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Geometry and curvature of spin networks

E Jonckheere, S G Schirmer, Frank C. Langbein

Published Jan 1, 2011• 10 citations
OpenAlex

Abstract

Abstract — A measure for the maximum quantum informa-tion transfer capacity (ITC) between nodes of a spin network is defined, and shown to induce a metric on a space of equivalence classes of nodes for homogeneous chains with XX and Heisen-berg couplings. The geometry and curvature of spin chains with respect of this metric are studied and compared to the physical network geometry. For general networks hierarchical clustering is used to elucidate the proximity of nodes with regard to the maximum ITC. Finally, it is shown how minimal control can be used to overcome intrinsic limitations and speed up information transfer. I.

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Abstract — A measure for the maximum quantum informa-tion transfer capacity (ITC) between nodes of a spin network is defined, and shown to induce a metric on a space of equivalence classes of nodes for homogeneous chains with XX and Heisen-berg couplings. The geometry and curvature of spin chains with respect of this metric are studied and compared to the physical network geometry. For general networks hierarchical clustering is used to elucidate the proximity of nodes with regard to the maximum ITC. Finally, it is shown how minimal control can be used to overcome intrinsic limitations and speed up information transfer. I.
A measure for the maximum quantum information transfer capacity (ITC) between nodes of a spin network is defined, and shown to induce a metric on a space of equivalence classes of nodes for homogeneous chains with XX and Heisen-berg couplings. The geometry and curvature of spin chains with respect of this metric are studied and compared to the physical network geometry. For general networks hierarchical clustering is used to elucidate the proximity of nodes with regard to the maximum ITC. Finally, it is shown how minimal control can be used to overcome intrinsic limitations and speed up information transfer.
We develop a geometric approach to spin networks with Heisenberg or XX coupling. Geometry is acquired by defin-ing a distance on the discrete set of spins. The key feature of the geometry of such networks is their Gauss curvature κ, viewed here as the ability to isometrically embed the chain in the standard Riemannian manifold of curvature κ. Here we focus on spin rings. Even though their visual geometry is trivial, it turns out that the geometry they acquire from the quantum mechanical distance is far from trivial. Index Terms — Spin chains, coarse geometry, curvature, Riemannian spaces, Feynman path integral.
We develop a geometric approach to spin networks with Heisenberg or XX coupling. Geometry is acquired by defining a distance on the discrete set of spins. The key feature of the geometry of such networks is their Gauss curvature κ, viewed here as the ability to isometrically embed the chain in the standard Riemannian manifold of curvature κ. Here we focus on spin rings. Even though their visual geometry is trivial, it turns out that the geometry they acquire from the quantum mechanical distance is far from trivial.
We study the evolution of N arbitrary spins described by the long-range Ising-type interaction. This fact allows us to obtain the Fubiny-Study metric of the manifold defined by this evolution. We explore the topology of this manifold and shown that it corresponds to a sphere. With the help of this fact the manifold geometry is determined. Also we calculate the Riemann curvature of manifold and represent it by means of the speed of evolution. This is important for an experimental measurement of the curvature. The conditions for achieving the minimal and maximal values of the speed of evolution are obtained. Finally, we examine the influence of an external magnetic field on the metric of manifold and on the speed of evolution. We obtain the conditions for achieving the minimal possible speed of evolution. For some predefined initial states the orientations of magnetic fields for reaching the minimal and maximal values of the speed are found.
To utilize a scalable quantum network and perform a quantum state transfer within distant arbitrary nodes, coherence and control of the dynamics of couplings between the information units must be achieved as a prerequisite ingredient for quantum information processing within a hierarchical structure. Graph theoretic approach provides a powerful tool for the characterization of quantum networks with non-trivial clustering properties. By encoding the topological features of the underlying quantum graphs, relations between the quantum complexity measures are presented revealing the intricate links between a quantum and a classical networks dynamics.