Voronoi Diagrams for Pure 1-qubit Quantum States

TL;DR

This paper develops Voronoi diagrams for pure 1-qubit quantum states on the Bloch sphere using Fubini-Study and Bures distances, connecting them to diagrams for mixed states via quantum divergence limits.

Contribution

It introduces Voronoi diagrams for pure quantum states on the Bloch sphere, extending existing divergence-based diagrams for mixed states.

Findings

Voronoi diagrams for pure states are equivalent to ordinary diagrams on the sphere.

Limit of divergence-based diagrams for mixed states yields diagrams for pure states.

New definitions enable analysis of pure states with existing divergence tools.

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.