%!TEX root = clock_main.tex
\begin{frame}{Asymmetric Delay Attack}
  % \begin{block}{We have assumed that delays were symmetric:}
  % \begin{center}
  % % \vfill
  % \vspace*{-15pt}
  % \def\svgwidth{0.8\textwidth}
  % \hspace*{45pt}\import{images/offset_schematic/}{offset_concept.pdf_tex}
  % \fbox{$\delta$ = midpoint between two coincidence peaks}
  % \end{center}
  % \end{block}
  % \begin{block}{}
  \begin{center}
    \includegraphics[width=\textwidth]{images/asymmetric_delay_expt_setup/clocksyncsetup_assymetric.pdf}
  \end{center}
  % \end{block}
  \vfill
  % \begin{block}{}
  \begin{center}
  \def\svgwidth{0.8\textwidth}
  \hspace*{45pt}\import{images/offset_schematic/}{offset_concept_assymetric.pdf_tex}
  \bigbreak
  \fbox{$\delta\neq$ midpoint between two coincidence peaks}
  \end{center}
  % \end{block}
% \footnotetext[1]{Detecting Circulators using Geometric Phases (arXiv:1808.09019)}
  \note{
    So far, we have assumed that the delay between Alice and Bob is the same in both directions.
    % We histogram t and t' and assume that the midpoint between the peaks is the clock error.

    However, this can be manipulated.

    For e,g., two circulators can be used to introduce different propagation delays, L and L', depending on propagation direction.

    In this case, the true offset delta is no longer at the mid pt between the peaks and cannot be determined independently of L and L'.

    The circulators do not project polarization states, 
    but does transform them. The question is: 
    can an adversary deploy them without disturbing the distributed entanglement.

    If so, the protocol is compromised.

    We first demonstrate the synchronization error introduced in this type of attack. 
    % Using polarization-independent circulators also means that this attack cannot be directly detected with a Bell measurement. 
    % Separately, we have published work on the arXiv describing how the circulators change the geometric phase of the photons, but we have yet to figure out an experiment capable of extracting the GP from the overall phase change imposed by the circulators.
  }
\end{frame}

\begin{frame}{Estimated offset (peak-midpoint) shifts\\
{\small under asymmetric delay attack}}
  \begin{center}
    \includegraphics[width=\textwidth]{images/asymmetric_delay_g2/g2.eps}
  \end{center}
  \note{
  We repeat the synchronization procedure,

  the control experiment with two symmetric delays shows the same offset, 

  while for an asymmetric delay,
  the peaks, together with the midpt, shift from their original position,

  indicating that the protocol is compromised.

  % Graphs a and b shows that despite a change in RTT, corresponding to the inter-peak separation, the midpoint remains the same.
  % However, when the same RTT is distributed in an asymmetric fashion, the midpoint shifts,
  % indicating an error in time synchronization.
  }
\end{frame}

\begin{frame}{Creating Channel Asymmetry\\ 
{\small with Faraday Rotation (FR)}}
  \begin{left}
    \textbf{NOT time-reversible}~(FR~in~circulators)\\
    \includegraphics[width=0.7\textwidth]{images/faraday_effect/faraday_effect.png}
    % \hspace{1cm}
    % \begin{right}
    \includegraphics[width=0.2\textwidth]{images/faraday_effect/circulator_boxed.png}
    % \end{right}
  \end{left}\\
    \begin{left}
    \textbf{Time-reversible}~(e.g. waveplates)\\
    \includegraphics[width=0.7\textwidth]{images/faraday_effect/reciprocal.png}
  \end{left}
\footnotetext[1]{Image Credit: FOSCO}
\note{
  To examine if the circulators can be detected,
  we consider how they work internally.

  Circulators distinguish propagation directions using an effect known as Faraday Rotation.

  This rotation is not time-reversible, 
  which means a beam propagating in the reversed direction will not return to its original polarization state. 

  This irreversibility is exploited in the design of a circulator to distinguish and reroute photons traveling in opposite directions.

  This contrasts with other types of polarization transformations, where the inverse transformation occurs for the time-reversed process.

  % Combined with other optical elements, FR forms the basis for the circulator used in our asymmetric delay attack.

  Although FR does not project polarization, it does rotate it.
  For an adversary to deploy the circulator without disturbing the input polarization,
  she would have to perform a 180 rotation, or 360..
  % However, this effect does rotates polarization; naively, one expects that this would disturb the polarization entanglement measured in our protocol.
}
\end{frame}

\begin{frame}{Can we detect Faraday Rotation in Circulators?}
  \begin{center}
    \includegraphics[width=0.9\textwidth]{images/tomography/tomography_setup.pdf}
  \end{center}
    \centering
  % \vspace{-0.5cm}
    \fbox{$n\times180^\text{o}$ rotation: \quad 
    $\ket{\psi}_B\rightarrow\pm\ket{\psi}_B$ \quad 
    $\ket{\Psi_{AB}^-} \rightarrow \pm\ket{\Psi_{AB}^-}$}
    % \\$\ket{\Psi^-} \rightarrow = \frac{1}{\sqrt{2}}\left(\redpmket{H}\ket{V} - \redpmket{V}\ket{H}\right) 
    % = \pm\ket{\Psi^-}$}
    % \fbox{$180^\text{o}\times n$\quad rotation:\\ 
    % $\ket{\Psi^-} = 
    % \frac{1}{\sqrt{2}}\left(\ket{H}\ket{V} - \ket{V}\ket{H}\right)
    % \rightarrow
    % \frac{1}{\sqrt{2}}\left(\redpmket{H}\ket{V} - \redpmket{V}\ket{H}\right) = \pm\ket{\Psi^-}$}
    % \fbox{$180^\text{o}\times n$\quad rotation:\\ 
    % $\ket{\Psi^-} = 
    % \frac{1}{\sqrt{2}}\left(\ket{H}\ket{V} - \ket{V}\ket{H}\right)
    % \rightarrow
    % \frac{1}{\sqrt{2}}\left(\redpmket{H}\ket{V} - \redpmket{V}\ket{H}\right) = \pm\ket{\Psi^-}$}
  \begin{center}
    \includegraphics[width=0.45\textwidth]{images/tomography/rho_before_real.pdf}
  % \end{center}
  % \begin{center}
    \includegraphics[width=0.45\textwidth]{images/tomography/rho_after_real.pdf}
  \end{center}
  \vfill
  \note{
  To verify that circulators can be deployed without affecting overall entanglement, 
  we measure its density matrix before and after they are inserted into the synchronization channel.

  % we measure the entangled state before and after circulators are inserted into the synchronization channel.

  % To evade detection, the circulators need to rotate the polarization in the channel back to the same state, or a multiple of 180 degrees.

  We used commercially available polarization-insensitive circulators,
  which perform the same polarization rotation for any input polarization.

  The density matrices are essentially the same by up to 98.8 percent.

  In other words, the circulators cannot be detected by monitoring entanglement distributed through the channel.
  
  Coming up with an experimental arrangement that detects the circulators remains an open question for us.
  }
\end{frame}