Geometry of the 3-Qubit State, Entanglement and Division Algebras

Details Geometry of the 3-Qubit State, Entanglement and Division Algebras Geometry of the 3-Qubit State, Entanglement and Division Algebras

2003-02-11

arxiv.org/abs/quant-ph/0302081v2

J. Phys. A: Math. Gen., 36, 8325 (2003)

Physics

Quantum Physics

12 pages, 2 figures, final version

Scientific paper

10.1088/0305-4470/36/30/309

We present a generalization to 3-qubits of the standard Bloch sphere representation for a single qubit and of the 7-dimensional sphere representation for 2 qubits presented in Mosseri {\it et al.}\cite{Mosseri2001}. The Hilbert space of the 3-qubit system is the 15-dimensional sphere $S^{15}$, which allows for a natural (last) Hopf fibration with $S^8$ as base and $S^7$ as fiber. A striking feature is, as in the case of 1 and 2 qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling 3 qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the 3-qubit state. Conjectures on the possibility to generalize the construction for higher qubit states are also discussed.

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