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

Abstract

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.

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

References in corpus (4)

• The Bloch Vector for N-Level Systems
• Degrees of controllability for quantum systems and applications to atomic systems
• Characterizing mixing and measurement in quantum mechanics
• Criteria for reachability of quantum states

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