I never heard of the CSHS Inequality until I read it in another thread.(adsbygoogle = window.adsbygoogle || []).push({});

The other interesting item was this: DrChinese said: ↑As a practical matter: most Bell tests do not use the 25% vs 33% standard for ruling out classical theories. Instead they use what is called the CHSH inequality. You can read about it in the references. It has an upper bound of 2.00 for classical (local realistic) theories, so predicts something below 2.00. That maps to the 33%. The quantum theoretical max is about 2.82. Bell tests usually give a value between 2.25 and 2.40.

I think an important part of that discussion is the more hits are ignored, the easier it is for local realistic theories to score over 2.00. So I just had to try. DrChinese said: ↑I usually talk about match % because it is easier to discuss and relate to the cos^2 formula. Then the formula is:

Matches/(Matches+NonMatches) where hits that cannot be paired are ignored. In real experiments, unpaired hits are usually noted and discussed.

For those familiar with C++, I offer this simulated experiment:When I ran this, I got these results:Code (Text):#include "stdafx.h"

#define _CRT_RAND_S

#include <stdlib.h>

#include <conio.h>

#include <math.h>

#include <atlstr.h>

using namespace System;

#define TRYCOUNT 1000

#define RADIANS 0.01745329251994329576923690768489

//

// Random number generators: range of 0.0 to 1.0.

double RNG()

{

unsigned int nRandom;

rand_s(&nRandom);

return ((double)nRandom) / UINT_MAX;

}

//

// We have a selection of 4 polarizer/splitters.

static double Splitters[4] = { 0.0, 45.0, 22.5, 67.5 };

//

// Here is the noiseless QM model.

void QMModel(double AngleA, double AngleB, int &ResultA, int &ResultB)

{

double Agree;

ResultA = ResultB = (RNG()<0.5) ? -1 : 1;

Agree = pow( (cos(RADIANS*(AngleA-AngleB))), 2);

if(RNG()>Agree) ResultA = -ResultA;

}

//

// Here is a local model that includes built-in misses.

// Result will be:

// +1: Detected by the "+" detector.

// -1: Detected by the "-" detector.

// 0: Not detected!

//

void LocalModel(double Hidden, double Angle, int &Result)

{

double Split;

Result = 1;

Split = cos(RADIANS*(Angle-Hidden));

if(Split<0.0) {

Split = -Split;

Result = -1;

}

if(RNG()>Split) Result = 0;

}

//

void ExpectedValues

(int nTries, int nAngleA, int nAngleB, double &EQM, double &ELocal)

{

int nQMDiscard, nQMAgree, nQMDisagree;

int nLocalDiscard, nLocalAgree, nLocalDisagree;

double AngleA, AngleB, Hidden;

int nTry, ResultA, ResultB, nMeasured, EVNumerator;

CStringA Report;

nQMDiscard = nLocalDiscard = 0;

nQMAgree = nLocalAgree = nQMDisagree = nLocalDisagree = 0;

AngleA = Splitters[nAngleA];

AngleB = Splitters[nAngleB];

for(nTry=0;nTry<nTries;nTry++) {

QMModel(AngleA,AngleB,ResultA,ResultB);

if((ResultA==0)||(ResultB==0)) nQMDiscard++;

else if((ResultA>0)&&(ResultB>0)) nQMAgree++;

else if((ResultA<0)&&(ResultB<0)) nQMAgree++;

else nQMDisagree++;

//

// In the local model, a hidden value is carried by both particles.

Hidden = 360.0 * RNG();

LocalModel(Hidden,AngleA,ResultA);

LocalModel(Hidden,AngleB,ResultB);

if((ResultA==0)||(ResultB==0)) nLocalDiscard++;

else if((ResultA>0)&&(ResultB>0)) nLocalAgree++;

else if((ResultA<0)&&(ResultB<0)) nLocalAgree++;

else nLocalDisagree++;

}

//

Report.Format(

"\nResults with splitters: A=%1.1f°, B%1.1f°\n",

AngleA, AngleB

);

puts(Report);

//

// Report noiseless QM results.

nMeasured = nQMAgree + nQMDisagree;

EVNumerator = nQMAgree - nQMDisagree;

EQM = ((double)EVNumerator)/nMeasured;

Report.Format(

" Noiseless QM Model:\n"

" Discard: %d/%d (%.2f%%)\n"

" Agree: %d/%d (%.2f%%)\n"

" Disagree: %d/%d (%.2f%%)\n"

