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

S. G. Schirmer·T. Zhang·J. V. Leahy
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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.

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S. G. Schirmer, T. Zhang, J. V. Leahy. 2003-11-26. Orbits of quantum states and geometry of Bloch vectors for $N$-level systems. https://doi.org/10.1088/0305-4470%2F37%2F4%2F022

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