Simulation for "Local origins of quantum correlations rooted in geometric algebra"
Author
Fred Diether
Title
Simulation for "Local origins of quantum correlations rooted in geometric algebra"
Description
A simulation of a local-realistic model of quantum correlations predicted by the entangled singlet state.
Category
Academic Articles & Supplements
Keywords
Clifford algebra, quantum correlations, geometric algebra, norm division algebras, local causality
URL
http://www.notebookarchive.org/2022-04-2tqt8f0/
DOI
https://notebookarchive.org/2022-04-2tqt8f0
Date Added
2022-04-06
Date Last Modified
2022-04-06
File Size
0.69 megabytes
Supplements
Rights
CC BY 4.0
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This file contains supplementary data for Joy Christian, “Local origins of quantum correlations rooted in geometric algebra,”
https://doi.org/10.48550/arXiv.2205.11372.
With 3D Vectors!
https://doi.org/10.48550/arXiv.2205.11372.
With 3D Vectors!
Simulation for "Local origins of quantum correlations rooted in geometric algebra"
Simulation for "Local origins of quantum correlations rooted in geometric algebra"
Created by Fred Diether, May 2022.
Load Clifford Package, Set Run Time Parameters, Initialize Arrays and Tables
Load Clifford Package, Set Run Time Parameters, Initialize Arrays and Tables
In[]:=
<<"clifford.m"m=40000;s3=ConstantArray[0,m];s4=ConstantArray[0,m];Ls1=ConstantArray[0,m];Ls2=ConstantArray[0,m];qa1=ConstantArray[0,m];qb1=ConstantArray[0,m];A=ConstantArray[0,m];B=ConstantArray[0,m];a1=ConstantArray[0,m];b1=ConstantArray[0,m];I3=Pseudoscalar[3];pc=ConstantArray[0,m];plotpc=Table[{0,0},m];
Generating Particle Data with Three Independent Do-Loops
Generating Particle Data with Three Independent Do-Loops
In[]:=
Do[s=RandomPoint[Sphere[]];(*Singlet3Dvector,commoncause*)s3[[l]]=s;s4[[l]]=-s;Ls1[[l]]=s.{i,j,k};(*Converttoquaternion,particlespintoA*)Ls2[[l]]=-s.{i,j,k},{l,m}](*ParticlespintoB--conservationofangularmomentum*)
In[]:=
Do[a=RandomPoint[Sphere[]];(*Detector3Dvector*)a1[[l]]=a;Da=a.{i,j,k};(*Converttoquaternion*)qa=QuaternionProduct[Da,Ls1[[l]]];(*Particlespin-detectorinteraction*)qa1[[l]]=qa;A[[l]]=a.Limit[s1,s1->Sign[a.s3[[l]]]a]+Total[Cross[a,Limit[s1,s1->Sign[a.s3[[l]]]a]]],{l,m}]
In[]:=
Do[b=RandomPoint[Sphere[]];(*Detector3Dvector*)b1[[l]]=b;Db=b.{i,j,k};(*Converttoquaternion*)qb=QuaternionProduct[Ls2[[l]],Db];(*Particlespin-detectorinteraction*)qb1[[l]]=qb;B[[l]]=b.Limit[s2,s2->Sign[b.s4[[l]]]b]+Total[Cross[b,Limit[s2,s2->Sign[b.s4[[l]]]b]]],{l,m}]
Verification of the Analytical 3-Sphere Model Prediction Based on
Geometric Algebra using Quaternions
Verification of the Analytical 3-Sphere Model Prediction Based on
Geometric Algebra using Quaternions
Geometric Algebra using Quaternions
In[]:=
Do[r0=ToBasis[(Re[-qa1[[l]]]Limit[Cross[s2,b1[[l]]],s2Sign[Re[qb1[[l]]]]b1[[l]]]+Re[qb1[[l]]]Limit[Cross[a1[[l]],s1],s1Sign[Re[-qa1[[l]]]]a1[[l]]]-Cross[Limit[Cross[a1[[l]],s1],s1Sign[Re[-qa1[[l]]]]a1[[l]]],Limit[Cross[s2,b1[[l]]],s2Sign[Re[qb1[[l]]]]b1[[l]]]])/(Sin[ArcCos[a1[[l]].b1[[l]]]])];Lr0=InnerProduct[I3,r0];r1=Im[qa1[[l]]];r2=Im[qb1[[l]]];qpc={(Re[qa1[[l]]]*Re[qb1[[l]]]-r1.r2)+Lr0};pc[[l]]=qpc;ϕA=ArcTan[a1[[l]][[1]],a1[[l]][[2]]];ϕB=ArcTan[b1[[l]][[2]],b1[[l]][[1]]];If[ϕA*ϕB>0,θ=ArcCos[a1[[l]].b1[[l]]]*180/π,θ=(2π-ArcCos[a1[[l]].b1[[l]]])*180/π];plotpc[[l]]={θ,qpc[[1]]},{l,m}]
In[]:=
AveA=N[Total[A]/m];AveB=N[Total[B]/m];Print["<A> = ",AveA];Print["<B> = ",AveB];meanpc=N[Mean[pc]];Print["Imaginary part vanishes, meanpc = ",meanpc];sim=ListPlot[plotpc,PlotMarkers{Automatic,Small},AspectRatio8/16,Ticks{{{0,0°},{45,45°},{90,90°},{135,135°},{180,180°},{225,225°},{270,270°},{315,315°},{360,360°}},Automatic},GridLinesAutomatic,AxesOrigin{0,-1.0}];p1=Plot[-1+2xDegree/π,{x,0,180},PlotStyle{Gray,Dashed}];p2=Plot[3-2xDegree/π,{x,180,360},PlotStyle{Gray,Dashed}];negcos=Plot[-Cos[xDegree],{x,0,360},PlotStyle{Magenta}];Show[sim,p1,p2,negcos]
<A> = 0.00455
<B> = -0.0067
Imaginary part vanishes, meanpc = {-0.000191771}
Out[]=
Blue is the correlation data, magenta is the negative cosine curve for an exact match.
Cite this as: Fred Diether, "Simulation for "Local origins of quantum correlations rooted in geometric algebra"" from the Notebook Archive (2022), https://notebookarchive.org/2022-04-2tqt8f0
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