From cf43061cd0ff25e00ff8df21c226844c1f96d995 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@rutgers.edu>
Date: Thu, 16 Feb 2023 11:34:29 -0500
Subject: As presented on 2022-02-16

---
 Jauslin_gmathphys_2023.tex | 552 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 552 insertions(+)
 create mode 100644 Jauslin_gmathphys_2023.tex

(limited to 'Jauslin_gmathphys_2023.tex')

diff --git a/Jauslin_gmathphys_2023.tex b/Jauslin_gmathphys_2023.tex
new file mode 100644
index 0000000..55e7fb5
--- /dev/null
+++ b/Jauslin_gmathphys_2023.tex
@@ -0,0 +1,552 @@
+\documentclass{ian-presentation}
+
+\usepackage[hidelinks]{hyperref}
+\usepackage{graphicx}
+\usepackage{xcolor}
+\usepackage{amsmath}
+\usepackage{array}
+\usepackage{ulem}
+
+
+\definecolor{highlight}{HTML}{981414}
+\long\def\high#1{{\color{highlight}#1}}
+
+\begin{document}
+\pagestyle{empty}
+\hbox{}\vfil
+\bf\Large
+\hfil Bell's inequalities\par
+\smallskip
+\hfil \large and non-locality\par
+\vfil
+\large
+\hfil Ian Jauslin
+\rm\normalsize
+\vfil
+Youtube channel: {\tt \href{https://www.youtube.com/@ianjauslin9430}{ianjauslin}}
+\hfill{\tt \href{http://ian.jauslin.org}{http://ian.jauslin.org}}
+\eject
+
+\setcounter{page}1
+\pagestyle{plain}
+
+\title{Introduction}
+\begin{itemize}
+  \item
+  Bell's inequalities:
+  \href{https://link.aps.org/pdf/10.1103/PhysicsPhysiqueFizika.1.195}{[Bell, 1964]},
+  \href{https://doi.org/10.1103/PhysRevLett.23.880}{[Clauser, Horne, Shimony, Holt, 1969]},\par
+  \href{https://cds.cern.ch/record/980036/files/197508125.pdf}{[Bell, 1975]}...
+
+  \item
+  Prove that quantum mechanics is a \high{non-local} theory.
+
+  \item
+  Actually, they are extremely general, and only assume some carefully chosen assumptions about the probabilistic nature of quantum theory.
+
+  \item
+  Often characterized as forbidding ``local hidden variable theories'', as we shall see, this is not exactly untrue, but is misleading.
+
+  \item
+  Nobel prize 2022: Aspect, Clauser, Zeilinger: experimental verification of the predictions of quantum mehcanics.
+
+  \item
+  Reference: \high{\href{http://www.scholarpedia.org/article/Bell\%27s_theorem}{{\it Scholarpedia} article by Goldstein, Norsen, Tausk, Zanghi.}}
+\end{itemize}
+
+\vfill
+\eject
+
+\hbox{}
+\vfill
+\hfil{\bf\Large Part I}\par\bigskip
+\hfil{\bf\Large Bell, 1964}
+\vfill
+\eject
+
+\title{Bell's theorem}
+\begin{itemize}
+  \item
+  Step 1: the EPR argument: \href{https://doi.org/10.1103/PhysRev.47.777}{[Einstein, Podolsky, Rosen, 1935]}.
+\end{itemize}
+\vfill
+\eject
+
+\hfil
+\includegraphics[trim=10 1in 2in 2in, clip, height=\textheight]{epr.pdf}
+\eject
+
+\title{EPR}
+\begin{itemize}
+  \item
+  Two observers at distance $\Delta x$ measure the spin of electrons.
+
+  \item
+  Electron spin: can be measured in any \high{direction} in $\mathcal S^2$, returns \high{$+1$ or $-1$}.
+
+  \item
+  Before measurement, the electrons are in an \high{entangled state}: they are \high{anticorrelated} (if one returns $+1$, the other returns $-1$).
+
+  \item
+  In the usual approach to quantum mechanics, observables \high{do not have values before they are measured}.
+
+  \item
+  The observers perform their measurement at the same time.
