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

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

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Bibliographic Details
Main Authors Schirmer, S. G, Zhang, T, Leahy, J. V
Format Journal Article
LanguageEnglish
Published 31.07.2003
Subjects
Physics - Quantum Physics
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Summary:J. Phys. A 37(4), 1389-1402 (2004) 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.
DOI:10.48550/arxiv.quant-ph/0308004