Advertisement

Algorithmic complexity of quantum capacity

  • Samad Khabbazi Oskouei
  • Stefano Mancini
Article
  • 76 Downloads

Abstract

We analyze the notion of quantum capacity from the perspective of algorithmic (descriptive) complexity. To this end, we resort to the concept of semi-computability in order to describe quantum states and quantum channel maps. We introduce algorithmic entropies (like algorithmic quantum coherent information) and derive relevant properties for them. Then we show that quantum capacity based on semi-computable concept equals the entropy rate of algorithmic coherent information, which in turn equals the standard quantum capacity. Thanks to this, we finally prove that the quantum capacity, for a given semi-computable channel, is limit computable.

Keywords

Algorithmic complexity Quantum entropies Quantum channels Quantum capacity 

References

  1. 1.
    Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Zvonkin, A.K., Levin, L.A.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russ. Math. Surv. 25, 83 (1970)CrossRefzbMATHGoogle Scholar
  3. 3.
    Gacs, P.: Quantum algorithmic entropy. J. Phys. A: Math. Theor. 34, 6859 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benatti, F., Oskouei, S.K., Deh Abad, A.S.: Gacs quantum algorithmic entropy in infinite dimensional Hilbert spaces. J. Math. Phys. 55, 082205 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Oskouei, S.K.: Gacs algorithmic complexity in infinite Hilbert spaces and its applications. Ph.D. dissertation, University of Tehran (2015)Google Scholar
  6. 6.
    Solomonoff, R.: A formal theory of inductive inference. Inf. Control 7, 224 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kolmogorov, A.: Three approaches to the quantitative definition of information. Probl. Inf. Transm. 1, 1 (1965)zbMATHGoogle Scholar
  8. 8.
    Chaitin, J.G.: On the length of programs for computing finite binary sequences. J. ACM 13, 547 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berthiaume, A., van Dam, W., Laplante, S.: Quantum Kolmogorov complexity. J. Comput. Syst. Sci. 63, 201 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Vitanyi, P.: Quantum Kolmogorov complexity based on classical descriptions. IEEE Trans. Inf. Theory 47, 2464 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mora, C.E., Briegel, H.J.: Algorithmic complexity and entanglement of quantum states. Phys. Rev. Lett. 95, 200503 (2005)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Wolf, M.M., Cubitt, T.S., Perez-Garcia, D.: Are problems in Quantum Information Theory (un)decidable? arxiv.org/abs/1111.5425 (2011)
  13. 13.
    Cubitt, T., et al.: Unbounded number of channel uses may be required to detect quantum capacity. Nat. Commun. 6, 6739 (2015)CrossRefGoogle Scholar
  14. 14.
    Stinespring, W.F.: Positive functions on \(C^*\)-algebras. Proc. Am. Math. Soc. 6, 211 (1955)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51, 44 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hayden, P., Horodecki, M., Winter, A., Yard, J.: A decoupling approach to the quantum capacity. Open Syst. Inf. Dyn. 15, 7 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lindblad, G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40, 147 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Davis, M., Sigal, R., Weyuker, E.J.: Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science. Academic Press, San Diego (1994)Google Scholar
  19. 19.
    Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIslamic Azad University, Varamin-Pishva BranchPishvaIran
  2. 2.School of Science and TechnologyUniversity of CamerinoCamerinoItaly
  3. 3.INFN–Sezione PerugiaPerugiaItaly

Personalised recommendations