@inproceedings{2008-QuantumVoronoi-ISIT
, author={Frank Nielsen and Richard Nock}
, title={Quantum Voronoi diagrams and Holevo channel capacity for $1$-qubit quantum states}
, booktitle={IEEE International Symposium on Information Theory (ISIT)}
, month={July}
, year={2008}
, pages={96-100}
, address={Toronto, Canada}
, publisher={IEEE}
, doi={10.1109/ISIT.2008.4594955}
, abstract={
  In this paper, we first introduce a smooth parametric family of Bregman-Csisz\'ar
 quantum entropies including the 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 Bregman generator acting on Hermitian
 density matrices. This implies that these dual diagrams can be derived from power
 diagrams of balls in the Laguerre geometry, and allows one to prove by equivalence
 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 defined there. We then show how to compute
 the Holevo channel capacity of $1$-qubit quantum states, and provide a practical
 approximation algorithm based on Bregman core-sets. Finally, we define the quantum
 sided centroids that provide practical upper bounds on the Holevo capacity in linear
 time.
  }
}