Geometry of entangled states, Bloch spheres and Hopf fibrations
Abstract
We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of highdimensional spheres by lower dimensional spheres. The singlequbit Hilbert space is the threedimensional 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 twoqubits case, the Hilbert space is a sevendimensional 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.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 November 2001
 DOI:
 10.1088/03054470/34/47/324
 arXiv:
 arXiv:quantph/0108137
 Bibcode:
 2001JPhA...3410243M
 Keywords:

 Quantum Physics;
 Condensed Matter
 EPrint:
 submitted to J. Phys. A