Rémy Mosseri and Rossen Dandoloff 2001 J. Phys. A: Math. Gen. 34 10243 doi:10.1088/0305-4470/34/47/324
Geometry of entangled states, Bloch spheres and Hopf fibrations
Rémy Mosseri1 and Rossen Dandoloff2
Show affiliationsWe discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of high-dimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is the three-dimensional sphere S 3. The S 2 base space of a suitably oriented S 3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the overall qubit phase degree of freedom. For the two-qubits case, the Hilbert space is a seven-dimensional sphere S 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed.
- PACS
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03.67.Mn Entanglement measures, witnesses, and other characterizations
- MSC
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81Rxx Groups and algebras in quantum theory
81P68 Quantum computation and quantum cryptography (See also 68Q05, 94A60)
- Subjects
- Dates
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Issue 47 (30 November 2001)
Received 3 September 2001
Published 16 November 2001
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Geometry of entangled states, Bloch spheres and Hopf fibrations
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