Skip to main content
SpringerLink
Log in

Fidelity and fidelity susceptibility based on Hilbert-Schmidt inner product

  • Article
  • Progress of Projects Supported by NSFC
  • Published:
Science China Physics, Mechanics and Astronomy Aims and scope Submit manuscript

Abstract

We reinvestigate the fidelity based on Hilbert-Schmidt inner product and give a simplified form. The geometric meaning of the fidelity is clarified. We then give the analytic expression of the fidelity susceptibility in both Hilbert and Liouville space. By using the reconstruction of symmetric logarithmic derivative in Liouville space, we present the time derivative of fidelity susceptibility with the normalized density vector representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Uhlmann A. The “transition probability” in the state space of a*-algebra. Rep Math Phys, 1976, 9: 273–279

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Alberti P, Uhlmann A. Transition probabilities on C*- and W*-algebra. In: Proceedings of the Second International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics. Leipzig: BSB B. G. Taubner-Verl., 1983

    Google Scholar 

  3. Alberti PM. A note on the transition probability over C*-algebras. Lett Math Phys, 1983, 7: 25–32

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Alberti P M, Uhlmann A. Stochastic linear maps and transition probability. Lett Math Phys, 1983, 7: 107–112

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Jozsa R. Fidelity for mixed quantum states. J Mod Opt, 1994, 41: 2315–2323

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Schumacher B. Quantum coding. Phys Rev A, 1995, 51: 2738–2747

    Article  MathSciNet  ADS  Google Scholar 

  7. Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000

    MATH  Google Scholar 

  8. Wang X, Yu C S, Yi X X. An alternative quantum fidelity for mixed states of qudits. Phys Lett A, 2008, 373: 58–60

    Article  ADS  MATH  Google Scholar 

  9. Mendoncça P, Napolitano R, Marchiolli M, et al. Alternative fidelity measure between quantum states. Phys Rev A, 2008, 78: 052330

    Article  ADS  Google Scholar 

  10. You W L, Li Y W, Gu S J. Fidelity, dynamic structure factor, and susceptibility in critical phenomena. Phys Rev E, 2007, 76: 022101

    Article  ADS  Google Scholar 

  11. Ma J, Xu L, Xiong H N, et al. Reduced fidelity susceptibility and its finite-size scaling behaviors in the Lipkin-Meshkov-Glick model. Phys Rev E, 2008, 78: 051126

    Article  ADS  Google Scholar 

  12. Fano U. Pressure broadening as a prototype of relaxation. Phys Rev, 1963, 131: 259–268

    Article  ADS  MATH  Google Scholar 

  13. Zwanzig R. On the identity of three generalized master equations. Physica, 1964, 30: 1109–1123; Zwanzig R. Ensemble method in the theory of irreversibility. J Chem Phys, 1960, 33: 1338–1341; Zwanzig R. Statistical mechanics of irreversibility. In: Brittin W E, Downs B W, Downs J, eds. Boulder Lecture Notes in Theoretical Physics. New York: Interscience, 1960. 3: 106–141

    Article  MathSciNet  ADS  Google Scholar 

  14. Redfield A G. On the theory of relaxation processes. IBM J Res Dev, 1957, 1: 19–31

    Article  Google Scholar 

  15. Jeener J. Superoperators in magnetic resonance. In: Waugh J S, ed. Advances in Magnetic Resonance. New York: Academic Press, 1982. 10: 1–38

    Google Scholar 

  16. Louisell W H. Quantum Statistical Properties of Radiation. New York: Wiley, 1973

    Google Scholar 

  17. Abragam A. The Principles of Nuclear Magnetism. London: Oxford Press, 1961

    Google Scholar 

  18. Goldman M. Quantum Description of High-Resolution NMR in Liquids. London: Clarendon Press, 1988

    Google Scholar 

  19. Ernst R R, Bodenbausen G, Wokaun A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions. London: Clarendon Press, 1987

    Google Scholar 

  20. Ben-Reuven A. Spectral line shapes in gases in the binary-collision approximation. Adv Chem Phys, 1975, 33: 235–293

    Article  ADS  Google Scholar 

  21. Mukamel S. Collisional broadening of spectral line shapes in twophoton and multiphoton processes. Phys Rep, 1982, 93: 1–60

    Article  ADS  Google Scholar 

  22. Mukamel S. Principles of Nonlinear Optical Spectroscopy. Oxford: Oxford University Press, 1995

    Google Scholar 

  23. Schlienz J, Mahler G. Description of entanglement. Phys Rev A, 1995, 52: 4396–4404

    Article  ADS  Google Scholar 

  24. Chruściński D, Kossakowski A. Non-Markovian quantum dynamics: Local versus nonlocal. Phys Rev Lett, 2010, 104, 070406

    Article  ADS  Google Scholar 

  25. Gorini V, Kossakowski A, Sudarshan E. Completely positive dynamical semigroups of N-level systems. JMath Phys, 1976, 17: 821–826; Lindblad G. On the generators of quantum dynamical semigroups. Commun Math Phys, 1976, 48: 119–130

    Article  MathSciNet  ADS  Google Scholar 

  26. Breuer H P. Genuine quantum trajectories for non-Markovian processes. Phys Rev A, 2004, 70: 012106

    Article  ADS  Google Scholar 

  27. Helstrom C W. Quantum Detection and Estimation Theory. New York: Academic, 1976; Holevo A S. Probabilistic and Statistical Aspects of Quantum Theory. Amsterdam: North-Holland, 1982

    Google Scholar 

  28. May V, Kühn O. Charge and Energy Transfer Dynamics in Molecular Systems. Weinheim: Wiley-VCH, 2004

    Google Scholar 

  29. Lu X M, Wang X G, Sun C P. Quantum Fisher information flow and non-Markovian processes of open systems. Phys Rev A, 2010,82: 042103

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, 310027, China

    Jing Liu

  2. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore, 117543, Singapore

    XiaoMing Lu

  3. Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, 310027, China

    Jian Ma & XiaoGuang Wang

Authors
  1. Jing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. XiaoMing Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. XiaoGuang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XiaoGuang Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, J., Lu, X., Ma, J. et al. Fidelity and fidelity susceptibility based on Hilbert-Schmidt inner product. Sci. China Phys. Mech. Astron. 55, 1529–1534 (2012). https://doi.org/10.1007/s11433-012-4852-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11433-012-4852-0

Keywords

Search

Navigation