The European Physical Journal Plus

, 129:191

Towards quantifying complexity with quantum mechanics

  • Ryan Tan
  • Daniel R. Terno
  • Jayne Thompson
  • Vlatko Vedral
  • Mile Gu
Regular Article
Part of the following topical collections:
  1. Focus Point on Quantum information and complexity

Abstract.

While we have intuitive notions of structure and complexity, the formalization of this intuition is non-trivial. The statistical complexity is a popular candidate. It is based on the idea that the complexity of a process can be quantified by the complexity of its simplest mathematical model —the model that requires the least past information for optimal future prediction. Here we review how such models, known as \( \epsilon\)-machines can be further simplified through quantum logic, and explore the resulting consequences for understanding complexity. In particular, we propose a new measure of complexity based on quantum \( \epsilon\)-machines. We apply this to a simple system undergoing constant thermalization. The resulting quantum measure of complexity aligns more closely with our intuition of how complexity should behave.

References

  1. 1.
    A.N. Kolmogorov, Theor. Comput. Sci. 207, 387 (1998)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    J. Ladyman, J. Lambert, K. Wiesner, Eur. J. Philos. Sci. 3, 33 (2013)CrossRefMATHGoogle Scholar
  3. 3.
    J.P. Crutchfield, K. Young, Phys. Rev. Lett. 63, 105 (1989)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    C.R. Shalizi, J.P. Crutchfield, J. Stat. Phys. 104, 817 (2001)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    C.-B. Li, H. Yang, T. Komatsuzaki, Proc. Natl. Acad. Sci. 105, 536 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    J.P. Crutchfield, D.P. Feldman, Phys. Rev. E 55, R1239 (1997) arXiv:9702191ADSCrossRefGoogle Scholar
  7. 7.
    K. Wiesner, M. Gu, E. Rieper, V. Vedral, Proc. R. Soc. A: Math. Phys. Eng. Sci. 468, 4058 (2012)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    W.G. Rory Cerbus, arXiv:1403.5356 (2014)
  9. 9.
    J.P. Crutchfield, Nat. Phys. 8, 17 (2012)CrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Gu, K. Wiesner, E. Rieper, V. Vedral, Nat. Commun. 3, 762 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    J.P. Crutchfield, Phys. D: Nonlinear Phenom. 75, 11 (1994)ADSCrossRefMATHGoogle Scholar
  12. 12.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010)Google Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Ryan Tan
    • 1
  • Daniel R. Terno
    • 2
  • Jayne Thompson
    • 1
  • Vlatko Vedral
    • 3
    • 1
    • 4
  • Mile Gu
    • 5
    • 1
  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore
  2. 2.Department of Physics and AstronomyMacquarie UniversitySydneyAustralia
  3. 3.Department of PhysicsUniversity of Oxford, Clarendon LaboratoryOxfordUK
  4. 4.Department of PhysicsNational University of SingaporeSingaporeSingapore
  5. 5.Centre for Quantum Information, Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina

Personalised recommendations