Markovian statistics on evolving systems

Original Paper

Abstract

A novel framework for the analysis of observation statistics on time discrete linear evolutions in Banach space is presented. The model differs from traditional models for stochastic processes and, in particular, clearly distinguishes between the deterministic evolution of a system and the stochastic nature of observations on the evolving system. General Markov chains are defined in this context and it is shown how typical traditional models of classical or quantum random walks and Markov processes fit into the framework and how a theory of quantum statistics (sensu Barndorff-Nielsen, Gill and Jupp) may be developed from it. The framework permits a general theory of joint observability of two or more observation variables which may be viewed as an extension of the Heisenberg uncertainty principle and, in particular, offers a novel mathematical perspective on the violation of Bell’s inequalities in quantum models. Main results include a general sampling theorem relative to Riesz evolution operators in the spirit of von Neumann’s mean ergodic theorem for normal operators in Hilbert space.

Keywords

Banach space Bell inequality Ergodic theorem Evolution Heisenberg uncertainty Hilbert space Markov chain Observable Quantum statistics Random walk Sampling Riesz operator Stochastic process 

Notes

Acknowledgements

The authors are grateful to the reviewers for their careful reading of the manuscript and their comments, which have improved the presentation.

References

  1. Aharonov D, Ambainis A, Kempe J, Vazirani U: Quantum walks on graphs. In: Proc. 33th STOC, ACM, New York, pp 60–69Google Scholar
  2. Aspect A, Dalibard J, Roger G (1982) Experimental tests of Bell’s inequalities using time-varying analyzers. Phys Rev Lett 49:1804MathSciNetCrossRefGoogle Scholar
  3. Barndorff-Nielsen OE, Gill RD, Jupp PE (2003) On quantum statistical inference. J R Stat Soc B 65:775–816MathSciNetCrossRefMATHGoogle Scholar
  4. Bell JS (1964) On the Einstein Podolsky Rosen paradox. Physics 1:195–200Google Scholar
  5. Bell JS (1966) On the problem of hidden variables in quantum mechanics. Rev Mod Phys 38:447–452MathSciNetCrossRefMATHGoogle Scholar
  6. Choi SPM, Yeung DY, Zhang NL (2000) Hidden-Markov decision processes for nonstationary sequential decision making. In: Sun R, Giles CL (eds) Sequence learning. Lecture notes in artificial intelligence, vol 1828. Springer, Berlin, pp 264–287Google Scholar
  7. Conway JB (1990) A course in functional analysis. Graduate texts in mathematics, vol 96, 2nd edn. Springer, New YorkGoogle Scholar
  8. Dharmadhikari SW (1965) A characterization of a class of functions of finite Markov chains. Ann Math Stat 36:524–528CrossRefMATHGoogle Scholar
  9. Dowson HR (1978) Spectral theory of linear operators. Academic Press, LondonMATHGoogle Scholar
  10. Elliot RJ, Aggoun L, Moore JB (1995) Hidden Markov models. Springer, BerlinGoogle Scholar
  11. Faigle U, Grabisch M (2012) Values for Markovian coalition processes. Econ Theory 51:505–538MathSciNetCrossRefMATHGoogle Scholar
  12. Faigle U, Kern W (1991) Note on the convergence of simulated annealing algorithms. SIAM J Control Optim 29:153159MathSciNetCrossRefMATHGoogle Scholar
  13. Faigle U, Schönhuth A (2007) Asymptotic mean stationarity of sources with finite evolution dimension. IEEE Trans Inf Theory 53:2342–2348MathSciNetCrossRefMATHGoogle Scholar
  14. Faigle U, Schönhuth A (2011) Efficient tests for equivalence of hidden Markov processes and quantum random walks. IEEE Trans Inf Theory 57:1746–1753MathSciNetCrossRefGoogle Scholar
  15. Feller W (1971) An introduction to probability theory and its applications II. Wiley, New YorkMATHGoogle Scholar
  16. Grabisch M (2016) Set functions, games and capacities in decision making. Springer, Berlin (ISBN 978-3-319-30690-2)MATHGoogle Scholar
  17. Gilbert EJ (1959) On the identifiability problem for functions of finite Markov chains. Ann Math Stat 30:688–697MathSciNetCrossRefMATHGoogle Scholar
  18. Gudder S (2008) Quantum Markov chains. J Math Phys 49:072105MathSciNetCrossRefMATHGoogle Scholar
  19. Heller A (1965) On stochastic processes derived from Markov chains. Ann Math Stat 36:1286–1291MathSciNetCrossRefMATHGoogle Scholar
  20. Hernandez-Lerma O, Lassere JB (2003) Markov chains and invariant probabilities theory. Birkaeuser, BaselCrossRefGoogle Scholar
  21. Ito H, Amari S-I, Kobayashi K (1992) Identifiability of hidden Markov information sources and their minimum degrees of freedom. IEEE Trans Inf Theory 38:324–333MathSciNetCrossRefMATHGoogle Scholar
  22. Jaeger H (2000) Observable operator models for discrete stochastic time series. Neural Comput 12:1371–1398CrossRefGoogle Scholar
  23. Kempe J (2003) Quantum random walks: an introductory overview. Contemp Phys 44:307–327CrossRefGoogle Scholar
  24. Kirkpatrick S, Gelatt CD Jr, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671680MathSciNetCrossRefMATHGoogle Scholar
  25. Markoff AA (1912) Wahrscheinlichkeitsrechnung. In: Teubner BG (ed) Leipzig (Übersetzung der 2. russischen Auflage)Google Scholar
  26. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087CrossRefGoogle Scholar
  27. Nielsen M, Chuang I (2000) Quantum computation and quantum information. Cambridge University Press, CambridgeMATHGoogle Scholar
  28. Portugal R, Santos RAM, Fernandes TD, Goncalves DN (2015) The staggered quantum walk model. Quant Inf Process. arXiv:1505.04761
  29. Szegedy M (2004) Quantum speed-up of Markov chain based algorithms. In: Proceedings 45th Symposium on Foundations of Computer Science, pp 32–41Google Scholar
  30. Temme K, Osborne TJ, Vollbrecht KG, Verstraete F (2011) Quantum metropolis sampling. Nature 471:87–90CrossRefGoogle Scholar
  31. Vidyasagar M (2011) The complete realization problem for hidden Markov models: a survey and some new results. Math Control Signals Syst 23(2011):1–65MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany
  2. 2.Department of MathematicsUniversity of California at RiversideRiversideUSA

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