Coincidence of Voronoi Diagrams in a Quantum State Space

1. Asian Conference on Quantum Information Science (AQIS 2007), Kyoto University, Japan Coincidence of Voronoi Diagrams in a Quantum State Space Kimikazu Kato1,2, Mayumi Oto3,, Hiroshi Imai1,4,5, Keiko Imai6 1Department of Computer Science, University of Tokyo 2Nihon Unisys, Ltd. 3Toshiba Corporation 4ERATO-SORST Quantum Computation and Information Project, JST 5Institute for Nano Quantum Information Electronics, University of Tokyo 6Department of Information and System Engineering, Chuo University Motivation Distances Geometric structure of quantum state space Quantum divergence (or relative entropy) We attempt to clarify the relation among the distances defined for a quantum state space Fubini Study distance (defined only for pure states) Is it possible to actually visualize a quantum state space? Numerical calculation of Holevo capacity Bures distance For one qubit system, there is an existing algorithm by [Oto et al. ’04] to compute the Holevo capacity. It uses Voronoi diagram in its process Holevo capacity Defined as a radius of the smallest enclosing ball of the image of the map with respect to the quantum divergence. Computational geometric point of view In classical information theory, Onishi and Imai analyzed the structure of Voronoi diagram [Onishi and Imai ’97] with respect to Kullback-Leibler divergence. This is a natural extension of it. Conclusion Whether a known geometric algorithm for Euclidean space can be applied for a general space is another interesting problem. Analysis on the Voronoi diagram gives a hint for this problem. We proved the following table which shows coincidences of Voronoi diagrams with respect to some distances. Preliminaries Voronoi diagram Definition For a given set of points (called sites), the Voronoi diagram is defined as: : not defined Roughly regions of the influence around each of sites ✔: equivalent to the divergence-Voronoi Voronoi diagram with 4 sites with respect to Euclidean distance ✖: not equivalent to the divergence-Voronoi Strictly Especially the coincidence of divergence-Voronoi diagram and Euclidean Voronoi diagram for one qubit states is a support for the effectiveness of the algorithm by [Oto et al. ’04]. What is it for? Standard tool for computer vision Generally used when you apply a discrete method to a continuous geometric object Future work Voronoi diagram for a quantum state space Voronoi diagrams with respect to the quantum divergence are defined as follows: As an application of this result, now we are trying to propose an algorithm to compute numerically the Holevo capacity of three or higher level quantum channel, which is - more robust and time effective than the algorithm by [Osawa and Nagaoka ’00], and - considered as an extension of the algorithm by [Oto et al. ’04] Voronoi diagram with respect to the divergence for pure states is naturally defined as a limit of the diagram for mixed states. Especially our main interest is: Does Welzl’s algorithm [Welzl ’91] for smallest enclosing ball problem in Euclidean space also work for such a distorted (pseudo-)distance as quantum divergence? Take limit Not defined on the boundary Naturally extended to the boundary