The Bloch Gyrovector
 JingLing Chen,
 Abraham A. Ungar
 … show all 2 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
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.
 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).
 K. Blum, Density Matrix Theory and Applications (Plenum, New York, 1996), 2nd edn.
 H. Urbantke, “Twolevel quantum systems: States, phases, and holonomy,”; Amer. J. Phys. 56, 503–509 (1991).
 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).
 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.
 F. J. Bloore, “Geometrical description of the convex sets of states for systems with spin1/2 and spin1,”; J. Phys. A 9, 2059–2067 (1976).
 K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, “Volume of the set of separable states,”; Phys. Rev. A (3) 58, 883–892 (1998).
 P. B. Slater, “A priori probabilities of separable quantum states,”; J. Phys. A 32, 5261–5275 (1999).
 P. B. Slater, “Comparative noninformativities of quantum priors based on monotone metrics,”; Phys. Lett. A 247, 1–8 (1998).
 A. Uhlmann, “Parallel transport and ‘quantum holonomy’ along density operators,”; Rep. Math. Phys. 24, 229–240 (1986).
 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.
 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).
 M. Hübner, “Computation of Uhlmann's parallel transport for density matrices and the Bures metric on threedimensional Hilbert space,”; Phys. Lett. A 179, 226–230 (1993).
 R. Jozsa, “Fidelity for mixed quantum states,”; J. Modern Opt. 41, 2315–2323 (1994).
 X. Wang, “Quantum teleportation of entangled coherent states,”; Phys. Rev. A (3) 64, 022302, 4 (2001).
 V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, “Quantifying entanglement,”; Phys. Rev. Lett. 78, 2275–2279 (1997).
 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
 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).
 A. Einstein, “Zur Elektrodynamik Bewegter Körper [On the electrodynamics of Moving bodies],”; Ann. Physik (Leipzig) 17, 891–921 (1905).
 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).
 V. Varičak, “Anwendung der Lobatschefskjschen Geometrie in der Relativtheorie,”; Physik. Z. 11, 93–96 (1910).
 V. Varičak, Darstellung der Redativitdtstheorie im dreidimensionalen Lobatchefskijschen Raume [Presentation of the Theory of Relativity in the Threedimensional Lobachevskian Space] (Zaklada, Zagreb, 1924).
 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).
 H. Goldstein, Classical Mechanics (AddisonWesley, Reading, Mass., 1980), 2nd edn.
 J. D. Jackson, Classical Electrodynamics (John Wiley, New York, 1975), second edn.
 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).
 B. Schumacher, “Quantum coding,”; Phys. Rev. A (3) 51, 2738–2747 (1995).
 J. Twamley, “Bures and statistical distance for squeezed thermal states,”; J. Phys. A 29, 3723–3731 (1996).
 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).
 C. A. Fuchs, Ph.D. thesis (University of Mexico, Albuquerque, NM, 1996); arXive eprint quantph/9601020.
 H. Scutaru, “Fidelity for displaced squeezed thermal states and the oscillator semigroup,”; J. Phys. A 31, 3659–3663 (1998).
 E. Knill and R. Laflamme, “Theory of quantum errorcorrecting codes,”; Phys. Rev. A (3) 55, 900–911 (1997).
 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 eprint quantph/ 9809010.
 S. Kakutani, “On equivalence of infinite product measures,”; Ann. of Math. (2) 49, 214–224 (1948).
 J. Dittmann and A. Uhlmann, “Connections and metrics respecting purification of quantum states,”; J. Math. Phys. 40, 3246–3267 (1999).
 D. Petz and C. Sudár, “Geometries of quantum states,”; J. Math. Phys. 37, 2662–2673 (1996).
 A. Uhlmann, “A gauge field governing parallel transport along mixed states,”; Lett. Math. Phys. 21, 229–236 (1991).
 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).
 M. Hübner, “Explicit computation of the Bures distance for density matrices,”; Phys. Lett. A 163, 239–242 (1992).
 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.
 J. Dittmann and G. Rudolph, “On a connection governing parallel transport along 2×2 density matrices,”; J. Geom. Phys. 10, 93–106 (1992).
 J. Dittmann and G. Rudolph, “A class of connections governing parallel transport along density matrices,”; J. Math. Phys. 33, 4148–4154 (1992).
 J. Dittmann, “On the Riemannian geometry of finitedimensional mixed states,”; Sem. Sophus Lie 3, 73–87 (1993).
 J. Dittmann, “Explicit formulae for the Bures metric,”; J. Phys. A 32, 2663–2670 (1999).
 A. A. Ungar, “Thomas rotation and the parametrization of the Lorentz transformation group,”; Found. Phys. Lett. 1, 57–89 (1988).
 A. A. Ungar, “Seeing the Möbius disctransformation like never before,”; Comput. Math. Appl. 42 (2002), in print.
 A. A. Ungar, “Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics,”; Found. Phys. 27, 881–951 (1997).
 L. Fuchs, Abelian Groups (Publishing House of the Hungarian Academy of Sciences, Budapest, 1958).
 B. R. Ebanks and C. T. Ng, “On generalized cocycle equations,”; Aequationes Math. 46, 76–90 (1993).
 H. Pollatsek, “Quantum error correction: classic group theory meets a quantum challenge,”; Amer Math. Monthly 108, 932–962 (2001).
 X.B. Wang, L. C. Kwek, and C. H. Oh, “Bures fidelity for diagonalizable quadratic Hamiltonians in multimode systems,”; J. Phys. A 33, 4925–4934 (2000).
 J.L. Chen, L. Fu, A. Ungar, and X.G. Zhao, “Alternative fidelity measure between two states of an instate quantum system,”; Phys. Rev. A (3) 65 (2002), in print.
 Title
 The Bloch Gyrovector
 Journal

Foundations of Physics
Volume 32, Issue 4 , pp 531565
 Cover Date
 20020401
 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