1.

A. A. Ungar, *Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces* (Kluwer Academic, Dordrecht/Boston/ London, 2001).

2.

K. Blum, *Density Matrix Theory and Applications* (Plenum, New York, 1996), 2nd edn.

3.

H. Urbantke, “Two-level quantum systems: States, phases, and holonomy,”; *Amer. J. Phys.*
**56**, 503–509 (1991).

4.

J.-L. Chen, L. Fu, A. Ungar, and X.-G. Zhao, “Geometric observation of Bures fidelity between two states of a qubit,”; *Phys. Rev. A (3)*
**65**, 024303, 3 (2002).

5.

J.-L. Chen, L. Fu, A. Ungar, and X.-G. Zhao, “Degree of entanglement for two qubits,”; *Phys. Rev. A (3)*
**65** (2002), in print.

6.

F. J. Bloore, “Geometrical description of the convex sets of states for systems with spin-1/2 and spin-1,”; *J. Phys. A*
**9**, 2059–2067 (1976).

7.

K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, “Volume of the set of separable states,”; *Phys. Rev. A (3)*
**58**, 883–892 (1998).

8.

P. B. Slater, “A priori probabilities of separable quantum states,”; *J. Phys. A*
**32**, 5261–5275 (1999).

9.

P. B. Slater, “Comparative noninformativities of quantum priors based on monotone metrics,”; *Phys. Lett. A*
**247**, 1–8 (1998).

10.

A. Uhlmann, “Parallel transport and ‘quantum holonomy’ along density operators,”; *Rep. Math. Phys.*
**24**, 229–240 (1986).

11.

A. Uhlmann, Parallel lifts and holonomy along density operators: computable examples using oi(3)-orbits, in *Symmetries in Science, VI (Bregenz, 1992)* (Plenum, New York, 1993), pp. 741–748.

12.

A. Uhlmann, Parallel transport and holonomy along density operators, in *Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics (Clausthal, 1986)* (Teaneck, NJ), pp. 246–254 (World Scientific, Singapore, 1987).

13.

M. Hübner, “Computation of Uhlmann's parallel transport for density matrices and the Bures metric on three-dimensional Hilbert space,”; *Phys. Lett. A*
**179**, 226–230 (1993).

14.

R. Jozsa, “Fidelity for mixed quantum states,”; *J. Modern Opt.*
**41**, 2315–2323 (1994).

15.

X. Wang, “Quantum teleportation of entangled coherent states,”; *Phys. Rev. A (3)*
**64**, 022302, 4 (2001).

16.

V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,”; *Phys. Rev. Lett.*
**78**, 2275–2279 (1997).

17.

M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information* (Cambridge University Press, Cambridge, 2000).

18.

V. Fock, *The Theory of Space, Time and Gravitation* (MacMillan, New York, 1964), 2nd revised edn., translated from the Russian by N. Kemmer (a Pergamon Press book).

19.

A. Einstein, “Zur Elektrodynamik Bewegter Körper [On the electrodynamics of Moving bodies],”; *Ann. Physik (Leipzig)*
**17**, 891–921 (1905).

20.

A. Einstein, *The Collected Papers of Albert Einstein, Vol. 2: The Swiss Years: Writings, 1900–1909*, John Stachel, ed., translation from the German by Anna Beck (Princeton University Press, Princeton, NJ, 1989).

21.

V. Varičak, “Anwendung der Lobatschefskjschen Geometrie in der Relativtheorie,”; *Physik. Z.*
**11**, 93–96 (1910).

22.

V. Varičak, *Darstellung der Redativitdtstheorie im dreidimensionalen Lobatchefskijschen Raume [Presentation of the Theory of Relativity in the Three-dimensional Lobachevskian Space]* (Zaklada, Zagreb, 1924).

23.

J.-l. Chen and A. A. Ungar, “From the group SL(2, c) to gyrogroups and gyrovector spaces and hyperbolic geometry,”; *Found. Phys.*
**31**, 1611–1639 (2001).

24.

H. Goldstein, *Classical Mechanics* (Addison-Wesley, Reading, Mass., 1980), 2nd edn.

25.

J. D. Jackson, *Classical Electrodynamics* (John Wiley, New York, 1975), second edn.

26.

D. Bures, “An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite *W**-algebras,”; *Trans. Amer. Math. Soc.*
**135**, 199–212 (1969).

