Foundations of Physics

, Volume 45, Issue 1, pp 62–74 | Cite as

Imperfect Cloning Operations in Algebraic Quantum Theory

  • Yuichiro Kitajima


No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal \(\epsilon \)-imperfect cloning operation which tolerates a finite loss \(\epsilon \) of fidelity in the cloned state, and show that an individual system’s algebra of observables is abelian if and only if there is a universal \(\epsilon \)-imperfect cloning operation in the case where the loss of fidelity is less than \(1/4\). Therefore in this case no universal \(\epsilon \)-imperfect cloning operation is possible in algebraic quantum theory.


No-cloning theorem Fidelity Completely positive map  Algebraic quantum field theory 



The author wishes to thank Izumi Ojima and Yutaka Shikano for helpful comments on an earlier draft. The author is supported by the JSPS KAKENHI, No.23701009 and the John Templeton Foundation Grant ID 35771.


  1. 1.
    Alberti, P.M.: A note on the transition probability over C*-algebras. Lett. Math. Phys. 7, 25–32 (1983)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alberti, P.M., Uhlmann, A.: Stochastic linear maps and transition probability. Lett. Math. Phys. 7, 107–112 (1983)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Araki, H., Raggio, G.A.: A remark on transition probability. Lett. Math. Phys. 6, 237–240 (1982)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bures, D.: An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite W*-algebras. Trans. Am. Math. Soc. 135, 199–212 (1969)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bužek, V., Hillery, M.: Quantum copying: Beyond the no-cloning theorem. Phys. Rev. A. 54, 1844–1852 (1996)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Clifton, R., Bub, J., Halvorson, H.: Characterizing quantum theory in terms of information-theoretic constraints. Found. Phys. 33, 1561–1591 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Dieks, D.: Communication by EPR devices. Phys. Lett. 92, 271–272 (1982)CrossRefGoogle Scholar
  8. 8.
    Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Horuzhy, S.S.: Introduction to Algebraic Quantum Field Theory. Springer, Berlin (1990)zbMATHGoogle Scholar
  10. 10.
    Kadison, R. V. and Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary theory. American Mathematical Society (1983)Google Scholar
  11. 11.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, Vol. Advanced theory. American Mathematical Society, II (1983)Google Scholar
  12. 12.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  13. 13.
    Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  14. 14.
    Promislow, D.: The Kakutani theorem for tensor products of W*-algebras. Pac. J. Math. 36, 507–514 (1971)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Raggio, G.A.: Comparison of Uhlmann’s transition probability with the one induced by the natural positive cone of von Neumann algebras in standard form. Lett. Math. Phys. 6, 233–236 (1982)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Roberts, J.E., Roepstorff, G.: Some basic concepts of algebraic quantum theory. Commun. Math. Phys. 11, 321–338 (1969)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Roos, H.: Independence of local algebras in quantum field theory. Commun. Math. Phys. 16, 238–246 (1970)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (2002)zbMATHGoogle Scholar
  19. 19.
    Uhlmann, A.: The “transition probability” in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Uhlmann, A.: The transition probability for states of *-algebras. Ann. Phys. 497, 524–532 (1985)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of Industrial TechnologyNihon UniversityNarashinoJapan

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