Foundations of Physics

, Volume 32, Issue 4, pp 531–565 | Cite as

The Bloch Gyrovector

  • Jing-Ling Chen
  • Abraham A. Ungar

Abstract

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.

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

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Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Jing-Ling Chen
    • 1
  • Abraham A. Ungar
    • 2
  1. 1.Laboratory of Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingPeople's Republic of China
  2. 2.Department of MathematicsNorth Dakota State UniversityFargo

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