+  \high{If the world were local}, then, since the observers are spatially separated, one measurement cannot affect the other.
+
+  \item
+  But the outcomes are \high{perfectly anticorrelated}.
+  Therefore, the outcomes had to be \high{determined before} the measurement was done (``hidden variables'').
+\end{itemize}
+\vfill
+\eject
+
+\title{Bell's theorem}
+\begin{itemize}
+  \item
+  Step 1: the EPR argument: \href{https://doi.org/10.1103/PhysRev.47.777}{[Einstein, Podolsky, Rosen, 1935]}:
+  $$
+  \boxed{\mathrm{QM\ is\ local}\quad\Longrightarrow\quad \mathrm{observables\ have\ predetermined\ values}}
+  $$
+
+  \item
+  Step 2: Bell's inequality.
+\end{itemize}
+\vfill
+\eject
+
+\hfil
+\includegraphics[trim=10 1in 2in 2in, clip, height=\textheight]{epr.pdf}
+\eject
+
+\title{Bell's inequality (pigeonhole)}
+\vskip-15pt
+\begin{itemize}
+  \item
+  In the EPR setting, the observers each make three measurements (e.g. choosing three different directions of spin).
+  The outcomes of the measurements are \high{random}.
+
+  \item
+  We \high{assume} that the outcomes of the measurements \high{exist} independently of the measurement (\high{predetermined values}).
+  In other words, we can represent the outcome of measurements as random variables that exist \high{simultaneously}:
+  $$
+    Z_1^{(A)},Z_2^{(A)},Z_3^{(A)}\in\{-1,+1\}
+    ,\quad
+    Z_1^{(B)},Z_2^{(B)},Z_3^{(B)}\in\{-1,+1\}
+  $$
+  that are distributed according to a probability distribution $\mathbb P$.
+
+  \item
+  We assume \high{perfect anticorrelation}:
+  $$
+    \mathbb P(Z_i^{(A)}\neq Z_i^{(B)})=1
+    .
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Bell's inequality (pigeonhole)}
+\begin{itemize}
+  \item
+  By the pigeonhole principle, at least two of the measurements must give the same answer:
+  $$
+    \mathbb P(Z_1^{(A)}= Z_2^{(A)})
+    +
+    \mathbb P(Z_1^{(A)}= Z_3^{(A)})
+    +
+    \mathbb P(Z_2^{(A)}= Z_3^{(A)})
+    \geqslant 1
+  $$
+
+  \item
+  Since $A$ and $B$ are anticorrelated: $\mathbb P(Z_i^{(A)}=Z_j^{(A)})=\mathbb P(Z_i^{(A)}\neq Z_j^{(B)})$, so
+  $$\boxed{
+    \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+    +
+    \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+    +
+    \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+    \geqslant 1
+  }$$
+\end{itemize}
+\vfill
+\eject
+
+\title{Bell's theorem}
+\begin{itemize}
+  \item
+  Step 1: the EPR argument: \href{https://doi.org/10.1103/PhysRev.47.777}{[Einstein, Podolsky, Rosen, 1935]}:
+  $$
+  \boxed{\mathrm{QM\ is\ local}\quad\Longrightarrow\quad \mathrm{spins\ have\ predetermined\ values}}
+  $$
+
+  \item
+  Step 2: Bell's inequality:
+  $$\boxed{
+    \begin{array}{l}
+      \mathrm{spins\ have\ predetermined\ values}
+      \Longrightarrow\\
+      \hskip30pt\Longrightarrow
+      \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+      +
+      \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+      +
+      \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+      \geqslant 1
+    \end{array}
+  }$$
+
+  \item
+  Step 3: According to the laws of quantum mechanics,
+\end{itemize}
+\vfill
+\eject
+
+\title{Quantum mechanical prediction}
+\begin{itemize}
+  \item
+  Suppose the three measurements are done at angles $\frac{2\pi}3$ from each other, then, for $i\neq j$,
+  $$
+    \mathbb P(Z_i^{(A)}\neq Z_j^{(B)})=\frac{1+\cos\frac{2\pi}3}2=\frac 14
+    .
