Multiparty entanglement in graph states

Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.

Figure 1

Example for a ${\sigma }_x$ measurement at vertex $1$ in graph No. 1, which is followed by a ${\sigma }_z$ measurement at vertex 2. In graph No. 1 a ${\sigma }_x$ measurement is performed at the vertex $1$. For the application of the rule in Eq. (37), vertex 2 was chosen as the special neighbor $b_0$, yielding graph No. 2 up to a local unitary $U_{x,\pm }^{\left(1\right)}={\left(\pm i{\sigma }_y^{\left(2\right)}\right)}^{1∕2}$. As stated in Table , the subsequent ${\sigma }_z$ measurement on the new graph state is therefore essentially another ${\sigma }_x$ measurement, now at vertex $2$ with a single neighbor $b_0=5$. The final graph is then graph No. 3.

Figure 2

A single ${\sigma }_y$ measurement at an arbitrary vertex in the complete graph No. 7 suffices to disentangle the corresponding state. Similarly, a single ${\sigma }_z$ measurement at the central vertex in graph Nos. 1–6 or a single ${\sigma }_x$ measurement at the noncentral vertices is a disentangling measurement. This is due to the fact that all graphs (Nos. 1–7) are locally equivalent by local unitaries, which transform the measurement basis correspondingly.

Figure 3

Whereas graph No. 1 is 2-colorable, the resulting graph No. 2 after a ${\sigma }_y$ measurement at the vertex $∎$ is not $2$-colorable. Also, none of the $132$ (or $3$) representatives in the corresponding equivalence class (if graph isomorphisms are included) is 2-colorable.

Figure 4

List of connected graphs with up to six vertices that are not equivalent under LU transformations and graph isomorphisms.

Figure 5

List of connected graphs with seven vertices that are not equivalent under LU transformations and graph isomorphisms.

Figure 6

An example for an successive application of the LU rule, which exhibits the whole equivalence class associated with graph No. 1. The rule is successively applied to that vertex of the precessor, which is written above the arrows of the following diagram: $\text{No.}1\stackrel{3}{\to }\text{No.}2\stackrel{2}{\to }\text{No.}3\stackrel{3}{\to }\text{No.}4\stackrel{1}{\to }\text{No.}5\stackrel{3}{\to }\text{No.}6\stackrel{1}{\to }\text{No.}7\stackrel{3}{\to }\text{No.}8\stackrel{4}{\to }\text{No.}9\stackrel{1}{\to }\text{No.}10\stackrel{2}{\to }\text{No.}11$.

Figure 7

An example of an equivalence class, which is a proper subset of class No. 4 in Fig. 4.

Figure 8

The Petersen graph. The depicted labeled graph is not LU equivalent to the graph which is obtained from it by exchanging the labels at each end of the five “spokes,” i.e., the graph isomorphism which permutes the vertices $1,2,3,4$, and $5$ with $6,7,8,9$, and $10$, respectively.

Figure 9

A sufficient condition for a graph to have maximal Schmidt rank.

Figure 10

The situation before and after the LOCC simulation for adding or deleting an edge $\{a_1,b_1\}$: the graph state vector $\mid G⟩$ can be transformed by $(A,B)$-local operations and classical communication with probability $1$ into the state vector $\mid G^{\prime }⟩$, where the edge between the partitions $A_1$ and $B_1$ is added or deleted. This is possible if one allows for an additional maximally entangled state ∎—∎ between $A_2$ and $B_2$. After the LOCC operation the resource is consumed, i.e., the state of $(A_2,B_2)$ is a pure product state ∎ ∎.

Figure 11

Graph No. 1 represents a tree. Its bipartitioning $(A,B)$, for which in graph No. 2 the vertices in $A$ are depicted by large boxes $∎$, is neither a minimal vertex cover nor yields maximal partial rank. Instead, the set of vertices $A$, represented by large boxes $∎$ in graph No. 3, is a minimal vertex cover with maximal partial rank. Here, the edges within the set $A$ are drawn by thin lines in order to illustrate the resulting graph $G_{A{A}^c}$ between $A$ and its complement, as considered in Sec. 3B.

Figure 12

An example for the $(4,5,3)$-cluster state and its resulting graph $G_{A{A}^c}$ between $A$ and its complement as considered in Sec. 3B. Here, the vertices in $A$ are depicted by small boxes $∎$.

Figure 13

An example for the $(7,5,5)$-cluster state and its resulting graph $G_{A{A}^c}$ between $A$ and its complement as considered in Sec. 3B. Here, the vertices in $A$ are depicted by small boxes $∎$. The picture gives a rotated view on the cluster considered in the proof for the case, that $X$, $Y$, and $Z$ are odd integers. The front plane, consisting of the vertices $1$ until $35$, is the $y\text{−}z$ plane $A_X$ in the proof.

Figure 14

Graph No. 1 is an entangled ring on $18$ vertices. Graph No. 2 represents the resulting graph between the partitions $A$, whose vertices are depicted by boxes, and the partition $B$, whose vertices are depicted by discs.

Figure 15

Resource graph state for the concatenated $[7,1,3]$-CSS code.

Figure 16

The graph associated with the QFT on 3 qubits in the one-way quantum computer is represented in graph No. 1, where the boxes denote the input (left) and output (right) vertices. Graph No. 3 is obtained from the first after performing all Pauli measurements according to the protocol in Ref. 3, except from the ${\sigma }_x$ measurements at the input vertices. More precisely, it is obtained from graph No. 1 after ${\sigma }_y$ measurements on the vertices $22,23,24,26,27,28,30,31,32$ and ${\sigma }_x$ measurements on the vertices $2,4,7,9,11,13,15,18,20$ have been performed.