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Init and final.
2020-12-23 21:45:08 +00:00
% Default to the notebook output style
% Inherit from the specified cell style.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
% Nicer default font (+ math font) than Computer Modern for most use cases
\usepackage{mathpazo}
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts ![](path) syntax from Markdown).
\usepackage{graphicx}
% We will generate all images so they have a width \maxwidth. This means
% that they will get their normal width if they fit onto the page, but
% are scaled down if they would overflow the margins.
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
\else\Gin@nat@width\fi}
\makeatother
\let\Oldincludegraphics\includegraphics
% Set max figure width to be 80% of text width, for now hardcoded.
\renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}}
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionLabelFormat{nolabel}{}
\captionsetup{labelformat=nolabel}
\usepackage{adjustbox} % Used to constrain images to a maximum size
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
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\begin{document}
\maketitle
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}1}]:} \PY{c+c1}{\PYZsh{} EPR Paradox Example}
\PY{c+c1}{\PYZsh{} A neutral pi meson (pi0) decays into an electron/positron (e\PYZhy{}/e+) }
\PY{c+c1}{\PYZsh{} pair.}
\PY{c+c1}{\PYZsh{} pi0 \PYZhy{}\PYZhy{}\PYZgt{} e+ + e\PYZhy{} (electron\PYZhy{}positron pair)}
\PY{c+c1}{\PYZsh{} pi0 has angular momentum l = s = 0}
\PY{c+c1}{\PYZsh{} Align electron and positron detectors in opposite directions.}
\PY{c+c1}{\PYZsh{} | pi0 |}
\PY{c+c1}{\PYZsh{} |e\PYZhy{} \PYZlt{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{} \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZgt{} e+|}
\PY{c+c1}{\PYZsh{} Where hbar = 1, the measurement of the spin in some direction is }
\PY{c+c1}{\PYZsh{} +/\PYZhy{}1 with spin state [1 0] (up) or [0 1] (down).}
\PY{c+c1}{\PYZsh{} The Pauli exclusion principle with conserved angular momentum 0}
\PY{c+c1}{\PYZsh{} dictates this system must be in the singlet state }
\PY{c+c1}{\PYZsh{} chi = [1/sqrt(2) (|up+\PYZgt{}|down\PYZhy{}\PYZgt{} \PYZhy{} |down+\PYZgt{}|up\PYZhy{}\PYZgt{})].}
\PY{c+c1}{\PYZsh{} In this state, if the positron is measured to have spin [1 0], the }
\PY{c+c1}{\PYZsh{} electron must have spin [0 1], or vice versa. There is an equal }
\PY{c+c1}{\PYZsh{} probability to find either state during the first measurement.}
\PY{c+c1}{\PYZsh{} This view is consistent with the realist view. The realist view could }
\PY{c+c1}{\PYZsh{} hold that the electron and position had those angular momenta }
\PY{c+c1}{\PYZsh{} from creation.}
\PY{c+c1}{\PYZsh{} EPR assumes influences cannot propagate faster than the speed of }
\PY{c+c1}{\PYZsh{} light. \PYZdq{}Wave function collapse\PYZdq{} is apparently instantaneous, however.}
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{patches} \PY{k}{as} \PY{n+nn}{mpatches}
\PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}2}]:} \PY{c+c1}{\PYZsh{} If information about the measurement of the wave function propagated}
\PY{c+c1}{\PYZsh{} at a finite speed, the particles could conceivably be measured such}
\PY{c+c1}{\PYZsh{} that both are equally likely to hold either spin up or spin down.}
\PY{c+c1}{\PYZsh{} What would happen if the measurements were uncorrelated?}
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{elepatch} \PY{o}{=} \PY{n}{mpatches}\PY{o}{.}\PY{n}{Patch}\PY{p}{(}\PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{blue}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{e\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{pospatch} \PY{o}{=} \PY{n}{mpatches}\PY{o}{.}\PY{n}{Patch}\PY{p}{(}\PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{orange}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{e+}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{}plt.legend(handles=[elepatch,pospatch])}
\PY{n}{plt}\PY{o}{.}\PY{n}{text}\PY{p}{(}\PY{l+m+mf}{0.5}\PY{p}{,}\PY{l+m+mi}{565}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{e\PYZhy{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{size}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{text}\PY{p}{(}\PY{l+m+mf}{3.5}\PY{p}{,}\PY{l+m+mi}{565}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{e+}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{size}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Uncorrelated Spins}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{600}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{,}\PY{l+m+mf}{3.125}\PY{p}{,}\PY{l+m+mf}{3.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{down}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{up}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{down}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{up}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_1_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}3}]:} \PY{c+c1}{\PYZsh{} How many violations of angular momentum would be measured?}
\PY{c+c1}{\PYZsh{} From running several simulations, it\PYZsq{}s evident a violation in the }
\PY{c+c1}{\PYZsh{} conservation of angular momentum would be measured half of the time.}
\PY{c+c1}{\PYZsh{} We can conclude that the information that the entangled particles are in}