+  $$
+
+  \item
+  Therefore,
+  $$
+    \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+    +
+    \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+    +
+    \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+    =\frac34<1
+    .
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Bell's theorem}
+\begin{itemize}
+  \item
+  Step 1: the EPR argument: \href{https://doi.org/10.1103/PhysRev.47.777}{[Einstein, Podolsky, Rosen, 1935]}:
+  $$
+  \boxed{\mathrm{QM\ is\ local}\quad\Longrightarrow\quad \mathrm{spins\ have\ predetermined\ values}}
+  $$
+
+  \item
+  Step 2: Bell's inequality:
+  $$\boxed{
+    \begin{array}{l}
+      \mathrm{spins\ have\ predetermined\ values}
+      \Longrightarrow\\
+      \hskip30pt\Longrightarrow
+      \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+      +
+      \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+      +
+      \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+      \geqslant 1
+    \end{array}
+  }$$
+
+  \item
+  Step 3: According to the laws of quantum mechanics,
+  $$\boxed{
+    \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+    +
+    \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+    +
+    \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+    \not\geqslant 1
+  }$$
+\end{itemize}
+\vfill
+\eject
+
+\title{Bell's theorem}
+\begin{itemize}
+  \item
+  Step 1: the EPR argument: \href{https://doi.org/10.1103/PhysRev.47.777}{[Einstein, Podolsky, Rosen, 1935]}:
+  $$
+  \boxed{\mathrm{QM\ is\ \high{not}\ local}\quad\Longleftarrow\quad \mathrm{spins\ \high{do\ not}\ have\ predetermined\ values}}
+  $$
+
+  \item
+  Step 2: Bell's inequality:
+  $$\boxed{
+    \begin{array}{l}
+      \mathrm{spins\ \high{do\ not}\ have\ predetermined\ values}
+      \Longleftarrow\\
+      \hskip30pt\Longleftarrow
+      \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+      +
+      \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+      +
+      \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+      \high{\not\geqslant} 1
+    \end{array}
+  }$$
+
+  \item
+  Step 3: According to the laws of quantum mechanics,
+  $$\boxed{
+    \mathbb P(Z_1^{(A)}\neq Z_2^{(B)})
+    +
+    \mathbb P(Z_1^{(A)}\neq Z_3^{(B)})
+    +
+    \mathbb P(Z_2^{(A)}\neq Z_3^{(B)})
+    \not\geqslant 1
+  }$$
+\end{itemize}
+\vfill
+\eject
+
+\title{Reactions to Bell's theorem}
+\begin{itemize}
+  \item
+  The violation of Bell's inequality shows that the values of the spin in different directions do not \high{simultaneously} exist.
+  More generally, the values of \high{non-commuting operators} do not simultaneously exist.
+
+  \item
+  This has lead many to state that Bell's theorem shows there are no \high{hidden variables} in quantum mechanics.
+  This is \high{not true}: it is only true that there non-commuting observables do not simultaneously have values (``no non-contextual hidden variables'').
+
+  \item
+  There \high{is} a theory of quantum mechanics with ``hidden variables'': \high{Bohmian}.
+
+  \item
+  Others have said that Bell's theorem shows there are no \high{local hidden variables} in quantum mechanics.
+  While this is technically true, it misses the point: there is no \high{locality} in quantum mechanics.
+\end{itemize}
+\vfill
+\eject
+
+\title{More general statement}
+\begin{itemize}
+  \item
+  The theorem discussed earlier requires \high{perfect anticorrelation} between spins in an entangled singlet.
+  What if quantum mechanics is only very slightly wrong, and the anticorrelation is not exactly perfect?
+
+  \item
+  The proof of Bell's theorem goes through showing there are no pre-existing values, which has caused some confusion.
+  Can we prove the theorem \high{without any reference to pre-existing values}?
+\end{itemize}
+\vfill
+\eject
+
+
+\hbox{}
+\vfill
+\hfil{\bf\Large Part II}\par\bigskip
+\hfil{\bf\Large Bell, revisited}
+\vfill
+\eject
+
+\hfil
+\includegraphics[trim=10 1in 1in 2in, clip, height=\textheight]{setup.pdf}
+\eject
+
+\title{General setup}
+\begin{itemize}
+  \item
+  Observers $A$ and $B$
+
+  \item
+  Each observer can set a tunable parameter on their measurement device: $\alpha,\beta$ (for instance, the direction of spin).
