fixed one proof
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@ -363,45 +363,87 @@ graph.
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\end{equation}
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\end{lemma}
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\begin{proof}
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FIXME: This
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The proof is done using mathematical induction over the edges $\{i,j\} \in
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E$. A similar proof can be found in \cite{hein_eisert_briegel2008}.
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\textbf{Base Case:} If $E = \{\}$ the stabilizers are $K_G^{(i)} = X_i$
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and stabilize the state $\ket{+}$.
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\textbf{Inductive Step:} Let $G' := (V, E \setminus \{\{l,j\}\})$.
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By the induction hypothesis the state $\ket{\bar{G}'}$ is stabilized
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by $K_{G'}^{(i)}$. Applying a $CZ_{l,j}$ to the state $\ket{\bar{G}'}$
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now transforms the stabilizers to
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Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before.
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Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$. In the following
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discussion the direction $\prod\limits_{\{l,k\} \in E} :=
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\prod\limits_{\{l,k\} \in E, l < k}$ is introduced as the graph is
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undirected and edges must not be handled twice. Set $\ket{\tilde{G}} :=
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\left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
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\begin{equation}
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\begin{aligned}
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K_G^{(i)} \ket{\tilde{G}}
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& = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)
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\left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right) \ket{+} \\
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& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,k\} \in E}
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\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
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& = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,k\} \in E}
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\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
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\end{aligned}
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\begin{aligned}
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S^{(i)} &= CZ_{l,j} K_{G'}^{(i)} CZ_{l,j}.\\
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\end{aligned}
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\end{equation}
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As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next
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step is a bit tricky: A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$
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giving no phase or into a $\ket{1}\bra{1}_j$ yielding a phase of $-1$. If
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there is no projector on $j$ the $Z_j$ can be commuted to the next
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projector. It is guaranteed that a projector on $j$ exists by the
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definition of $\ket{\tilde{G}}$.
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Note that $CZ_{l,j}$ commutes with $K_{G'}^{(i)}$ for $l \neq i \neq j$.
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Further $CZ_{l,j} = CZ_{j,l}$. Consider now the case $l = i$:
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\begin{equation}
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\begin{aligned}
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K_G^{(i)} \ket{\tilde{G}}
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& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l} + \delta_{j,k}}\ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
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& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
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& = +1 \ket{\tilde{G}}
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\end{aligned}
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\begin{aligned}
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S^{(i)} &= CZ_{i,j} K_{G'}^{(i)} CZ_{i,j} \\
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&= \left( \ket{1}\bra{1}_j \otimes Z_i + \ket{0}\bra{0}_j \otimes I_i \right)
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X_i \prod\limits_{l \in n'_i} Z_l
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\left( \ket{1}\bra{1}_j \otimes Z_i + \ket{0}\bra{0}_j \otimes I_i \right)\\
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&= \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
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&\mbox{ }+ \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
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&\mbox{ }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
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&\mbox{ }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
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&= \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
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&\mbox{ }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
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&= \left(\left(-\ket{1}\bra{1}_j + \ket{1}\bra{1}_j\right) \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
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&= \left(Z_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
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&= X_i \prod\limits_{l \in n_i} Z_l \\
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&= K_G^{(i)}
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\end{aligned}
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\end{equation}
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The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions
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of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
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%FIXME: This
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%Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before.
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%Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$. In the following
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%discussion the direction $\prod\limits_{\{l,k\} \in E} :=
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%\prod\limits_{\{l,k\} \in E, l < k}$ is introduced as the graph is
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%undirected and edges must not be handled twice. Set $\ket{\tilde{G}} :=
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%\left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
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%\begin{equation}
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% \begin{aligned}
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% K_G^{(i)} \ket{\tilde{G}}
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% & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)
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% \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right) \ket{+} \\
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% & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,k\} \in E}
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% \left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
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% & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,k\} \in E}
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% \left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
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% \end{aligned}
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%\end{equation}
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%As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next
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%step is a bit tricky: A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$
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%giving no phase or into a $\ket{1}\bra{1}_j$ yielding a phase of $-1$. If
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%there is no projector on $j$ the $Z_j$ can be commuted to the next
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%projector. It is guaranteed that a projector on $j$ exists by the
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%definition of $\ket{\tilde{G}}$.
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%\begin{equation}
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% \begin{aligned}
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% K_G^{(i)} \ket{\tilde{G}}
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% & = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l} + \delta_{j,k}}\ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
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% & = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
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% & = +1 \ket{\tilde{G}}
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% \end{aligned}
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%\end{equation}
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%The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions
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%of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
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\end{proof}
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\subsubsection{Dynamics of the VOP-free Graph States}
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