Annales Henri Poincaré

, Volume 19, Issue 4, pp 1215–1257 | Cite as

Quantum Speed Limit Versus Classical Displacement Energy

  • Laurent Charles
  • Leonid Polterovich


We discuss a link between symplectic displacement energy, a fundamental notion of symplectic topology, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes. The link is provided by the quantum-classical correspondence formalized within the framework of the Berezin–Toeplitz quantization.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ando, T.: Comparison of norms \(|||f(A)-f(B)|||\) and \(|||f(|A-B|)|||\). Math. Z. 197, 403–409 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andersson, O., Heydari, H.: Quantum speed limits and optimal Hamiltonians for driven systems in mixed states. J. Phys. A Math. Theor. 47(21), 215301 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds and \({\rm gl}(N), N\rightarrow \infty \) limits. Commun. Math. Phys. 165, 281–296 (1994)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Borthwick, D., Uribe, A.: Almost complex structures and geometric quantization. Math. Res. Lett. 3, 845–861 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boutet, de Monvel, L., Guillemin, V.: The Spectral Theory of Toeplitz Operators, Vol. 99, Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1981)Google Scholar
  6. 6.
    Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegő. In Journées: Équations aux Dérivées Partielles de Rennes (1975), pp. 123–164. Astérisque, No. 34–35. Soc. Math. France, Paris, 1976Google Scholar
  7. 7.
    Charles, L.: Berezin–Toeplitz operators, a semi-classical approach. Commun. Math. Phys. 239, 1–28 (2003)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Charles, L.: Quasimodes and Bohr–Sommerfeld conditions for the Toeplitz operators. Commun. PDE. 28, 1527–1566 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Charles, L.: On the quantization of compact symplectic manifold. J. Geom. Anal. 26, 2664–2710 (2016) and arXiv:1409.8507, v2, (2017)
  10. 10.
    Charles, L., Polterovich, L.: Sharp correspondence principle and quantum measurements. Algebra i Analiz 29, 237–278 (2017)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Chekanov, Yu.V.: Hofer’s symplectic energy and Lagrangian intersections. In Contact and symplectic geometry, (Cambridge, 1994), pp.296–306, Publ. Newton Inst., 8, Cambridge University Press, Cambridge (1996)Google Scholar
  12. 12.
    Combescure, M., Robert, D.: Coherent States and Applications in Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht (2012)zbMATHGoogle Scholar
  13. 13.
    Ferraro, A., Paris, M.: Non-classicality criteria from phase-space representations and information-theoretical constraints are maximally inequivalent. Phys. Rev. Lett 108(26), 260403 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Giraud, O., Braun, P., Braun, D.: Classicality of spin states. Phys. Rev. A 78(4), 042112 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Guillemin, V.: Star products on compact pre-quantizable symplectic manifolds. Lett. Math. Phys. 35, 85–89 (1995)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hairer, E., Wanner, G.: Analysis by Its History. Undergraduate Texts in Mathematics. Readings in Mathematics, p. x+374. Springer, New York (1996)Google Scholar
  18. 18.
    Hayashi, M.: Quantum Information. An Introduction. Springer, Berlin (2006)zbMATHGoogle Scholar
  19. 19.
    Hofer, H.: On the topological properties of symplectic maps. Proc. R. Soc. Edinb. Sect. A 115, 25–38 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hörmander, L.: Fourier integral operators. I. Acta Math. 127, 79–183 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Katok, A.: Ergodic perturbations of degenerate integrable Hamiltonian systems. Math. USSR Izv. 7, 535–571 (1973)CrossRefzbMATHGoogle Scholar
  23. 23.
    Lalonde, F., McDuff, D.: The geometry of symplectic energy. Ann. Math. 2(141), 349–371 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2012)CrossRefGoogle Scholar
  25. 25.
    Lloyd, S.: Coherent quantum feedback. Phys. Rev. A (3) 62(2), 022108 (2000)ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Ma, X., Marinescu, G.: Holomorphic Morse inequalities and Bergman kernels. Volume 254 Progress in Mathematics. Birkhäuser Verlag, Basel (2007)Google Scholar
  27. 27.
    Margolus, N., Levitin, L.B.: The maximum speed of dynamical evolution. Phys. D Nonlinear Phenom. 120, 188–195 (1998)ADSCrossRefGoogle Scholar
  28. 28.
    Mandelstam, L., Tamm, I.: The uncertainty relation between energy and time in nonrelativistic quantum mechanics. J. Phys. (USSR) 9(249), 1 (1945)Google Scholar
  29. 29.
    Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Ostrover, Y., Wagner, R.: On the extremality of Hofer’s metric on the group of Hamiltonian diffeomorphisms. Int. Math. Res. Not. 35, 2123–2141 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Polterovich, L., Rosen, D.: Function Theory on Symplectic Manifolds. CRM Monograph Series, vol. 34. American Mathematical Society, Providence, RI (2014)CrossRefzbMATHGoogle Scholar
  32. 32.
    Teschl, G.: Ordinary Differential Equations and Dynamical Systems, vol. 140. American Mathematical Society, Providence, RI (2012)zbMATHGoogle Scholar
  33. 33.
    Uhlmann, A.: An energy dispersion estimate. Phys. Lett. A 161(4), 329–331 (1992)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.UMR 7586, Institut de Mathématiques, de Jussieu-Paris Rive GaucheSorbonne UniversitésParisFrance
  2. 2.Faculty of Exact Sciences, School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations