On a Geometric Structure of Pure Multi-qubit Quantum States and Its Applicability to a Numerical Computation

TL;DR

This paper investigates the geometric structure of multi-qubit quantum states, showing that a property useful for capacity calculation in one-qubit states does not extend to higher dimensions, impacting the method's applicability.

Contribution

It provides a mathematical proof that Voronoi diagrams with respect to Euclidean distance and quantum divergence do not coincide for higher-dimensional quantum states, unlike the one-qubit case.

Findings

Voronoi diagrams do not coincide in higher dimensions

The method for Holevo capacity calculation in one-qubit states is not directly applicable to multi-qubit states

Mathematical proof of the non-coincidence of diagrams in higher dimensions

Abstract

For one-qubit pure quantum states, it is already proved that the Voronoi diagrams with respect to two distances -- Euclidean distance and the quantum divergence -- coincide. This fact is a support for a known method to calculate the Holevo capacity. To consider an applicability of this method to quantum states of a higher level system, it is essential to check if the coincidence of the Voronoi diagrams also occurs. In this paper, we show a negative result for that expectation. In other words, we mathematically prove that those diagrams no longer coincide in a higher dimension. That indicates that the method used in one-qubit case to calculate the Holevo capacity might not be effective in a higher dimension.