

http://plato.stanford.edu/entries/bell-theorem/#2 for photons
and
https://drive.google.com/file/d/0B67qmv ... iZjBi/view
for spin 1/2

So a classical local realistic model doesn't need to use outcomes either for its prediction. But we do have simulations using outcomes for those not stuck in a flat right-handedland perspective. But it is measurement outcomes in real experiments that matter the most. They confirm both the quantum theory prediction and Joy's classical local realistic model prediction.

[Xray]

If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

Dear jreed (who, with due respect and imho, does not write here -- in this instance -- like Richard Gill at all),

In the interest of precision and clarity (you quote Bell's eqn (11) from Bertlmann's socks; as underlined above), let us rewrite it as:

P(A,B|a, b, lambda) = P1(A|a, lambda)*P2(B|b, lambda') (1?)

Reason: For clarity in EPRB (the subject of Bell's study), lambda and lambda' (lambda-prime) are correlated by the conservation of total angular momentum such that, in each particle-pair:

lambda + lambda' = 0; or, lambda = - lambda'. (2)

Now: Let the detectors (polariser-analysers) be Alpha and Beta. Then, given that a and b are unit-vectors in 3-space, Alpha and Beta are correlated by the function:

C(Alpha, Beta) = a.b. (3)

So, given the correlation of the particle-pairs by (2), and of the detectors by (3): The detector-outputs A and B must be correlated! So the correct formula is NOT (1?) -- nor Bell's (11) -- but:

P(A,B|a, b, lambda) = P1(A|a, lambda)*P2(B|b, lambda', A) = P1(A|a, lambda; B)*P2(B|b, lambda'). (4)

Much interesting discussion follows but (for now) note how the anti-correlation of each particle-pair in (2) and the direct correlation of the detectors in (3) yield the Expectation E immediately:

E(a, b) = - a.b. (5)

Bell's problem was that he thought that Einstein-locality required (1?). It does not! Einstein-locality and common-sense reality implies (4): Which will be experimentally confirmed via (5)!

Incidentally, with reference to the CHSH limit: A wholly classical experiment (a filtered version of EPRB) would yield:

E(a, b; classical]) = - a.b/2. (6)

So there should be no surprise that a non-filtered (wholly quantum; more tightly correlated) experiment would yield:

E(a, b) > - a.b/2; (7)

to thereby breach CHSH locally and realistically!

HTH. With best regards,

Xray

[Joy Christian]

If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

Dear jreed (who, with due respect and imho, does not write here -- in this instance -- like Richard Gill at all.

Nonsense. "Richard Gill" is written all over John Reed's post above. All John Reed has to do to dispel this accusation is to confirm that his post above was written by him alone, without any interference from Gill. He has to simply state that Gill did not approach him by private email, reffered to this discussion on this forum, and dictated the above post, after John Reed made a real boo-boo in his previous attempt to derive the bound of 2 on CHSH: viewtopic.php?f=6&t=199&p=5572#p5537.

[FrediFizzx]

If you want to find out where CHSH comes from, you should read the article by Bell, "Bertlmann's socks and the nature of reality". It's in his book. This explains it all very well. I've read it and it makes sense, but I won't try to give all the details here. Basically it says that for certain measurements the joint probability P(A,B|a,b,lambda) can be written as:
P1(A|a,lambda)*P2(B|b,lambda) where lambda is a hidden variable. The rest follows from this.

Dear jreed (who, with due respect and imho, does not write here -- in this instance -- like Richard Gill at all.

Nonsense. "Richard Gill" is written all over John Reed's post above. All John Reed has to do to dispel this accusation is to confirm that his post above was written by him alone, without any interference from Gill. He has to simply state that Gill did not approach him by private email, reffered to this discussion on this forum, and dictated the above post, after John Reed made a real boo-boo in his previous attempt to derive the bound of 2 on CHSH: viewtopic.php?f=6&t=199&p=5572#p5537.

I can confirm that Gill has been trying to post as a guest poster for quite some time now what John posted. Guys, please post your own ideas and content. Ok, let's get back on topic.