Generalized quantum no-go theorems of pure states

  • Hui-Ran Li
  • Ming-Xing LuoEmail author
  • Hong Lai


Various results of the no-cloning theorem, no-deleting theorem and no-superposing theorem in quantum mechanics have been proved using the superposition principle and the linearity of quantum operations. In this paper, we investigate general transformations forbidden by quantum mechanics in order to unify these theorems. First, we prove that any useful information cannot be created from an unknown pure state which is randomly chosen from a Hilbert space according to the Harr measure. And then, we propose a unified no-go theorem based on a generalized no-superposing result. The new theorem includes the no-cloning theorem, no-anticloning theorem, no-partial-erasure theorem, no-splitting theorem, no-superposing theorem or no-encoding theorem as a special case. Moreover, it implies various new results. Third, we extend the new theorem into another form that includes the no-deleting theorem as a special case.


Quantum no-go theorem Generalized no-go theorem Quantum no-superposing theorem Pure state 



We thank Luming Duan and Yaoyun Shi. This work was supported by the National Natural Science Foundation of China (Nos. 61772437, 61702427), Sichuan Youth Science and Technique Foundation (No. 2017JQ0048) and Fundamental Research Funds for the Central Universities (No. XDJK2016C043), Chuying Fellowship and the Doctoral Program of Higher Education (No. SWU115091).


