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.
We refine recent local unitary entanglement classification for symmetric pure states of
n qubits (that is, states invariant under permutations of qubits) using local unitary stabilizer subgroups and Majorana configurations. Stabilizer subgroups carry more entanglement distinguishing power than do the stabilizer subalgebras used in our previous work. We extend to mixed states recent results about local operations on pure symmetric states by showing that if two symmetric density operators are equivalent by a local unitary operation, then they are equivalent via a local unitary operation that is the {\em same} in each qubit. A geometric consequence, used in our entanglement classification, is that two symmetric pure states are local unitary equivalent if and only if their Majorana configurations can be interchanged by a rotation of the Bloch sphere.
We study particle-loss resistant entanglement within the framework of stabilizer and graph states. A pure state is
m-resistant if it remains entangled after the loss of any
m particles and becomes fully separable after the loss of any
m+1 particles. The smallest previously unresolved qubit case was the existence of a five-qubit
1-resistant pure state, which is resolved here by the five-cycle graph state
∣C5⟩. A stabilizer-subgroup method is also developed for verifying
m-resistance in graph states, using local stabilizers to certify full separability and exact negative partial transpose~(NPT) witnesses to certify entanglement.
Applying this to all graph states associated with non-isomorphic graphs on five, six, and seven vertices, we obtain a graph state classification up to local Clifford equivalence, which also classifies stabilizer states
up to local Clifford equivalence. Thus, the five-qubit
1-resistant stabilizer states are exactly the local Clifford class of
C5. Six-qubit
2-resistant stabilizer states exist in three distinct local Clifford classes, whereas no seven-qubit stabilizer state is
m-resistant for any nonzero admissible
m. Finally, we prove that the cycle graph states
∣CN⟩ with
N≥7 are not
m-resistant for any
0≤m≤N−2.
The class of entangled N-qubit states known as graph states, and the corresponding stabilizer groups of N-qubit Pauli observables, have found a wide range of applications in quantum information processing and the foundations of quantum mechanics. A review of the properties of graph states is given and core spaces of graph states are introduced and discussed. A bonding model of entanglement for generalized graph states is then presented, in which the presence or absence of a bond between two qubits unequivocally specifies whether or not they are entangled. A physical interpretation of these bonds is given, along with a characterization of how they can be created or destroyed by entangling unitary operations and how they can be destroyed by local Pauli measurements. It is shown that local unitary operations do not affect the bond structure of a graph state, and therefore that if two graph states have nonisomorphic bond structures, then local unitary operations and/or reordering of qubits cannot change one into the other. Color multigraphs are introduced to depict the bond structures of graph states and to make some of their properties more apparent.