Abstract:
A qubit is a two state quantum system, completely described by the qubit density matrix  $\rho(\mathbf{v})$  parametrized by the Bloch vector  $\mathbf{v}$  varying in the unit ball of the Euclidean 3-space.  The only well-known structure of the space of all qubit density matrices is the convex structure.  Qubit density matrices give rise to the  trace distance  and  Bures fidelity  between two qubit density matrices.  Much to their chagrin, Nielsen and Chuang admit:

"Unfortunately, no similarly [alluding to the  trace distance  and its Euclidean geometric interpretation] clear geometric interpretation is known for the [Bures] fidelity between two states of a qubit".  Nielsen and Chuang [1, p. 410], 2000.

Surprisingly, Bures fidelity does have a novel rich geometric and algebraic structure, but it lies in the hyperbolic geometry of Bolyai and Lobachevsky rather than in Euclidean geometry [2].

Following [3, 4, 5] I will introduce a novel "gyrovector space" approach to the classical hyperbolic geometry of Bolyai and Lobachevsky which, unexpectedly, turns out to be fully analogous to the common vector space approach to Euclidean geometry.  I will then demonstrate that

(i) Bloch vector is not a vector but, rather, a  gyrovector  (that is, a hyperbolic vector); and that

(ii) the space of all qubit density matrices possesses the same novel, rich, nonassociative algebraic structure that regulates (a) hyperbolic geometry and (b) Einstein's special relativity theory.

In particular, I will show that Bures fidelity has a clear hyperbolic geometric interpretation, and indicate further applications of hyperbolic geometry in quantum information and computation.

References
[1] Michael A. Nielsen and Isaac L. Chuang.  Quantum computation and quantum information.  Cambridge University Press, Cambridge, 2000.

[2] J.-L. Chen, L. Fu, A. A. Ungar, and X.-G. Zhao, "Geometric observation of Bures fidelity between two states of a qubit,"  Phys. Rev. A (3), vol. 65, no. 2, pp. 024303/1-3, 2002.

[3] Abraham A. Ungar.  Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces, volume 117 of  Fundamental Theories of Physics.  Kluwer Academic Publishers Group, Dordrecht, 2001.

[4] Scott Walter.  Book Review:  Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces, by Abraham A. Ungar.  Found. Phys., 32(2):327-330, 2002.

[5] Abraham A. Ungar,  Analytic Hyperbolic Geometry: Mathematical Foundations and Applications.  Singapore: World Scientific, 2005.