  1. 1.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802 (1982)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Diecks, D.: Communication by EPR devices. Phys. Lett. A 92, 271 (1982)ADSCrossRefGoogle Scholar
  3. 3.
    Yuen, H.P.: Amplification of quantum states and noiseless photon amplifiers. Phys. Lett. A 113, 405–407 (1986)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, Norwell (1995)zbMATHGoogle Scholar
  5. 5.
    Barnum, H., Caves, C.M., Fuchs, C.A., Jozsa, R., Schumacher, B.: Noncommuting mixed states cannot be broadcast. Phys. Rev. Lett. 76, 2818 (1996)ADSCrossRefGoogle Scholar
  6. 6.
    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Generalized no-broadcasting theorem. Phys. Rev. Lett. 99, 240501 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    Kalev, A., Hen, I.: No-broadcasting theorem and its classical counterpart. Phys. Rev. Lett. 100, 210502 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Piani, M., Horodecki, P., Horodecki, R.: No-local-broadcasting theorem for multipartite quantum correlations. Phys. Rev. Lett. 100, 090502 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    Luo, S.L.: On quantum no-broadcasting. Lett. Math. Phys. 92, 143 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Luo, S.L., Sun, W.: Decomposition of bipartite states with applications to quantum no-broadcasting theorems. Phys. Rev. A 82, 012338 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Greenberger, D.M., Horne, M.A., Zeilinger, A.: Bell’s theorem without inequalities. In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe, pp. 69–72. Springer, Dordrecht (1989)Google Scholar
  14. 14.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gisin, N.: Bell’s inequality holds for all non-product states. Phys. Lett. A 154, 201 (1991)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Branciard, C., Rosset, D., Gisin, N., Pironio, S.: Bilocal versus nonbilocal correlations in entanglement-swapping experiments. Phys. Rev. A 85, 032119 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    Chaves, R.: Polynomial bell inequalities. Phys. Rev. Lett. 116, 010402 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Luo, M.: Computationally efficient nonlinear Bell inequalities for general quantum networks. Phys. Rev. Lett. 120, 140402 (2018)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lindblad, G.: A general no-cloning theorem. Lett. Math. Phys. 47, 189 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Cloning and broadcasting in generic probabilistic theories. arXiv:quant-ph/0611295 (2006)
  21. 21.
    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Generalized no-broadcasting theorem. Phys. Rev. Lett. 99, 240501 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    Kaniowski, K., Lubnauer, K., Łuczak, A.: Multicloning and multibroadcasting in operator algebras. Q. J. Math. 66, 191–192 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Stinespring, W.F.: Positive functions on \(C^*\)-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Choi, M.-D.: Completely positive maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kraus, K.: States, Effects and Operations: Fundamental Notions of Quantum Theory. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Miyadera, T., Imai, H.: No-cloning theorem on quantum logics. J. Math. Phys. 50, 1063 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Niestegge, G.: Non-classical conditional probability and the quantum no-cloning theorem. Phys. Scr. 90, 095101 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    D’Ariano, G.M., Perinotti, P.: Quantum no-stretching: a geometrical interpretation of the no-cloning theorem. Phys. Lett. A 373, 2416–2419 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Abramsky, S.: No-cloning in categorical quantum mechanics. In: Gay, S., Mackie, I. (eds.) Semantic Techniques in Quantum Computation, pp. 1–28. Cambridge University Press, Cambridge (2010)Google Scholar
  30. 30.
    Bennett, C.H., Brassard, G.: In: Proceedings IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, p. 175. IEEE, New York (1984)Google Scholar
  31. 31.
    Shor, P.W., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441 (2000)ADSCrossRefGoogle Scholar
  32. 32.
    Lo, H.-K., Chau, H.F.: Unconditional security of quantum key distribution over arbitrarily long distances. Science 283, 2050–2056 (1999)ADSCrossRefGoogle Scholar
  33. 33.
    Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using \(d\)-level systems. Phys. Rev. Lett. 88(12), 127902 (2002)ADSCrossRefGoogle Scholar
  34. 34.
    Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Dužek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81(3), 1301 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Towards practical and fast quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Pati, A.K., Braunstein, S.L.: Impossibility of deleting an unknown quantum state. Nature 404, 104 (2000)Google Scholar
  37. 37.
    Oszmaniec, M., Grudka, A., Horodecki, M., Wójcik, A.: Creation of superposition of unknown quantum states. Phys. Rev. Lett. 116, 110403 (2016)ADSCrossRefGoogle Scholar
  38. 38.
    Alvarez-Rodriguez, U., Sanz, M., Lamata, L., Solano, E.: The forbidden quantum adder. Sci. Rep. 5, 11983 (2015)ADSCrossRefGoogle Scholar
  39. 39.
    Luo, M.L., Li, H.R., Lai, H., Wang, X.: Unified quantum no-go theorems and transforming of quantum pure states in a restricted set. Quantum Inf. Process. 16, 297 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pati, A.K.: General impossible operations in quantum information. Phys. Rev. A 66, 062319 (2002)ADSCrossRefGoogle Scholar
  42. 42.
    Pati, A.K., Sanders, B.C.: No-partial erasure of quantum information. Phys. Lett. A 359, 31–36 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    Zhou, D.L., Zeng, B., You, L.: Quantum information cannot be split into complementary parts. Phys. Lett. A 352, 41 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Riesz, F.C.: Sur les operations fonctionelles lineaires. R. Acad. Sci. Paris. 149, 974–977 (1909)zbMATHGoogle Scholar
  45. 45.
    Duan, L.M., Guo, G.C.: A probabilistic cloning machine for replicating two non-orthogonal states. Phys. Lett. A 243, 261 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Duan, L.M., Guo, G.C.: Probabilistic cloning and identification of linearly independent quantum states. Phys. Rev. Lett. 80, 4999 (1998)ADSCrossRefGoogle Scholar
  47. 47.
    Chefles, A., Barnett, S.M.: Quantum state separation, unambiguous discrimination and exact cloning. J. Phys. A 31, 10097 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Chefles, A.: Unambiguous discrimination between linearly independent quantum states. Phys. Lett. A 239, 339–347 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Chefles, A., Jozsa, R., Winter, A.: On the existence of physical transformations between sets of quantum states. Int. J. Quantum Inf. 02, 11–21 (2004)CrossRefzbMATHGoogle Scholar
  50. 50.
    Braunstein, S.L., Pati, A.K.: Quantum information cannot be completely hidden in correlations: implications for the black-hole information paradox. Phys. Rev. Lett. 98, 080502 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Cleve, R., Gottesman, D., Lo, H.-K.: How to share a quantum secret? Phys. Rev. Lett. 83, 648 (1999)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Information Security and National Computing Grid LaboratorySouthwest Jiaotong UniversityChengduChina
  2. 2.School of Computer and Information ScienceSouthwest UniversityChongqingChina

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