Voronoi Diagrams and a Numerical Estimation of a Quantum Channel Capacity

TL;DR

This paper introduces a geometric interpretation of quantum states using Voronoi diagrams, revealing structural coincidences across different distances for one-qubit states and higher-level systems, aiding in quantum channel capacity estimation.

Contribution

It provides a new geometric framework for quantum states with insights into Voronoi diagram coincidences across various distances, facilitating numerical estimation of quantum channel capacities.

Findings

Voronoi diagrams for one-qubit pure states coincide across several distances.

Coincidence of diagrams extends to higher-level pure states for divergence, Fubini-Study, and Bures distances.

Differences in diagram coincidences impact numerical methods for quantum channel capacity estimation.

Abstract

We give a new geometric interpretation of quantum pure states. Using Voronoi diagrams, we reinterpret the structure of the space of pure states as a subspace of the quantum state space. In addition to the known coincidence of some Voronoi diagrams for one-qubit pure states, we will show that even for mixed one-qubit states, as far as sites are given as pure states, the Voronoi diagram with respect to some distances -- the divergence, the Bures distance, and the Euclidean distance -- are all the same. As to higher level pure quantum states, for the divergence, the Fubini-Study distance, and the Bures distance, the coincidence of the diagrams still holds, while the coincidence of the diagrams with respect to the divergence and the Euclidean distance no longer holds. That fact has a significant meaning when we try to apply the method used for a numerical estimation of a one-qubit quantum channel capacity to a higher level system.