# A Reduced Graph States

#### Johny Boy

Summary
I have a paper query regarding a fundamental result on graph states, more specifically a result that characterizes a reduced graph state in a particular basis.
Reduced graph states are characterized as follows from page 46 of this paper:

Proposition:
Let $A \subseteq V$ be a subset of vertices for a graph $G = (V,E)$ and $B = V\setminus A$ the corresponding complement in $V$. The reduced state $\rho_{G}^{A}:= tr_{B}(|G\rangle\langle G|)$ is given by $$\rho^{A}_{G} = \frac{1}{2^{|A|}}\sum_{\sigma \in \mathcal{S_{A}}}\sigma,~~~~~~~~~~~~~~~~~~(1)$$where $\mathcal{S}_{A}:=\{ \sigma \in \mathcal{S}| \text{supp}(\sigma) \subseteq A \}$ denotes the subgroup of stabilizer elements $\sigma \in \mathcal{S}$ for $|G\rangle$ with support on the set of vertices within $A$. $\rho_{G}^{A}$ is up to some factor a projection, i.e.

$$(\rho_{G}^{A})^2 = \frac{|\mathcal{S}_{A}|}{2^{|A|}}\rho_{G}^{A}~~~~~~~~~~~~~~~~~~~(2)$$It projects onto the subspace in $\mathbf{H}^{A}$ spanned by the vectors $$|\mathbf{\Gamma}'B'\rangle_{G[A]} = \sigma_{z}^{\mathbf{\Gamma}'B'}|G[A]\rangle~~~~~~~(B' \subseteq B)~~~~~~~~~(3)$$where $G[A] = G\setminus B$ is the subgraph of $G$ induced by $A$ and $\mathbf{\Gamma}':=\mathbf{\Gamma}^{AB}$ denotes the $|A| \times |B|-$off diagonal sub-matrix of the adjacency matrix $\mathbf{\Gamma}$ for $G$ that represents the edges between $A$ and $B$:
\begin{align} \begin{pmatrix} \mathbf{\Gamma}_{A} & \mathbf{\Gamma}_{AB} \\ \mathbf{\Gamma}^{T}_{AB} & \mathbf{\Gamma}_{B} \end{pmatrix} = \mathbf{\Gamma}. \end{align}
In this basis, $\rho_{G}^{A}$ can be written as $$\rho_{G}^{A} = \frac{1}{2^{|B|}}\sum_{B' \subseteq B}| \mathbf{\Gamma}' B' \rangle_{G[A]} \langle \mathbf{\Gamma}'B'|.~~~~~~~~~~~~~(4)$$
Question:
The results of equation (1) and (2) are understood. I'm trying to understand the motivation for defining the basis states $| \mathbf{\Gamma}' B' \rangle_{G[A]}$ as shown in equation (3). As I understand, $\mathbf{\Gamma} B'$ in the exponent of equation (3), is some string in $\{0,1\}^{|B'|}$. In this way they would show that there are sufficient permutations of $\mathbf{\Gamma}' B'$, where $B' \subseteq B$, to produce orthogonal states $| \mathbf{\Gamma}' B' \rangle_{G[A]}$ which spans a subspace of $\mathbf{H}^{A} \subseteq (\mathbb{C})^V$. Explicitly how is the term '$| \mathbf{\Gamma}' B' \rangle_{G[A]}$' defined? I don't really understand why the exponent is chosen as $\mathbf{\Gamma} B'$ in to begin with to characterize the basis states?

Thanks for any assistance.

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