Quantum Information Processing

, Volume 9, Issue 2, pp 295–305 | Cite as

The diagonalization method in quantum recursion theory

  • Karl Svozil


As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to be modified. Quantum diagonalization involves unitary operators whose eigenvalues are different from one.


Quantum information Quantum recursion theory Halting problem 


03.67.Hk 03.65.Ud 


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  1. 1.
    Rogers H. Jr: Theory of Recursive Functions and Effective Computability. MacGraw-Hill, New York (1967)zbMATHGoogle Scholar
  2. 2.
    Davis M.: The Undecidable. Basic Papers on Undecidable, Unsolvable Problems and Computable Functions. Raven Press, Hewlett (1965)Google Scholar
  3. 3.
    Barwise J.: Handbook of Mathematical Logic. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  4. 4.
    Enderton H.: A Mathematical Introduction to Logic, 2nd edn. Academic Press, San Diego (2001)zbMATHGoogle Scholar
  5. 5.
    Odifreddi P.: Classical Recursion Theory, vol 1. North-Holland, Amsterdam (1989)Google Scholar
  6. 6.
    Boolos G.S., Burgess J.P., Jeffrey R.C.: Computability and Logic, 5th edn. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  7. 7.
    Landauer, R.: Information is physical. Phys. Today 44, 23–29. (1991)Google Scholar
  8. 8.
    Bridgman, P.W.: A physicist’s second reaction to Mengenlehre. Scripta Mathematica 2, 101–117, 224–234, cf. R. Landauer [50] (1934)Google Scholar
  9. 9.
    Olszewski A., Woleński J., Janusz R.: Church’s Thesis After 70 Years. Ontos, Berlin (2006)zbMATHGoogle Scholar
  10. 10.
    Feynman R.P.: Simulating physics with computers. Int. J. Theo. Phys. 21, 467–488 (1982)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Fredkin, E., Toffoli, T.: Conservative logic. Int. J. Theo. Phys. 21, 219–253, reprinted in [51 part I, Chap. 3]. (1982)
  12. 12.
    Leff H.S., Rex A.F.: Maxwells Demon. Princeton University Press, Princeton (1990)CrossRefGoogle Scholar
  13. 13.
    Feynman, R.P.: The Feynman lectures on computation. In: Hey, A.J.G., Allen, R.W. (eds.) Addison-Wesley Publishing Company, Reading (1996)Google Scholar
  14. 14.
    Birkhoff G., von Neumann J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)CrossRefGoogle Scholar
  15. 15.
    Kochen, S., Specker, E.P.: Logical structures arising in quantum theory. In: Symposium on the Theory of Models. Proceedings of the 1963 International Symposium at Berkeley, pp. 177–189, reprinted in [52, pp. 209–221] (1965)Google Scholar
  16. 16.
    Kochen, S., Specker, E.P.: The calculus of partial propositional functions. In: Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, Jerusalem, pp. 45–57, reprinted in [52, pp. 222–234] (1965)Google Scholar
  17. 17.
    Foulis, D.J., Piron, C., Randall, C.H.: Realism, operationalism, and quantum mechanics. Found. Phys. 13, 813–841, invited papers dedicated to Günther Ludwig. (1983)Google Scholar
  18. 18.
    Randall, C.H., Foulis, D.J.: Properties and operational propositions in quantum mechanics. Found. Phys. 13, 843–857, invited papers dedicated to Günther Ludwig. (1983)
  19. 19.
    Bournez, O., Campagnolo, M.L.: A survey on continuous time computations. In: Cooper, S., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms. Changing Conceptions of What is Computable. Springer, New York, pp. 383–423. (2008)
  20. 20.
    Deutsch, D.: Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proceedings of the royal society of London. Math. Phys. Sci. A (1934–1990) 400, 97–117. (1985)
  21. 21.
    Gruska J.: Quantum Computing. McGraw-Hill, London (1999)Google Scholar
  22. 22.
    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  23. 23.
    Lo H.-K., Popescu S., Spiller T.: Introduction to Quantum Computation and Information. World Scientific Publishing Company, Singapore (2001)Google Scholar
  24. 24.
    Brylinski R.K., Chen G., Brylinski B.K.: Mathematics of Quantum Computation. Chapman & Hall/CRC Press, London (2002)zbMATHGoogle Scholar
  25. 25.
    Hayashi M.: Quantum Information. An Introduction. Springer, Berlin (2006)Google Scholar
  26. 26.
    Imai H., Hayashi M.: Quantum Computation and Information. From Theory to Experiment. Springer, Berlin (2006)zbMATHCrossRefGoogle Scholar
