Quantum Voronoi diagrams and Holevo channel capacity for 1-qubit quantum states

In this paper, we first introduce a smooth parametric family of Bregman-Csiszar 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 yield practical upper bounds on the Holevo capacity in linear time.