Title :
Quantum Voronoi diagrams and Holevo channel capacity for 1-qubit quantum states
Author :
Nielsen, Frank ; Nock, Richard
Author_Institution :
Ecole Polytech., Sony Comput. Sci. Labs., Palaiseau
Abstract :
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.
Keywords :
channel capacity; computational geometry; matrix algebra; quantum communication; 1-qubit quantum states; 3D Bloch ball representation; Bregman generator; Bregman-Csiszar quantum entropies; Burg quantum entropies; Euclidean Voronoi diagram; Hermitian density matrices; Holevo channel capacity; Laguerre geometry; Neumann quantum entropies; power diagrams; practical approximation algorithm; quantum Voronoi diagrams; quantum sided centroids; Approximation algorithms; Channel capacity; Computer science; Entropy; Geometry; Information analysis; Laboratories; Matrix decomposition; Quantum computing; Quantum mechanics;
Conference_Titel :
Information Theory, 2008. ISIT 2008. IEEE International Symposium on
Conference_Location :
Toronto, ON
Print_ISBN :
978-1-4244-2256-2
Electronic_ISBN :
978-1-4244-2257-9
DOI :
10.1109/ISIT.2008.4594955