, Volume 32, Issue 4, pp 531565
First online:
The Bloch Gyrovector
 JingLing ChenAffiliated withLaboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics
 , Abraham A. UngarAffiliated withDepartment of Mathematics, North Dakota State University
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Hyperbolic vectors are called gyrovectors. We show that the Bloch vector of quantum mechanics is a gyrovector. The Bures fidelity between two states of a qubit is generated by two Bloch vectors. Treating these as gyrovectors rather than vectors results in our novel expression for the Bures fidelity, expressed in terms of its two generating Bloch gyrovectors. Taming the Thomas precession of Einstein's special theory of relativity led to the advent of the theory of gyrogroups and gyrovector spaces. Gyrovector spaces, in turn, form the setting for various models of the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for the standard model of Euclidean geometry. It is the recent advent of the theory of gyrogroups and gyrovector spaces that allows the Bures fidelity to be studied in its natural context, hyperbolic geometry, resulting in our new representation of the Bures fidelity, that reveals simplicity, elegance, and hyperbolic geometric significance.
 Title
 The Bloch Gyrovector
 Journal

Foundations of Physics
Volume 32, Issue 4 , pp 531565
 Cover Date
 200204
 DOI
 10.1023/A:1015032332156
 Print ISSN
 00159018
 Online ISSN
 15729516
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Bloch vector
 Bures fidelity
 Einstein's addition: gyrovector
 hyperbolic geometry
 Industry Sectors
 Authors

 JingLing Chen ^{(1)}
 Abraham A. Ungar ^{(2)}
 Author Affiliations

 1. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009(26), Beijing, 100088, People's Republic of China
 2. Department of Mathematics, North Dakota State University, Fargo, North Dakota, 58105