Spin graphs for quantum communication and ground state entanglement

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In this talk, two specific directions of research in quantum information are presented which could potentially gain from graph theory. The first is the study of quantum communication using systems of perpetually interacting qubits (or spins) as a databus. After introducing the topic through the simplest examples of linear chains of spins as transmitters of quantum information, we briefly mention existing work on quantum communication through more general graphs of spins. We then explain why the transmission of quantum information between vertices of a graph in the case of an isotropic Heisenberg interaction (between the spins placed at these vertices) depends on the Laplacian of the graph. How the quality of communication varies when starts cutting edges after starting a fully connected graph will be discussed *. Another direction is related to the entanglement naturally present in the ground state of a graph of perpetually interacting spins: various specific examples --- fully connected, star and tree will be discussed. Some easily solvable interactions obtained by putting higher spins in specific vertices of simple graphs will also be discussed. In the end we also present an example where one can get a \'graph independent\' ground state by placing qudits with exchange interactions on an arbitrary graph. (* The first part of the talk is based on ongoing work with Simone Severini, Stefano Mancini and Andrea Casaccino, while the second part is based on work with Vladimir Korepin and Christopher Hadley)