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The Bloch Gyrovector

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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.

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REFERENCES

  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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

  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.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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).

    Google Scholar 

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

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Authors and 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

    Jing-Ling Chen

  2. Department of Mathematics, North Dakota State University, Fargo, North Dakota, 58105

    Abraham A. Ungar

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  1. Jing-Ling Chen
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  2. Abraham A. Ungar
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Chen, JL., Ungar, A.A. The Bloch Gyrovector. Foundations of Physics 32, 531–565 (2002). https://doi.org/10.1023/A:1015032332156

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