27.

B. Schumacher, “Quantum coding,”; *Phys. Rev. A (3)*
**51**, 2738–2747 (1995).

28.

J. Twamley, “Bures and statistical distance for squeezed thermal states,”; *J. Phys. A*
**29**, 3723–3731 (1996).

29.

M. B. Ruskai, “Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy,”; *Rev. Math. Phys.*
**6** (5A), 1147–1161 (1994), with an appendix on applications to logarithmic Sobolev inequalities (special issue dedicated to Elliott H. Lieb).

30.

C. A. Fuchs, Ph.D. thesis (University of Mexico, Albuquerque, NM, 1996); arXive e-print quantph/9601020.

31.

H. Scutaru, “Fidelity for displaced squeezed thermal states and the oscillator semigroup,”; *J. Phys. A*
**31**, 3659–3663 (1998).

32.

E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,”; *Phys. Rev. A (3)*
**55**, 900–911 (1997).

33.

H. Barnum, E. Knill, and M. A. Nielsen, “On quantum fidelities and channel capacities,”; *IEEE Trans. Inform. Theory*
**46**(4), 1317–1329 (2000); see arXive e-print quant-ph/ 9809010.

34.

S. Kakutani, “On equivalence of infinite product measures,”; *Ann. of Math. (2)*
**49**, 214–224 (1948).

35.

J. Dittmann and A. Uhlmann, “Connections and metrics respecting purification of quantum states,”; *J. Math. Phys.*
**40**, 3246–3267 (1999).

36.

D. Petz and C. Sudár, “Geometries of quantum states,”; *J. Math. Phys.*
**37**, 2662–2673 (1996).

37.

A. Uhlmann, “A gauge field governing parallel transport along mixed states,”; *Lett. Math. Phys.*
**21**, 229–236 (1991).

38.

A. Uhlmann, “Density operators as an arena for differential geometry,”; in *Proceedings of the XXV Symposium on Mathematical Physics (Toruń, 1992)*
**33**, 253–263 (1993).

39.

M. Hübner, “Explicit computation of the Bures distance for density matrices,”; *Phys. Lett. A*
**163**, 239–242 (1992).

40.

M. Hübner, “The Bures metric and Uhlmann's transition probability: explicit results,”; in *Classical and Quantum Systems (Goslar, 1991)* (World Scientific, Singapore, 1993), pp. 406–409.

41.

J. Dittmann and G. Rudolph, “On a connection governing parallel transport along 2×2 density matrices,”; *J. Geom. Phys.*
**10**, 93–106 (1992).

42.

J. Dittmann and G. Rudolph, “A class of connections governing parallel transport along density matrices,”; *J. Math. Phys.*
**33**, 4148–4154 (1992).

43.

J. Dittmann, “On the Riemannian geometry of finite-dimensional mixed states,”; *Sem. Sophus Lie*
**3**, 73–87 (1993).

44.

J. Dittmann, “Explicit formulae for the Bures metric,”; *J. Phys. A*
**32**, 2663–2670 (1999).

45.

A. A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,”; *Found. Phys. Lett.*
**1**, 57–89 (1988).

46.

A. A. Ungar, “Seeing the Möbius disc-transformation like never before,”; *Comput. Math. Appl.*
**42** (2002), in print.

47.

A. A. Ungar, “Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics,”; *Found. Phys.*
**27**, 881–951 (1997).

48.

L. Fuchs, *Abelian Groups* (Publishing House of the Hungarian Academy of Sciences, Budapest, 1958).

49.

B. R. Ebanks and C. T. Ng, “On generalized cocycle equations,”; *Aequationes Math.*
**46**, 76–90 (1993).

50.

H. Pollatsek, “Quantum error correction: classic group theory meets a quantum challenge,”; *Amer Math. Monthly*
**108**, 932–962 (2001).

51.

X.-B. Wang, L. C. Kwek, and C. H. Oh, “Bures fidelity for diagonalizable quadratic Hamiltonians in multi-mode systems,”; *J. Phys. A*
**33**, 4925–4934 (2000).

52.

J.-L. Chen, L. Fu, A. Ungar, and X.-G. Zhao, “Alternative fidelity measure between two states of an i*n*-state quantum system,”; *Phys. Rev. A (3)*
**65** (2002), in print.