@inproceedings{2008-QuantumVoronoiDiagram-EuroCG
, author={Frank Nielsen and Richard Nock}
, title={Quantum Voronoi diagrams}
, booktitle={European Workshop on Computational Geometry (EuroCG)}
, month={March}
, year={2008}
, pages={225-228}
, address={Nancy, France}
, publisher={IEEE}
, abstract={
   We introduce the smooth parametric family of Bregman-Csiszar quantum entropies
 including the usual von Neumann and Burg quantum entropies. We then describe the
 dualistic nature of Voronoi diagrams for 1-qubit quantum states inside the 3D Bloch
 ball representation. We show that these diagrams can be computed as Bregman Voronoi
 diagrams for the corresponding entropic Bregman generator acting on Hermitian density
 matrices. This implies that these dual diagrams can be derived from equivalent power
 diagrams of balls in the Laguerre geometry, and allows one to prove by extension
 that the von Neumann quantum Voronoi diagram on the degenerated Bloch sphere of
 pure quantum states coincides with the ordinary Euclidean Voronoi diagram, bypassing
 the fact that the quantum divergence is not properly defined there.
   }
}