Optimal Counterfeiting Attacks and Generalizations for Wiesner’s Quantum Money

  • Abel Molina
  • Thomas Vidick
  • John Watrous
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7582)


We present an analysis of Wiesner’s quantum money scheme, as well as some natural generalizations of it, based on semidefinite programming. For Wiesner’s original scheme, it is determined that the optimal probability for a counterfeiter to create two copies of a bank note from one, where both copies pass the bank’s test for validity, is (3/4) n for n being the number of qubits used for each note. Generalizations in which other ensembles of states are substituted for the one considered by Wiesner are also discussed, including a scheme recently proposed by Pastawski, Yao, Jiang, Lukin, and Cirac, as well as schemes based on higher dimensional quantum systems. In addition, we introduce a variant of Wiesner’s quantum money in which the verification protocol for bank notes involves only classical communication with the bank. We show that the optimal probability with which a counterfeiter can succeed in two independent verification attempts, given access to a single valid n-qubit bank note, is \((3/4+\sqrt{2}/8)^n\). We also analyze extensions of this variant to higher-dimensional schemes.


Dual Problem Success Probability Hermitian Operator Quantum Cloning Counterfeit Attack 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aaronson, S.: Quantum copy-protection and quantum money. In: Proceedings of the 24th Annual IEEE Conference on Computational Complexity, pp. 229–242 (2009)Google Scholar
  2. 2.
    Aaronson, S.: On the security of private-key quantum money (in preparation, 2012)Google Scholar
  3. 3.
    Aaronson, S., Christiano, P.: Quantum money from hidden subspaces. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (2012)Google Scholar
  4. 4.
    Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: Proceedings of the 22nd Annual IEEE Conference on Computational Complexity, pp. 129–140 (2007)Google Scholar
  5. 5.
    Audenaert, K., De Moor, B.: Optimizing completely positive maps using semidefinite programming. Physical Review A 65, 30302 (2002)CrossRefGoogle Scholar
  6. 6.
    Bennett, C., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, pp. 175–179 (1984)Google Scholar
  7. 7.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)Google Scholar
  8. 8.
    Bruß, D., Cinchetti, M., D’Ariano, G., Macchiavello, C.: Phase covariant quantum cloning. Physical Review A 62, 012302 (2000)CrossRefGoogle Scholar
  9. 9.
    Bužek, V., Hillery, M.: Quantum copying: Beyond the no-cloning theorem. Physical Review A 54(3), 1844–1852 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cerf, N., Fiurášek, J.: Optical quantum cloning. Progress in Optics, ch.6, vol. 49, pp. 455–545. Elsevier (2006)Google Scholar
  11. 11.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra and Its Applications 10(3), 285–290 (1975)zbMATHCrossRefGoogle Scholar
  12. 12.
    de Klerk, E.: Aspects of Semidefinite Programming – Interior Point Algorithms and Selected Applications. Applied Optimization, vol. 65. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  13. 13.
    Eggeling, T., Werner, R.: Separability properties of tripartite states with U ⊗ U ⊗ U symmetry. Physical Review A 63(4), 042111 (2001)Google Scholar
  14. 14.
    Eldar, Y., Megretski, A., Verghese, G.: Designing optimal quantum detectors via semidefinite programming. IEEE Transactions on Information Theory 49(4), 1007–1012 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Farhi, E., Gosset, D., Hassidim, A., Lutomirski, A., Shor, P.: Quantum money from knots. Available as e-Print 1004.5127 (2010)Google Scholar
  16. 16.
    Gavinsky, D.: Quantum money with classical verification. Available as e-Print 1109.0372 (2011)Google Scholar
  17. 17.
    Gottesman, D.: Uncloneable encryption. Available as e-Print quant-ph/0210062 (2002)Google Scholar
  18. 18.
    Gutoski, G., Watrous, J.: Toward a general theory of quantum games. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 565–574 (2007)Google Scholar
  19. 19.
    Jamiołkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics 3(4), 275–278 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lo, H., Spiller, T., Popescu, S.: Introduction to Quantum Computation and Information. World Scientific Publishing Company (1998)Google Scholar
  21. 21.
    Lovász, L.: Semidefinite programs and combinatorial optimization. Recent Advances in Algorithms and Combinatorics (2003)Google Scholar
  22. 22.
    Lutomirski, A.: An online attack against Wiesner’s quantum money. Available as e-Print 1010.0256 (2010)Google Scholar
  23. 23.
    Lutomirski, A., Aaronson, S., Farhi, E., Gosset, D., Hassidim, A., Kelner, J., Shor, P.: Breaking and making quantum money: toward a new quantum cryptographic protocol. In: Proceedings of Innovations in Computer Science (ICS), pp. 20–31 (2010)Google Scholar
  24. 24.
    Mayers, D.: Unconditional security in quantum cryptography. Journal of the ACM 48, 351–406 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mittal, R., Szegedy, M.: Product Rules in Semidefinite Programming. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) FCT 2007. LNCS, vol. 4639, pp. 435–445. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  26. 26.
    Molina, A., Watrous, J.: Hedging bets with correlated quantum strategies. Available as e-Print 1104.1140 (2011)Google Scholar
  27. 27.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)Google Scholar
  28. 28.
    Pastawski, F., Yao, N.Y., Jiang, L., Lukin, M.D., Cirac, J.I.: Unforgeable noise-tolerant quantum tokens. Available as e-Print 1112.5456 (2011)Google Scholar
  29. 29.
    Renes, J., Blume-Kohout, R., Scott, A., Caves, C.: Symmetric informationally complete quantum measurements. Journal of Mathematical Physics 45, 2171–2180 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Shor, P., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Physical Review Letters 85(2), 441–444 (2000)CrossRefGoogle Scholar
  31. 31.
    Tomamichel, M., Renner, R.: Uncertainty relation for smooth entropies. Physical Review Letters 106, 110506 (2011)CrossRefGoogle Scholar
  32. 32.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Vidick, T., Wehner, S.: Does ignorance of the whole imply ignorance of the parts? Large violations of noncontextuality in quantum theory. Physical Review Letters 107, 030402 (2011)CrossRefGoogle Scholar
  34. 34.
    Watrous, J.: Lecture notes on Theory of Quantum Information (2011),
  35. 35.
    Werner, R.: Optimal cloning of pure states. Physical Review A 58, 1827–1832 (1998)CrossRefGoogle Scholar
  36. 36.
    Wiesner, S.: Conjugate coding. SIGACT News 15(1), 78–88 (1983)CrossRefGoogle Scholar
  37. 37.
    Wootters, W., Zurek, W.: A single quantum state cannot be cloned. Nature 299, 802–803 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Abel Molina
    • 1
  • Thomas Vidick
    • 2
  • John Watrous
    • 1
  1. 1.Institute for Quantum Computing and School of Computer ScienceUniversity of WaterlooCanada
  2. 2.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyUSA

Personalised recommendations