The geometry of entanglement: metrics, connections and the geometric phase

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dc.creatorLevay, Peter
dc.date2003-06-17
dc.date.accessioned2026-07-07T10:51:06Z
dc.date.available2026-07-07T10:51:06Z
dc.descriptionUsing the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.
dc.description42 pages
dc.identifierhttps://arxiv.org/abs/quant-ph/0306115
dc.identifierhttp://arxiv.org/abs/quant-ph/0306115
dc.identifierJ.Phys.A37:1821-1842,2004
dc.identifierdoi:10.1088/0305-4470/37/5/024
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/184749
dc.subjectQuantum Physics
dc.subjectHigh Energy Physics - Theory
dc.titleThe geometry of entanglement: metrics, connections and the geometric phase
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