Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0607029 · International Symposium on Voronoi Diagrams in Science and Engineering

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

Kimikazu Kato

Published Dec 1, 2006• 5 citations
arXivHTMLPDF

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.

DOI
10.1109/isvd.2006.27

Related papers

Voronoi Diagrams and a Numerical Estimation of a Quantum Channel Capacity

Kimikazu Kato, Mayumi Oto, Imai, Hiroshi +1 more

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.

arXiv:quant-ph/0611146•Nov 14, 2006•5 citations

Voronoi Diagrams for Pure 1-qubit Quantum States

Kato, Kimikazu, Oto, Mayumi, Imai, Hiroshi +1 more

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.

arXiv:quant-ph/0604101•Apr 14, 2006•18 citations

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

Frank Nielsen, Richard Nock

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.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2008•7 citations

Computational Geometry Analysis of Quantum State Space and Its Applications.

Kimikazu Kato, Mayumi Oto, Hiroshi Imai +1 more

We show some coincidences of Voronoi diagrams in a quantum state space with respect to some distances. This means a computational geometry interpretation of a structure of a quantum state space. More properly, we investigate the Voronoi diagrams with respect to the divergence, Fubini-Study distance, Bures distance, geodesic distance and Euclidean distance.

•Jan 1, 2008•0 citations

Voronoi Diagrams for Quantum States and Its Application to a Numerical Estimation of a Quantum Channel Capacity

Kimikazu Kato

In quantum information theory, a geometric approach, known as "quantum information geometry," has been considered as a powerful method. In this thesis, we give a computational geometric interpretation to the geometric structure of a quantum system. Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states. With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space. Additionally, as an application, we show an effective method to compute the capacity of a quantum channel.

arXiv:0803.3109•Mar 21, 2008•3 citations

Qubit channels that achieve capacity with two states

Dominic William Berry

This paper considers a class of qubit channels for which three states are always sufficient to achieve the Holevo capacity. For these channels, it is known that there are cases where two orthogonal states are sufficient, two nonorthogonal states are required, or three states are necessary. Here a systematic theory is given which provides criteria to distinguish cases where two states are sufficient, and determine whether these two states should be orthogonal or nonorthogonal. In addition, we prove a theorem on the form of the optimal ensemble when three states are required, and present efficient methods of calculating the Holevo capacity.

arXiv:quant-ph/0412113•Physical Review A•Mar 22, 2005•15 citations
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.
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.
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.
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.
In quantum information theory, a geometric approach, known as "quantum information geometry," has been considered as a powerful method. In this thesis, we give a computational geometric interpretation to the geometric structure of a quantum system. Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states. With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space. Additionally, as an application, we show an effective method to compute the capacity of a quantum channel.
This paper considers a class of qubit channels for which three states are always sufficient to achieve the Holevo capacity. For these channels, it is known that there are cases where two orthogonal states are sufficient, two nonorthogonal states are required, or three states are necessary. Here a systematic theory is given which provides criteria to distinguish cases where two states are sufficient, and determine whether these two states should be orthogonal or nonorthogonal. In addition, we prove a theorem on the form of the optimal ensemble when three states are required, and present efficient methods of calculating the Holevo capacity.