Foundations of Physics

, Volume 41, Issue 3, pp 450–465 | Cite as

Trigonometry of Quantum States

Article

Abstract

Recently the geometry of quantum states has been under considerable development. Every good geometry deserves, if possible, an accompanying trigonometry. I will here introduce such a trigonometry to accompany the geometry of quantum states.

Keywords

Entanglement Entropy Decoherence Fidelity Stratification 

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References

  1. 1.
    Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States. Cambridge Press, Cambridge (2008) Google Scholar
  2. 2.
    Kus, M., Zyczkowski, K.: Geometry of entangled states. Phys. Rev. A 63, 032307 (2001) CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Gustafson, K., Rao, D.: Numerical Range. Springer, New York (1997) CrossRefGoogle Scholar
  4. 4.
    Gustafson, K.: Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra. World Scientific, Singapore (1997) MATHCrossRefGoogle Scholar
  5. 5.
    Gustafson, K.: Noncommutative trigonometry. Oper. Theory, Adv. Appl. 167, 127–155 (2006) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gustafson, K.: A min–max theorem. Am. Math. Soc. Not. 15, 799 (1968) Google Scholar
  7. 7.
    Gustafson, K.: Operator trigonometry of Hotelling correlation, Frobenius condition, Penrose twistor. Linear Algebra Appl. 430, 2762–2770 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gustafson, K.: The trigonometry of twistors and elementary particles. In: Accardi, L., Adenier, G., Fuchs, C., Jaeger, G., Khrennikov, A., Larsson, J., Stenholm, S. (eds.) Foundations of Probability and Physics-5, Vaxjo, August 2008. Am. Inst. Phys. Conf. Proceedings, vol. 1101, pp. 65–73. AIP, New York (2009) Google Scholar
  9. 9.
    Nielsen, M., Kempe, J.: Separable states are more disordered globally than locally. Phys. Rev. Lett. 86, 5184–5187 (2001) CrossRefADSGoogle Scholar
  10. 10.
    Yu, T., Eberly, J.H.: Sudden death of entanglement: classical noise effects. Opt. Commun. 264, 393–397 (2006) CrossRefADSGoogle Scholar
  11. 11.
    Jaeger, G., Ann, K.: Entanglement sudden death in qubit–qutrit systems. Phys. Lett. A 372, 579–583 (2008) CrossRefMathSciNetADSMATHGoogle Scholar
  12. 12.
    Gustafson, K.: An extended operator trigonometry. Linear Algebra Appl. 319, 117–135 (2000) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gustafson, K.: Operator trigonometry of statistics and econometrics. Linear Algebra Appl. 354, 141–158 (2002) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gustafson, K.: The trigonometry of matrix statistics. Int. Stat. Rev. 74, 187–202 (2006) CrossRefGoogle Scholar
  15. 15.
    Levay, P.: The geometry of entanglement: metrics, connections and the geometric phase. J. Phys. A 37, 1821–1841 (2004) MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Mendl, C., Wolf, M.: Unital quantum channels–convex structure and revivals of Birkhoff’s theorem. Comm. Math. Phys. 289, 1057–1086 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Uhlmann, A.: The ‘transition probability’ in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976) MATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994) MATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  20. 20.
    Knill, M., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55, 900–911 (1997) CrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Gustafson, K.: Unpublished notes Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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