### Abstract

Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of S^{7}over the quaternionic projective space HP^{1} ≃ S^{4} with an SU (2) ≃ S^{3} 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 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 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 language | English |
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Pages (from-to) | 1821-1841 |

Number of pages | 21 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 37 |

Issue number | 5 |

DOIs | |

Publication status | Published - Feb 6 2004 |

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### ASJC Scopus subject areas

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

### Cite this

**The geometry of entanglement : Metrics, connections and the geometric phase.** / Lévay, P.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 37, no. 5, pp. 1821-1841. https://doi.org/10.1088/0305-4470/37/5/024

}

TY - JOUR

T1 - The geometry of entanglement

T2 - Metrics, connections and the geometric phase

AU - Lévay, P.

PY - 2004/2/6

Y1 - 2004/2/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=1142277203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1142277203&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/37/5/024

DO - 10.1088/0305-4470/37/5/024

M3 - Article

AN - SCOPUS:1142277203

VL - 37

SP - 1821

EP - 1841

JO - Journal Physics D: Applied Physics

JF - Journal Physics D: Applied Physics

SN - 0022-3727

IS - 5

ER -