Orbits of quantum states and geometry of Bloch vectors for $N$-level systems

Details Orbits of quantum states and geometry of Bloch vectors for $N$-level systems Orbits of quantum states and geometry of Bloch vectors for $N$-level systems

2003-07-31

arxiv.org/abs/quant-ph/0308004v2

J. Phys. A 37(4), 1389-1402 (2004)

Physics

Quantum Physics

15 pages LaTeX, 3 figures, reformatted, slightly modified version, corrected eq.(3), to appear in J. Physics A

Scientific paper

10.1088/0305-4470/37/4/022

Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than two. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n-level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in R^{n^2-1} only for n=2. In higher dimensions the coherence vector only defines a non-surjective embedding into a closed ball. A detailed analysis of the three-level case is presented. Finally, a refined stratification in terms of symplectic orbits is considered.

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