" Expected: %d/%d (%.2f%%)\n",

nQMDiscard, nTries, (nQMDiscard*100.0)/nTries,

nQMAgree, nMeasured, (nQMAgree*100.0)/nMeasured,

nQMDisagree, nMeasured, (nQMDisagree*100.0)/nMeasured,

EVNumerator, nMeasured, 100.0*EQM

);

puts(Report);

//

// Report local results.

nMeasured = nLocalAgree + nLocalDisagree;

EVNumerator = nLocalAgree - nLocalDisagree;

ELocal = ((double)EVNumerator)/nMeasured;

Report.Format(

" Local Model:\n"

" Discard: %d/%d (%.2f%%)\n"

" Agree: %d/%d (%.2f%%)\n"

" Disagree: %d/%d (%.2f%%)\n"

" Expected: %d/%d (%.2f%%)\n",

nLocalDiscard, nTries, (nLocalDiscard*100.0)/nTries,

nLocalAgree, nMeasured, (nLocalAgree*100.0)/nMeasured,

nLocalDisagree, nMeasured, (nLocalDisagree*100.0)/nMeasured,

EVNumerator, nMeasured, 100.0*ELocal

);

puts(Report);

}

//

int main()

{

double dQM_CSHS, dLl_CSHS;

double EQMAB, EQMAb, EQMaB, EQMab, ELlAB, ELlAb, ELlaB, ELlab;

CStringA Report;

//

// Perform the experiment four times with the polarizing splitters

// at different setting combinations.

ExpectedValues(TRYCOUNT, 0, 2, EQMAB, ELlAB);

ExpectedValues(TRYCOUNT, 0, 3, EQMAb, ELlAb);

ExpectedValues(TRYCOUNT, 1, 2, EQMaB, ELlaB);

ExpectedValues(TRYCOUNT, 1, 3, EQMab, ELlab);

//

// Tally and report the results.

dQM_CSHS = EQMAB - EQMAb + EQMaB + EQMab;

dLl_CSHS = ELlAB - ELlAb + ELlaB + ELlab;

Report.Format(

"\nQM CSHS = %.4f\nLocal CSHS = %.4f\n",

dQM_CSHS,dLl_CSHS

);

puts(Report);

_getch();

return 0;

}It is claimed (wikipedia) that the angles I picked, 0, 45, 22.5, and 67.5, produce the greatest deviation from the classical results. Is this why I got such as high value for the QM model (2.8) or did I mess something up?Code (Text):

Results with splitters: A=0.0°, B22.5°

Noiseless QM Model:

Discard: 0/1000 (0.00%)

Agree: 833/1000 (83.30%)

Disagree: 167/1000 (16.70%)

Expected: 666/1000 (66.60%)

Local Model:

Discard: 542/1000 (54.20%)

Agree: 455/458 (99.34%)

Disagree: 3/458 (0.66%)

Expected: 452/458 (98.69%)

Results with splitters: A=0.0°, B67.5°

Noiseless QM Model:

Discard: 0/1000 (0.00%)

Agree: 138/1000 (13.80%)

Disagree: 862/1000 (86.20%)

Expected: -724/1000 (-72.40%)

Local Model:

Discard: 676/1000 (67.60%)

Agree: 247/324 (76.23%)

Disagree: 77/324 (23.77%)

Expected: 170/324 (52.47%)

Results with splitters: A=45.0°, B22.5°

Noiseless QM Model:

Discard: 0/1000 (0.00%)

Agree: 855/1000 (85.50%)

Disagree: 145/1000 (14.50%)

Expected: 710/1000 (71.00%)

Local Model:

Discard: 542/1000 (54.20%)

Agree: 457/458 (99.78%)

Disagree: 1/458 (0.22%)

Expected: 456/458 (99.56%)

Results with splitters: A=45.0°, B67.5°

Noiseless QM Model:

Discard: 0/1000 (0.00%)

Agree: 853/1000 (85.30%)

Disagree: 147/1000 (14.70%)

Expected: 706/1000 (70.60%)

Local Model:

Discard: 548/1000 (54.80%)

Agree: 446/452 (98.67%)

Disagree: 6/452 (1.33%)

Expected: 440/452 (97.35%)

QM CSHS = 2.8060

Local CSHS = 2.4313

The 2.4313 value is slightly lucky. When I set "TRIES" to 100000, I got value closer to 2.40. In any case, assuming the code is good, it demonstrates that an experimental value of 2.4 might be discounted if the number of discards is as much as the number of hits.

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# CSHS Inequality - and noise

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