Geometry of the three-qubit state, entanglement and division algebras

Bogdan A. Bernevig, Han Dong Chen

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


We present a generalization to three qubits of the standard Bloch sphere representation for a single qubit and of the seven-dimensional sphere representation for two qubits presented in Mosseri et al (Mosseri R and Dandoloff R 2001 J. Phys. A: Math. Gen. 34 10243). The Hubert space of the three-qubit system is the 15-dimensional sphere S15, which allows for a natural (last) Hopf fibration with S8 as base and S 7 as fibre. A striking feature is, as in the case of one and two qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling three qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the three-qubit state. Conjectures on the possibility of generalizing the construction for higher qubit states are also discussed.

Original languageEnglish (US)
Pages (from-to)8325-8339
Number of pages15
JournalJournal of Physics A: Mathematical and General
Issue number30
StatePublished - Aug 1 2003
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy


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