Voronoi Diagrams for Pure 1-qubit Quantum States

Abstract

1-qubit quantum states form a space called the three-dimensional Bloch ball. To compute Holevo capacity, Voronoi diagrams in the Bloch ball with respect to the quantum divergence have been used as a powerful tool. These diagrams basically treat mixed quantum states corresponding to points in the interior of the Bloch ball. Due to the existence of logarithm in the quantum divergence, the diagrams are not defined on pure quantum states corresponding to points on the two-dimensional sphere. This paper first defines the Voronoi diagrams for pure quantum states on the Bloch sphere by the Fubini-Study distance and the Bures distance. We also introduce other Voronoi diagrams on the sphere obtained by taking a limit of Voronoi diagrams for mixed quantum states by the quantum divergences in the Bloch ball. These diagrams are shown to be equivalent to the ordinary Voronoi diagram on the sphere.

7 pages, 2 figures. Talk presented at "International Symposium on Voronoi Diagrams in Science and Engineering 2005"

Cited by in corpus (4)

• Voronoi Diagrams and a Numerical Estimation of a Quantum Channel Capacity
• On a Geometric Structure of Pure Multi-qubit Quantum States and Its Applicability to a Numerical Computation
• Voronoi Diagrams for Quantum States and Its Application to a Numerical Estimation of a Quantum Channel Capacity
• Degeneracy of Angular Voronoi Diagram