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

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50 Citations (Scopus)


Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of S7over the quaternionic projective space HP1 ≃ S4 with an SU (2) ≃ S3 fibre, the geometry of entanglement for two qubits is investigated. The relationship between base and fibre 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 HP1 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 obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anholonomy of the connection and entanglement via the geometric phase are shown. Connections with important notions such as the Bures metric and Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.

Original languageEnglish
Pages (from-to)1821-1841
Number of pages21
JournalJournal of Physics A: Mathematical and General
Issue number5
Publication statusPublished - Feb 6 2004

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)

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