# Query on Reduced Graph States

Reduced graph states are characterized as follows (from page 46 of this paper): 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 $$\rho_{G}^{A}:= \mathrm{tr}_{B}(|G\rangle\langle G|)$$ 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.