From bbbac43c826844ac5f560f968a803336ef9be1bc Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Daniel=20Kn=C3=BCttel?= <daniel.knuettel@daknuett.eu>
Date: Wed, 11 Mar 2020 10:48:51 +0100
Subject: [PATCH] some fixos

---
 presentation/main.tex | 26 ++++++++++++++++----------
 1 file changed, 16 insertions(+), 10 deletions(-)

diff --git a/presentation/main.tex b/presentation/main.tex
index f462672..43d330c 100644
--- a/presentation/main.tex
+++ b/presentation/main.tex
@@ -486,10 +486,10 @@
                 To compute the probability to measure a result of $s=0$ one can use the trace formula
                 \begin{equation}
                 \begin{aligned}
-                    P(s=0) &= \Tr(\frac{I + g_a}{2} \ket{\psi}\bra{\psi}) \\
-                            &= \Tr(\frac{I + g_a}{2} S^{(j)} \ket{\psi}\bra{\psi}) \\
-                            &= \Tr(S^{(j)}\frac{I - g_a}{2} \ket{\psi}\bra{\psi}) \\
-                            &= \Tr(\frac{I - g_a}{2} \ket{\psi}\bra{\psi}) \\
+                    P(s=0) &=  \left|\Tr(\frac{I + g_a}{2} \ket{\psi}\bra{\psi})\right| \\
+                           &= \left|\Tr(\frac{I + g_a}{2} S^{(j)} \ket{\psi}\bra{\psi})\right| \\
+                           &= \left|\Tr(S^{(j)}\frac{I - g_a}{2} \ket{\psi}\bra{\psi})\right| \\
+                           &= \left|\Tr(\frac{I - g_a}{2} \ket{\psi}\bra{\psi})\right| \\
                             &= P(s=1)\\
                 \end{aligned}
                 \end{equation}
@@ -498,7 +498,6 @@
                 trace and absorbed into the $\bra{\psi}$.
             }
     \end{itemize}
-
 \end{frame}
 }
 
@@ -773,19 +772,26 @@
         
             \begin{equation}
             \begin{aligned}
-                P_a \ket{G} &= \left(\prod\limits{i \neq a} o_i\right) P_a o_a \ket{\bar{G}} \\
-                            &= \left(\prod\limits{i \neq a} o_i\right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
-                            &=  \left(\prod\limits{i} o_i \right) \tilde{P}_a \ket{\bar{G}}\\
+                P_a \ket{G} &= \left(\prod\limits_{i \neq a} o_i\right) P_a o_a \ket{\bar{G}} \\
+                            &= \left(\prod\limits_{i \neq a} o_i\right) o_a o_a^\dagger P_a o_a \ket{\bar{G}} \\
+                            &=  \left(\prod\limits_{i} o_i \right) \tilde{P}_a \ket{\bar{G}}\\
             \end{aligned}
             \end{equation}
 
             With a new projector $\tilde{P}_a = \frac{I + g_a}{2}$.
         }
+    \end{itemize}
+    
+\end{frame}
+}
+{
+\begin{frame}{Measurements on Graph States}
+    \begin{itemize}
         \item{The anticommuting stabilizers are given by
             \begin{itemize}
                 \item{$A_X = \{K_G^{(i)} | i \in n_a\}$ for $g_a = \pm X_a$,}
-                \item{$A_Y = \{K_G^{(i)} | i \in n_a \cup a\}$ for $g_a = \pm Y_a$ and}
-                \item{$A_Z = \{K_G^{(a)}$ for $g_a = Z_a$.}
+                \item{$A_Y = \{K_G^{(i)} | i \in n_a \cup \{a\}\}$ for $g_a = \pm Y_a$ and}
+                \item{$A_Z = \{K_G^{(a)}\}$ for $g_a = \pm Z_a$.}
             \end{itemize}}
         \item{This can be used to compute the probability amplitudes and update $(G,V,E)$ after
             the measurement.}