Multiparty entanglement in graph states

TLDR

Graph states are multiparticle entangled states represented by mathematical graphs, where vertices correspond to quantum spin systems and edges to Ising interactions, and they play a central role in quantum error correction, multiparty communication, and one‑way quantum computing. The study aims to characterize and quantify genuine multiparticle entanglement of graph states using the Schmidt measure. The authors derive upper and lower bounds for this measure expressed in graph‑theoretical terms. They demonstrate that for several graph classes—including trees, cluster states, error‑correction codes, and the quantum Fourier transform—the bounds coincide, and they fully characterize all graphs up to seven vertices up to local‑unitary equivalence and graph isomorphism.

Abstract

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.