+
+  \item
+  Outcomes of a measurement: random variables $X_A,X_B\in[-1,1]$, distributed according to a \high{joint} probability distribution $\mathbb P_{\alpha,\beta}$ that \high{depends on $\alpha,\beta$} (the outcomes of measurements with different parameters $\alpha,\beta$ are not required to simultaneously exist).
+\end{itemize}
+\vfill
+\eject
+
+\title{Locality}
+\begin{itemize}
+  \item
+  Naive definition:
+  $$
+    \mathbb P_{\alpha,\beta}(X_A,X_B)
+    =\mathbb P_\alpha^{(A)}(X_A)
+    \mathbb P_\beta^{(B)}(X_B)
+  $$
+
+  \item
+  This does not allow for any correlation between $A$ and $B$ (in particular, it excludes the anticorrelation considered earlier).
+\end{itemize}
+\vfill
+\eject
+
+\title{General setup}
+\begin{itemize}
+  \item
+  Observers $A$ and $B$
+
+  \item
+  Parameters $\alpha,\beta$.
+
+  \item
+  Outcomes of a measurement: random variables $X_A,X_B\in[-1,1]$, $\sim\mathbb P_{\alpha,\beta}$.
+
+  \item
+  Allow for an extra parameter $\lambda$, whose value is shared by both $A$ and $B$ and may affect the outcome of the measurements.
+  For instance, $\lambda$ could be a shared state of the electrons (e.g. an anticorrelated entangled state).
+  $\lambda$ is a random variable distributed according to $\mathbb Q$ (independent of $\alpha,\beta$).
+
+  \item
+  Locality:
+  $$
+    \mathbb P_{\alpha,\beta}(X_A,X_B|\lambda)
+    =\mathbb P_\alpha^{(A)}(X_A|\lambda)
+    \mathbb P_\beta^{(B)}(X_B|\lambda)
+  $$
+\end{itemize}
+\vfill
+\eject
+
+\title{Bell's inequality}
+\vfill
+{\bf Theorem}: Under these assumptions, $\forall\alpha,\alpha',\beta,\beta'$,
+$$
+  |\mathbb E_{\alpha,\beta}(X_AX_B)
+  -\mathbb E_{\alpha,\beta'}(X_AX_B)|
+  +
+  |\mathbb E_{\alpha',\beta}(X_AX_B)
+  +\mathbb E_{\alpha',\beta'}(X_AX_B)|
+  \leqslant 2
+$$
+where $\mathbb E_{\alpha,\beta}$ is the expectation value in the probability distribution $\mathbb P_{\alpha,\beta}$.
+\vfill
+\eject
+
+\title{Proof of Bell's inequality}
+$$
+  \mathbb E_{\alpha,\beta}(X_AX_B)
+  =\int d\mathbb Q(\lambda)\ \mathbb E_{\alpha,\beta}(X_AX_B|\lambda)
+$$
+by the locality condition,
+$$
+  \mathbb E_{\alpha,\beta}(X_AX_B|\lambda)
+  =
+  \mathbb E_{\alpha}^{(A)}(X_A|\lambda)
+  \mathbb E_{\beta}^{(B)}(X_B|\lambda)
+$$
+so
+$$
+  |\mathbb E_{\alpha,\beta}(X_AX_B)
+  \pm\mathbb E_{\alpha,\beta'}(X_AX_B)|
+  \leqslant
+  \int d\mathbb Q(\lambda)\ 
+  \left|\mathbb E_\alpha^{(A)}(X_A|\lambda)\right|
+  \left|\mathbb E_\beta^{(B)}(X_B|\lambda)\pm\mathbb E_{\beta'}^{(B)}(X_B|\lambda)\right|
+  .
+$$
+Since $|X_A|\leqslant 1$, $|\mathbb E_\alpha^{(A)}(X_A|\lambda)|\leqslant 1$.
