Group structures and representations of graph states

A special configuration of graph state stabilizers, which contains only Pauli ${\sigma }_X$ operators, is studied. The vertex sets $\xi$ associated with such configurations are defined as what we call $X$ chains of graph states. The $X$ chains of a general graph state can be determined efficiently. They form a group structure such that one can obtain the explicit representation of graph states in the $X$ basis via the so-called $X$-chain factorization diagram. We show that graph states with different $X$-chain groups can have different probability distributions of $X$-measurement outcomes, which allows one to distinguish certain graph states with $X$ measurements. We provide an approach to find the Schmidt decomposition of graph states in the $X$ basis. The existence of $X$ chains in a subsystem facilitates error correction in the entanglement localization of graph states. In all of these applications, the difficulty of the task decreases with increasing number of $X$ chains. Furthermore, we show that the overlap of two graph states can be efficiently determined via $X$ chains, while its computational complexity with other known methods increases exponentially.

Figure 1

Correlation indices and $X$ resources. (a) Three-vertex star graph. (b) The mapping from $X$ resources to correlation indices is illustrated in the incidence structure [13] of the graph $S_3$. The upper line is the correlation index, while the lower lines are the vertex subsets (including the empty set). The arrows go from lower vertex subsets $\xi$ to the upper vertices corresponding to the nonzero entries of their correlation index $c_{\xi }$. For example, the vertex set $\{1,2,3\}$ points to the vertices $\{2,3\}$, indicating that the correlation index of $\{1,2,3\}$ is $c_{\{1,2,3\}}=\{2,3\}$. In particular, the vertex sets $\varnothing$ and $\{2,3\}$ are $X$ chains (see Definition 5) since their correlation index is $\varnothing$. The resources in the sets ${\mathcal{X}}_G^{\left(\varnothing \right)}=\{\varnothing ,\{2,3\}\}, {\mathcal{X}}_G^{\left\{1\right\}}=\{\left\{2\right\},\left\{3\right\}\}, {\mathcal{X}}_G^{\{2,3\}}=\{\left\{1\right\},\{1,2,3\}\}$, and ${\mathcal{X}}_G^{\{1,2,3\}}=\{\{1,2\},\{1,3\}\}$ are all “connected” by $\{2,3\}$ via the symmetric difference operation $\mathrm{\Delta }$. (c) Grouping of vertex subsets according to the correlation index. ${\mathrm{\Gamma }}_G$ and ${\mathcal{K}}_G$ are the $X$-chain group generators and correlation group generators, respectively.

Figure 2

Example for the determination of $X$-chain states (see Sec. 4 for details). (a) The graph state $|K_4^{\neg 1}\rangle$. (b) The factorization diagram of $|K_4^{\neg 1}\rangle$ (for an explanation, see Algorithm 1). (c),(d) The incidence structure of the $X$-chain generators and correlation group generators of $|K_4^{\neg 1}\rangle$. The choices of these generators are not unique and lead to different fundamental $X$-chain states $|i^{\left(x_{\mathrm{\Gamma }}\right)}\rangle$. However, they arrive at the identical set of $X$-chain states $\{|{\psi }_{\varnothing }\left(\xi \right)\rangle ,\xi \in \langle {\mathcal{K}}_G\rangle \}$.

Figure 3

$X$-chain factorization diagram of graph states: A graphical summary of Propositions 3 and 4, and Theorem 1. This diagram illustrates the algorithm for representing a graph state in the $X$ basis.

Figure 4

$Z$-balanced graph states (see Definition 7) up to five vertices. Each graph represents a graph isomorphic class. Each balanced graph state has at least one $X$ chain ${\gamma }^{-}$ with negative parity. In each graph, the ${\gamma }^{-}$-induced subgraph $G\left[{\gamma }^{-}\right]$ is highlighted in red with bold edges. Every highlighted ${\gamma }^{-}$-induced subgraph has an odd edge number.

Figure 5

Orthogonal graph states derived from the $Z$-balanced graph state $|C_3\rangle$. The graph states in each cell are orthogonal to each other. Their symmetric difference is identical to the cycle graph $C_3$, where $C_3$ is the first graph in Fig. 4.

Figure 6

Orthogonal graph states derived from the $Z$-balanced graph state $|C_5\rangle$. The graph states in each cell are orthogonal. Their symmetric difference is identical to the cycle graph $C_5$, where $C_5$ is the fifth graph in Fig. 4.

Figure 7

The $X$-chain factorization diagram of correlation states. A graphical summary of Theorem 2. The $\xi$ in $|{\psi }_{{\mathcal{K}}_1\cup {\mathcal{K}}_2}\left(\xi \right)\rangle$ are elements in the quotient group, $\xi \in \langle {\mathcal{K}}_G\rangle /\langle {\mathcal{K}}_1\cup {\mathcal{K}}_2\rangle$.

Figure 8

$A\rfloor B$ factorization of graph states. (a) The graph state $|G_{\mathrm{House}}\rangle$ corresponding to a “St. Nicholas's house” is divided in two subsystems, $A=\{1,2,3\}$ and $B=\{4,5\}$. (b) The binary representation of the $X$-chain factorization (the upper row) and $A\rfloor B$ factorization (the lower row). (c) The $A\rfloor B$-factorization diagram (see Algorithm 3) of the St. Nicholas's house graph state $|G_{\mathrm{House}}\rangle$.

Figure 9

$X$-chain factorization diagram for the Schmidt decomposition of graph states in the $X$ basis. A graphical summary of Lemmas 2 and 3, and Theorem 3.

Figure 10

An example of entanglement localization of graph states protected against errors. (a) Local $X$ measurements on subsystem $A$ project the graph state $|G\rangle$ onto the maximally entangled state $|{\varphi }_{A\rfloor B}^{\left(B\right)}\left(\xi \right)\rangle$ for subsystem $B$. Under the assumption of a single qubit error, the outcome $|m_X^{\left(A\right)}\rangle =|110\rangle$ indicates a $Z$ error on vertex 3. Alice sends Bob the corrected outcome $\left(111\right)$, such that Bob knows from the Schmidt decomposition that he possesses the state $|{\varphi }_{A\rfloor B}^{\left(B\right)}\left(\left\{4\right\}\right)\rangle$. (b) Binary representation and incidence structure after $A\rfloor B$ factorization.