Multiparty entanglement in graph states

By M. Hein, Jens Eisert, Hans J. Briegel. 2003

914 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/ea094fea73d137d237181d554a6ec63b46b1bc44

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Citations in the corpus

Quantum Entanglement in Neural Network States (2017, methodology): “It serves as a simple toy model for studying SPT phases and has important applications in measurement-based quantum computation [121–123].”
Experimental quantum teleportation of a two-qubit composite system (2006, methodology): “In addition, whereas a modified six-photon interferometer can be used to prepare a six star-ring cluster stat”
Linear optical quantum computing with photonic qubits (2005, background): “All the other neighbours become single connected qubits to the neighbour that inherited the bonds (Raussendorf et al. 2003; Hein et al. 2004).”
Graphical calculus for Gaussian pure states (2010, methodology): “A similar connection between complete graphs and star graphs is known for qubit cluster states [33, 34].”
Towards the fast scrambling conjecture (2011, background): “For a subset S ⊆ V of spins, the entanglement entropy of the density operator ΨS(tent) = trSc |Ψ(tent)〉〈Ψ(tent)| has a simple formula in terms of the submatrix AdjS of the adjacency matrix of G that selects the rows of S and the columns of Sc [52]:”
Entanglement in many-body systems (2007, background): “…on the factorizability of a given many particle state into smaller parts. k-separability and -producibility both do not distinguish between different k-particle entanglement classes (as e.g. the k-particle W-states and different k-particle graph states (Hein et al., 2004), like the GHZ state).”
Deterministic generation of all-photonic quantum repeaters from solid-state emitters (2016, neutral): “In particular, the fully interconnected web of photons at the center of an RGS can be obtained by performing an operation known as local complementation (LC)[14, 34] around the central “root” vertex of a tree, as shown in Fig.”
Experimental demonstration of a hyper-entangled ten-qubit Schr\ (2008, methodology): “the cat states, we note that this method can also be flexibly modified for the generation of other graph states (39) which are central resources in measurement-based quantum computing (see Appendix Fig.”
Index Theory of One Dimensional Quantum Walks and Cellular Automata (2009, background): “on, does not increase the localization region of an observable by more than two sites. We note that U 1 is known in the quantum information literature as the interaction used to generate graph states [15,9]. In Section 7, we will set up a general theory for answering the questions posed above—includ”
Universal Blind Quantum Computation (2008, background): “It is known that Z measurements change the underlying graph state into a new graph state [21].”