+\vfill
+\eject
+
+\title{Proof of Bell's inequality}
+If $x:=\mathbb E_\beta^{(B)}(X_B|\lambda)$ and $y:=\mathbb E_{\beta'}^{(B)}(X_B|\lambda)$, then
+$$
+  \begin{array}{>\displaystyle l}
+    |\mathbb E_{\alpha,\beta}(X_AX_B)
+    -\mathbb E_{\alpha,\beta'}(X_AX_B)|
+    +
+    |\mathbb E_{\alpha',\beta}(X_AX_B)
+    +\mathbb E_{\alpha',\beta'}(X_AX_B)|
+    \leqslant\\\hfill\leqslant
+    \int d\mathbb Q(\lambda)\ 
+    |x-y|+|x+y|
+  \end{array}
+$$
+and since $|x|\leqslant 1$ and $|y|\leqslant 1$,
+$$
+  |x-y|+|x+y|\leqslant 2
+  .
+$$
+\hfill$\square$
+\vfill
+\eject
+
+\title{Quantum mechanical prediction}
+\begin{itemize}
+  \item
+  $\alpha,\beta\in\mathcal S^2$: direction of spin:
+  $$
+    \mathbb E_{\alpha,\beta}(X_AX_B)=-\alpha\cdot\beta
+    .
+  $$
+
+  \item
+  $$
+  \begin{array}{>\displaystyle l}
+    |\mathbb E_{\alpha,\beta}(X_AX_B)
+    -\mathbb E_{\alpha,\beta'}(X_AX_B)|
+    +
+    |\mathbb E_{\alpha',\beta}(X_AX_B)
+    +\mathbb E_{\alpha',\beta'}(X_AX_B)|
+    =\\[0.3cm]\hfill=
+    |\alpha\cdot(\beta-\beta')|+|\alpha'\cdot(\beta+\beta')|
+  \end{array}
+  $$
+
+  \item Choose $\beta'\cdot\beta=0$, $\alpha=(\beta-\beta')/\sqrt2$, $\alpha'=(\beta+\beta')/\sqrt2$:
+  $$
+    |\mathbb E_{\alpha,\beta}(X_AX_B)
+    -\mathbb E_{\alpha,\beta'}(X_AX_B)|
+    +
+    |\mathbb E_{\alpha',\beta}(X_AX_B)
+    +\mathbb E_{\alpha',\beta'}(X_AX_B)|
+    =2\sqrt2>2
+    .
+  $$
+
+  \item
+  \high{Quantum mechanics violates Bell's inequality}.
+\end{itemize}
+\vfill
+\eject
+
+\title{General setup}
+\begin{itemize}
+  \item
+  Observers $A$ and $B$
+
+  \item
+  Parameters $\alpha,\beta$.
+
+  \item
+  Outcomes of a measurement: random variables $X_A,X_B\in[-1,1]$, $\sim\mathbb P_{\alpha,\beta}$.
+
+  \item
+  Allow for an extra parameter $\lambda$, distributed according to $\mathbb Q$ (independent of $\alpha,\beta$).
+
+  \item
+  \high{\sout{Locality}}.
+\end{itemize}
+\vfill
+\eject
+
+\title{EPR}
+{\bf Theorem}:
+If $X_A,X_B\in\{-1,1\}$, if
+$$
+  \mathbb P_{\alpha,\alpha}(X_A\neq X_B)=1
+$$
+and $\mathbb P$ is \high{local}, then $\forall\lambda,\alpha$, there exists $Z_\alpha(\lambda)\in\{-1,1\}$ such that
+$$
+  \mathbb P_{\alpha,\beta}(X_A=Z_\alpha(\lambda)|\lambda)
+  =\mathbb P_{\alpha,\beta}(X_B=-Z_\alpha(\lambda)|\lambda)
+  =1
+$$
+and
+$$
+  \mathbb P_{\alpha}^{(A)}(X_A=Z_\alpha(\lambda)|\lambda)
+  =\mathbb P_{\beta}^{(B)}(X_B=-Z_\alpha(\lambda)|\lambda)
+  =1
+  .
+$$
+
+
+\end{document}
-- 
cgit v1.2.3-70-g09d2