  27. 27.
    Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Dusek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. (2008)
  28. 28.
    Jaeger G.: Quantum Information an Overview. Springer, New York (2007)zbMATHGoogle Scholar
  29. 29.
    Mermin, N.D.: Quantum Computer Science. Cambridge University Press, Cambridge (2007)
  30. 30.
    Mermin, N.D.: From Cbits to Qbits: Teaching computer scientists quantum mechanics. Am. J. Phys. 71, 23–30. (2003)
  31. 31.
    Svozil, K.: Contexts in quantum, classical and partition logic. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures, pp. 551–586. Elsevier, Amsterdam. (2008)
  32. 32.
    Born, M.: Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37, 863–867. (1926)
  33. 33.
    Zeilinger, A.: The message of the quantum. Nature 438, 743. (2005)
  34. 34.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik 38, 173–198, English translation in [53], and in [2] (1931)Google Scholar
  35. 35.
    Anderson A.R.: St. Paul’s epistle to Titus. In: Martin, R.L. (ed.) The Paradox of the Liar. Yale University Press, New Haven. The Bible contains a passage which refers to Epimenides, a Crete living in the capital city of Cnossus: “One of themselves, a prophet of their own, said, ‘Cretans are always liars, evil beasts, lazy gluttons.’ ”,—St. Paul, Epistle to Titus I (12–13) (1970)Google Scholar
  36. 36.
    Cantor G.: Abhandlungen. Springer, Berlin (1932)Google Scholar
  37. 37.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London mathematical society, series 2. 42 and 43, 230–265 and 544–546, reprinted in [2] (1936)Google Scholar
  38. 38.
    Svozil K.: Consistent use of paradoxes in deriving contraints on the dynamics of physical systems and of no-go-theorems. Found. Phys. Lett. 8, 523–535 (1995)CrossRefGoogle Scholar
  39. 39.
    Diaconis, P., Holmes, S., Montgomery, R.: Dynamical bias in the coin toss. SIAM Rev. 49, 211–235. (2007)Google Scholar
  40. 40.
    Murnaghan F.D.: The Unitary and Rotation Groups. Spartan Books, Washington, DC (1962)zbMATHGoogle Scholar
  41. 41.
    Shankar R.: Principles of Quantum Mechanics, 2nd edn. Kluwer Academic/Plenum Publishers, New York (1994)zbMATHGoogle Scholar
  42. 42.
    Halmos P.R.: Finite-dimensional Vector Spaces. Springer, New York (1974)zbMATHGoogle Scholar
  43. 43.
    Reck M., Zeilinger A., Bernstein H.J., Bertani P.: Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61. (1994)Google Scholar
  44. 44.
    Reck, M., Zeilinger, A.: Quantum phase tracing of correlated photons in optical multiports. In: Martini, F.D., Denardo, G., Zeilinger, A., (eds.) Quantum Interferometry, pp. 170–177 (1994)Google Scholar
  45. 45.
    Zukowski, M., Zeilinger, A., Horne, M.A.: Realizable higher-dimensional two-particle entanglements via multiport beam splitters. Phys. Rev. A 55, 2564–2579. (1997)
  46. 46.
    Svozil, K. Noncontextuality in multipartite entanglement. J. Phys. A Math. Gen. 38, 5781–5798. (2005)
  47. 47.
    Greenberger D.M., Horne M.A., Zeilinger A.: Multiparticle interferometry and the superposition principle. Phys. Today 46, 22–29 (1993)CrossRefGoogle Scholar
  48. 48.
    Yurke, B., McCall, S.L., Klauder, J.R.: SU(2) and SU(1,1) interferometers. Phys. Rev. A 33, 4033–4054. (1986)
  49. 49.
    Campos, R.A., Saleh, B.E.A., Teich, M.C.: Fourth-order interference of joint single-photon wave packets in lossless optical systems. Phys. Rev. A 42, 4127–4137. (1990)
  50. 50.
    Landauer, R.: Advertisement for a paper I like. In: Casti, J.L., Traub, J.F. (eds.) On limits. Santa Fe Institute Report 94-10-056, Santa Fe, NM, p. 39. (1994)
  51. 51.
    Adamatzky A.: Collision-based Computing. Springer, London (2002)zbMATHGoogle Scholar
  52. 52.
    Specker E.: Selecta. Birkhäuser Verlag, Basel (1990)zbMATHGoogle Scholar
  53. 53.
    Gödel, K. In: Feferman, S., Dawson, J.W., Kleene, S.C., Moore, G.H., Solovay, R.M., van Heijenoort, J. (eds.). Collected Works. Publications 1929–1936, vol I. Oxford University Press, Oxford (1986)Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsVienna University of TechnologyViennaAustria

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