- 9

- 0

- 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:

*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.*

Proposition:

Proposition:

*$$(\rho_{G}^{A})^2 = \frac{|\mathcal{S}_{A}|}{2^{|A|}}\rho_{G}^{A}~~~~~~~~~~~~~~~~~~~(2)$$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 $$\rho_{G}^{A} = \frac{1}{2^{|B|}}\sum_{B' \subseteq B}| \mathbf{\Gamma}' B' \rangle_{G[A]} \langle \mathbf{\Gamma}'B'|.~~~~~~~~~~~~~(4)$$

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?

$$

\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.**