\PY{c+c1}{\PYZsh{} orthogonal spin states is instantaneously agreed once a measurement}
\PY{c+c1}{\PYZsh{} is made.}
\PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{trial} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}\PY{p}{:}
\PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}
\PY{k}{if} \PY{n}{elespin} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
\PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{else}\PY{p}{:}
\PY{n}{elespin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{if} \PY{n}{posspin} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
\PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{else}\PY{p}{:}
\PY{n}{posspin} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{elespin}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}
\PY{n}{posspin}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}
\PY{n}{chi\PYZus{}squared} \PY{o}{=} \PY{n}{elespin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1; 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{o}{*}\PY{n}{posspin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0; 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{posspin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{1; 0}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{o}{*}\PY{n}{elespin}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{matrix}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{0; 1}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{if} \PY{n}{chi\PYZus{}squared} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:}
\PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1}
\PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
\PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{l+m+mi}{1000}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
\PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{400}\PY{p}{,}\PY{l+m+mi}{600}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Conservation Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}3}]:} <matplotlib.figure.Figure at 0x7f22020419e8>
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_2_1.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{verbatim}
<matplotlib.figure.Figure at 0x7f22020419e8>
\end{verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}4}]:} \PY{c+c1}{\PYZsh{} Bell\PYZsq{}s Experiment took this a step further, to rule out locality }
\PY{c+c1}{\PYZsh{} completely. Establish the detectors to \PYZdq{}float\PYZdq{} such that they }
\PY{c+c1}{\PYZsh{} measure the components of the spins of the electron and positron }
\PY{c+c1}{\PYZsh{} along a unit vector a or b, with angles phi\PYZus{}a and phi\PYZus{}b, }
\PY{c+c1}{\PYZsh{} respectively. Compute a product P of the spins in units of hbar/2. }
\PY{c+c1}{\PYZsh{} This will give +/\PYZhy{}1. }
\PY{c+c1}{\PYZsh{} / pi0 \PYZbs{}}
\PY{c+c1}{\PYZsh{} /e\PYZhy{} \PYZlt{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{} \PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZgt{} e+\PYZbs{}}
\PY{c+c1}{\PYZsh{} QM predicts P(a,b) = \PYZhy{}a dot b, the expectation value of the product }
\PY{c+c1}{\PYZsh{} of the spins.}
\PY{c+c1}{\PYZsh{} In 1964, Bell derived the Bell inequality for a local hidden variable}
\PY{c+c1}{\PYZsh{} theory: abs(P(a,b) \PYZhy{} P(a,c)) \PYZlt{}= 1 + P(b,c)}
\PY{c+c1}{\PYZsh{} For any local hidden variable theory, the Bell inequality must hold. }
\PY{c+c1}{\PYZsh{} It introduces a third unit vector c, which is any other unit vector }
\PY{c+c1}{\PYZsh{} than a or b.}
\PY{c+c1}{\PYZsh{} Does the quantum mechanical prediction violate the Bell inequality?}
\PY{c+c1}{\PYZsh{} Testing several detector configurations in a plane, systematically }
\PY{c+c1}{\PYZsh{} from 0 to pi, we determine whether the QM prediction is consistent }
\PY{c+c1}{\PYZsh{} with a local hidden variable theory.}
\PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{trials} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{step\PYZus{}a} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{step\PYZus{}b} \PY{o+ow}{in} \PY{n+nb}{range} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{:}
\PY{n}{phi\PYZus{}a} \PY{o}{=} \PY{n}{step\PYZus{}a}\PY{o}{/}\PY{l+m+mi}{6}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
\PY{n}{phi\PYZus{}b} \PY{o}{=} \PY{n}{step\PYZus{}b}\PY{o}{/}\PY{l+m+mi}{6}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
\PY{n}{phi\PYZus{}c} \PY{o}{=} \PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{phi\PYZus{}b}
\PY{n}{P\PYZus{}ab} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}b}\PY{p}{)}
\PY{n}{P\PYZus{}ac} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
\PY{n}{P\PYZus{}bc} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}b} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
\PY{n}{bell\PYZus{}lhs} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{P\PYZus{}ab} \PY{o}{\PYZhy{}} \PY{n}{P\PYZus{}ac}\PY{p}{)}
\PY{n}{bell\PYZus{}rhs} \PY{o}{=} \PY{l+m+mi}{1} \PY{o}{+} \PY{n}{P\PYZus{}bc}
\PY{k}{if} \PY{n}{bell\PYZus{}lhs} \PY{o}{\PYZgt{}} \PY{n}{bell\PYZus{}rhs}\PY{p}{:}
\PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1}
\PY{n}{trials} \PY{o}{=} \PY{n}{trials} \PY{o}{+} \PY{l+m+mi}{1}
\PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
\PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{n}{trials}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
\PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{p}{,}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{10}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Hidden Locality Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}4}]:} <matplotlib.figure.Figure at 0x7f22020737f0>
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_3_1.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{verbatim}
<matplotlib.figure.Figure at 0x7f22020737f0>
\end{verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}5}]:} \PY{c+c1}{\PYZsh{} It is seen that the QM prediction disagrees with a local hidden }
\PY{c+c1}{\PYZsh{} variable theory in a significant number of configurations.}
\PY{c+c1}{\PYZsh{} On average, for random orientations between 0 and pi, how often? }
\PY{c+c1}{\PYZsh{} After running several trials, it appears to be about half of the, }
\PY{c+c1}{\PYZsh{} which is what one would expect from the quantum mechanical}
\PY{c+c1}{\PYZsh{} prediction.}
\PY{n}{violations} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{trials} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{rand\PYZus{}a} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{rand\PYZus{}b} \PY{o+ow}{in} \PY{n+nb}{range} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{)}\PY{p}{:}
\PY{n}{phi\PYZus{}a} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{rand}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
\PY{n}{phi\PYZus{}b} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{rand}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{np}\PY{o}{.}\PY{n}{pi}
\PY{n}{phi\PYZus{}c} \PY{o}{=} \PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{phi\PYZus{}b}
\PY{n}{P\PYZus{}ab} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}b}\PY{p}{)}
\PY{n}{P\PYZus{}ac} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}a} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
\PY{n}{P\PYZus{}bc} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1} \PY{o}{*} \PY{n}{np}\PY{o}{.}\PY{n}{cos}\PY{p}{(}\PY{n}{phi\PYZus{}b} \PY{o}{\PYZhy{}} \PY{n}{phi\PYZus{}c}\PY{p}{)}
\PY{n}{bell\PYZus{}lhs} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{P\PYZus{}ab} \PY{o}{\PYZhy{}} \PY{n}{P\PYZus{}ac}\PY{p}{)}
\PY{n}{bell\PYZus{}rhs} \PY{o}{=} \PY{l+m+mi}{1} \PY{o}{+} \PY{n}{P\PYZus{}bc}
\PY{k}{if} \PY{n}{bell\PYZus{}lhs} \PY{o}{\PYZgt{}} \PY{n}{bell\PYZus{}rhs}\PY{p}{:}
\PY{n}{violations} \PY{o}{=} \PY{n}{violations} \PY{o}{+} \PY{l+m+mi}{1}
\PY{n}{trials} \PY{o}{=} \PY{n}{trials} \PY{o}{+} \PY{l+m+mi}{1}
\PY{n}{zeroes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{violations}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
\PY{n}{ones} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{full}\PY{p}{(}\PY{p}{(}\PY{n}{trials}\PY{o}{\PYZhy{}}\PY{n}{violations}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{dtype}\PY{o}{=}\PY{n+nb}{int}\PY{p}{)}
\PY{n}{result} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{p}{(}\PY{n}{ones}\PY{p}{,}\PY{n}{zeroes}\PY{p}{)}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylim}\PY{p}{(}\PY{p}{[}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{p}{,}\PY{n}{trials}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{10}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlim}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xticks}\PY{p}{(}\PY{p}{[}\PY{l+m+mf}{0.125}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{]}\PY{p}{,}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{violation}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{adherence}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{tick\PYZus{}params}\PY{p}{(}\PY{n}{axis}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{both}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{labelsize}\PY{o}{=}\PY{l+m+mi}{15}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{suptitle}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Hidden Locality Violations}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{fontsize}\PY{o}{=}\PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{hist}\PY{p}{(}\PY{p}{[}\PY{n}{result}\PY{p}{]}\PY{p}{,}\PY{n}{bins}\PY{o}{=}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}5}]:} <matplotlib.figure.Figure at 0x7f220202b080>
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_4_1.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{verbatim}
<matplotlib.figure.Figure at 0x7f220202b080>
\end{verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}6}]:} \PY{c+c1}{\PYZsh{} The prediction (Bell\PYZsq{}s inequality) made by assuming a local hidden}
\PY{c+c1}{\PYZsh{} variable is violated by a significant number of the possibile }
\PY{c+c1}{\PYZsh{} bborientations. This simulation cannot determine which theory is}
\PY{c+c1}{\PYZsh{} correct, but the QM prediction has been confirmed through experiment.}
\PY{c+c1}{\PYZsh{} No hidden local variable holds actionable information about the }
\PY{c+c1}{\PYZsh{} state. Entangled states retain their entanglement in a non\PYZhy{}local }
\PY{c+c1}{\PYZsh{} nature.}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}7}]:} \PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
% Add a bibliography block to the postdoc
\end{document}
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