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%% strange-quantum.tex
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%% Made by Alex Dehnert
%% Login   <alex@dubrovnik-l.dehnert.arctic.org>
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%% Started on  Sat Aug 25 09:14:39 2007 Alex Dehnert
%% Last update Sat Aug 25 09:14:39 2007 Alex Dehnert
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\documentclass{article}
\usepackage{alexschool}
\newcommand{\qop}[1]{\ensuremath{\widehat{#1}}}

\standardmargins
\title{The Strange Quantum}
\author{Robert Spekkens (ISSYP lecturer), as filtered through Alex Dehnert}
\date{Thursday, August 24, 2007}
\begin{document}
\maketitle{}

\section{Measuring and Commutativity}

\begin{itemize}
\item Consider boxes that measure $\pm X$, $\pm Y$, $\pm Z$ spin.
\item Outcome of a second $\pm Z$ measurement is exactly same as first
\item Outcome of a $\pm Z \pm X$ sequence is random (all four
  possibilities equal)
\item Outcome of a $\pm Z \pm X \pm Z$ sequence is random (all
  \emph{eight} possibilities equal)
\item An intervening $X$ randomizes the $Z$ (equivalently, $[\qop{X},
  \qop{Z}] \not = \qop{0}]$)
\item If Alice has a $\pm Z$ box behind a wall, Bob (maybe) measures with a
  $\pm X$ or $\pm Z$ box, Alice can detect Bob with another $\pm Z$
  --- Bob has a low chance of measuring Alice's polarization without detection
\end{itemize}

\section{Cryptographic Applications}

\subsection{Quantum Counterfeit-Proof Money}

\begin{itemize}
\item Take a bill, put small storage chambers inside
\item Put randomly selected atoms with polarization
\item Save polarization data ($+Z-Z-X+X+X$)
\item To check a bill, merchant teleports atoms to the mint, which
  checks the polarization
\item To counterfeit, measure the atoms. However, counterfeiter needs to guess,
  and his guessing destroys the information.
\item Probability of copied money passing the test: $2^{-B}$, where $B$ = number of atoms
\item Probability of original passing the test: $\pfr{3}{4}^B$, where
  $B$ = number of atoms
\end{itemize}

\subsection{Quantum Detection of Eavesdroppers}

\begin{itemize}
\item Alice and Bob have a channel to communicate on
\item Eve wants to add a wiretap
\item Conventional Ciphers
  \begin{description}
  \item[Caesar] Easy to brute-force, or just use frequency analysis
  \item[Vernam (One time pad)] Impossible to break, but need a really
    long, random key (Information theoretic security)
  \end{description}
\item Quantum Key distribution
  \begin{itemize}
  \item Alice picks some bases at random ($\pm X$, $\pm Z$)
  \item Bob also picks some bases at random
  \item Alice sends those photons, and keeps track of what she saw
  \item Alice, Bob compare which pairs should have been measured equally
  \item Take a small subset of the successfully transmitted photons,
    and compare what they should have been
  \item If some differ, we know that the photon interacted with the
    environment (including possibly Eve)
  \item Need a source of randomness (but needn't share anything, I think)
  \end{itemize}
\end{itemize}

\section{Idea of Hidden Variable Models of QM}

\begin{itemize}
\item Toy world with restrictions on knowledge
\item Four states, can't know which --- can know that it is equally
  likely to be in either of two states (and know not in the other two states)
\item Can only check whether in one pair, or the other (eg, in (1 or
  2) or (3 or 4))
\item Updating the probability distribution
  \begin{itemize}
  \item $\pm Z \rightarrow (1\mbox{ or } 2)$
  \item $\pm X \rightarrow (1\mbox{ or } 3)$
  \item $\pm Z \pm X$ must just force the measurement to (1 or 2) (and
    let states jump around), b/c otherwise we could get
    maximal knowledge
  \item Half the time swap states (1 and 2), half the time don't do anything
  \end{itemize}

\end{itemize}

\section{Bell's Theorem}

Why any realistic account of quantum mechanics must be nonlocal

\begin{itemize}
\item Take a laser, nonlinear crystal, polarization to get entangled states
\item Send one photon to Alice, one to Bob
\item Measurements: S, T
\item Alice, Bob give an answer to S or T measurement, with required
  method of correlation
\item Some of these ``games'' can never be won (classically)
\item You can cheat in one of these guessing games by sending what the
  measurement was
\item However, photons can ``cheat'' this way even when they are too
  far for even light to get there in time
\item How does this work with both no faster-than-light information
  transfer, and the necessity of faster-than-light info transfer for
  entanglement to work
\end{itemize}

\end{document}