On Monday, February 6, 2017 at 5:57:47 AM UTC, Jos Bergervoet wrote:   
   > On 2/4/2017 5:43 PM, ben6993 wrote:   
   > .....   
      
   Thanks for the reply Jos.   
      
   > phi=45 gives exactly 0 for the correlation. (This was about   
   > measuring photon polarizations, resulting in "H" or "V".)   
      
   Yes, agreed.  Cos(2*45 degrees) = 0. I am too used to thinking about   
   electrons rather than photons in this context.  So the four cells of   
   the 2x2 table would all be 0.25. But for phi = say 22.5 degrees then   
   the minus cosine term needs to be in the (+, +) cell if you need to   
   make the correlation negative.   
      
   > For other angles, it depends on whether you start with   
   > anti-aligned or aligned photon pairs. (OP of this thread   
   > references a Wikipedia example with *aligned* photon pairs   
   > which makes the correlation positive for phi < 45 deg.)   
      
   Also agreed.  I want only to refer to anti-correlated singlet   
   electron-positron pairs as in Susskind's example.   
      
   >   
   > Be aware that for electrons the angular variation is only   
   > half of that for photons. So the correlation there is 0 for   
   > phi=90 degrees. If the electrons are initially in aloigned   
   > pairs (like the photons above) then we will have positive   
   > correlation between detector results for < 90 deg. angle   
   > between detector orientations, in particular we have full   
   > correlation for phi=0.   
      
   Agreed.  I do know that but when I quoted the 45 degree case above my   
   mind was still set on halving, which I did instead of doubling! Oops.   
      
   >   
   > If, however, the electrons come in anti-aligned pairs (as   
   > in the 'singlet state') then it is the opposite: negative   
   > correlation for less than 90 degrees orientation difference   
   > between the detectors and complete anti-correlation (-1)   
   > with angle phi = 0 between detectors.   
   >   
      
   This is the case I am interested in.   
      
   > I didn't look up the Susskind lecture you refer to, but   
   > for electrons, this second possibility is often used:   
   >    sqrt(1/2) * ( |+-> - |-+> )   
   >   
      
   OK   
      
   > It is of course also possible to start with the other   
   > possibility (aligned instead of anti-aligned), although   
   > it may be more difficult to prepare experimentally.   
   >   
      
   I have sometimes prepared aligned singlets in a computer simulation,   
   but I did not realise that it was possible to prepare them in the   
   lab.  I mean exactly aligned rather than aligned to be pointing in   
   the same hemisphere.   
      
   >   ..   
   > > For electrons the formula for cell (+ +) is different than for   
   > > photons being   
   > > 0.25*(1 - cos phi).   
   >   
   > In any case the variation with angle phi is with half   
   > the rate. And the sign may also be different if the   
   > prepared electron pairs are anti-aligned and you compare   
   > them with photon pairs that are created aligned.   
      
   Completely agree with all of this.   
      
   > > So did Susskind make a mistake?   
   >   
   > I expect it is a matter of comparing different cases..   
      
   >   
   > > What inequality did he break, if any?   
   >   
   > You already wrote that it was about Bell's inequality.   
   > That means he broke the inequality that describes the   
   > limits of possible classical behavior. What Bell's   
   > inequality does is telling you what is possible with   
   > classical physics. More is possible in QM.   
      
   I am quite convinced that Susskind did not break the inequality that   
   he set out to break which is AB' + BC' >= AC' using singlet electron   
   pairs.  Where a=0, b=45 and c=90 degrees.   
      
   He used QM calculations to find AB' = BC' = 0.073, where AC' = 0.25.   
   So it is certainly true that 0.073 + 0.073 is not >= 0.25.   
      
   But using his values one gets a correlation of +0.707 instead of   
   -0.707.  So he seems to have broken some inequality or other   
   involving AB = BC = 0.073 and AC = 0.25.   
      
   Also, I found the value 0.073 using a computer simulation where the   
   compute program knows the particles' hidden variables which are the   
   particles exact vectors (p). But I found that value 0.073 for cells   
   (+ +) and (- -) whereas Suskind found that value for cells (+ -) and   
   (- +).   
      
   I calculated p dot a and p dot b  for each particle pair and the   
   correlated these data.  The correlation was -0.707.   
   To get 0.073, I accumulated values where p dot a was positive for   
   particle 1 and p dot b was positive for the partner particle. So I   
   calculated with hidden variables the same values as did Sussking   
   using QM.   
      
   What neither Susskind nor I did was to calculate -0.707 using integer   
   values of A and B which of course is expected to be 0.5 in the long   
   run.  We both started out with integer values but then took   
   fractional values of them when projected onto exact detector vectors.   
   QM may not have done this as explicitly as I did but it found the   
   same result.  That is excellent for QM as it managed to do so without   
   using individual particles' hidden vectors.  But it did use   
   projection operators  to find 0.073 which gives a clue that QM was   
   doing something similar in intent.   
      
   What is puzzling me after my calculations which match Susskind's is   
   why finding 0.073 by QM leads one to think that you can get a   
   correlation of -0.707 for a 2x2 table of integer values of A and B?   
   The proportions in the 2X2 table are (or have the same value as)   
   accumulations of fractional loadings on exact detector vectors.  IMO   
   to break Bell's Inequality you need to correlate Alice's and Bob's   
   integer measurements direct as -0.707 and not mess with them first.   
   My calculation could not beat 0.5. Whatever led QM to think that a   
   direct correlation of integer measurements could break the barrier?   
      
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