Quantum Probabilistic Dyadic Second-Order Logic

  • Alexandru Baltag
  • Jort M. Bergfeld
  • Kohei Kishida
  • Joshua Sack
  • Sonja J. L. Smets
  • Shengyang Zhong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8071)


We propose an expressive but decidable logic for reasoning about quantum systems. The logic is endowed with tensor operators to capture properties of composite systems, and with probabilistic predication formulas P  ≥ r (s), saying that a quantum system in state s will yield the answer ‘yes’ (i.e. it will collapse to a state satisfying property P) with a probability at least r whenever a binary measurement of property P is performed. Besides first-order quantifiers ranging over quantum states, we have two second-order quantifiers, one ranging over quantum-testable properties, the other over quantum “actions”. We use this formalism to express the correctness of some quantum programs. We prove decidability, via translation into the first-order logic of real numbers.


Quantum System Function Symbol Quantum Logic Predicate Symbol Unary Predicate 
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.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th IEEE Conference on Logic in Computer Science (LiCS 2004), pp. 415–425. IEEE Press (2004)Google Scholar
  2. 2.
    Aerts, D.: Description of compound physical systems and logical interaction of physical systems. In: Beltrametti, E., van Fraassen, B. (eds.) Current Issues on Quantum Logic, pp. 381–405. Kluwer Academic (1981)Google Scholar
  3. 3.
    Baltag, A., Bergfeld, J., Kishida, K., Sack, J., Smets, S., Zhong, S.: PLQP & company: Decidable logics for quantum algorithms. Submitted to the International Journal of Theoretical Physics (2013)Google Scholar
  4. 4.
    Baltag, A., Bergfeld, J., Kishida, K., Sack, J., Smets, S., Zhong, S.: A Decidable Dynamic Logic for Quantum Reasoning. In: EPTCS (2012) (in print)Google Scholar
  5. 5.
    Baltag, A., Smets, S.: Complete Axiomatizations for Quantum Actions. International Journal of Theoretical Physics 44(12), 2267–2282 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Baltag, A., Smets, S.: LQP: The Dynamic Logic of Quantum Information. Mathematical Structures in Computer Science 16(3), 491–525 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Birkhoff, G., von Neumann, J.: The Logic of Quantum Mechanics. The Annals of Mathematics 37, 823–843 (1936)CrossRefGoogle Scholar
  8. 8.
    Chadha, R., Mateus, P., Sernadas, A., Sernadas, C.: Extending classical logic for reasoning about quantum systems. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Logic, pp. 325–371. Elsevier (2009)Google Scholar
  9. 9.
    Dalla Chiara, M.L., Giuntini, R., Greechie, R.: Reasoning in quantum theory: sharp and unsharp quantum logics. Trends in logic, vol. 22. Kluwer Acadamic Press, Dordrecht (2004)CrossRefGoogle Scholar
  10. 10.
    Dunn, J.M., Hagge, T.J., Moss, L.S., Wang, Z.: Quantum Logic as Motivated by Quantum Computing. The Journal of Symbolic Logic 70(2), 353–359 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Henkin, L.: Completeness in the Theory of Types. The Journal of Symbolic Logic 15, 81–91 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)Google Scholar
  13. 13.
    Piron, C.: Foundations of Quantum Physics. W.A. Benjamin Inc. (1976)Google Scholar
  14. 14.
    Rabin, M.: Decidability of second order theories and automata on infinite trees. Transactions of the American Mathematical Society, 1–35 (1969)Google Scholar
  15. 15.
    Randall, C., Foulis, D.: Tensor products of quantum logics do not exist. Notices Amer. Math. Soc. 26(6) (1979)Google Scholar
  16. 16.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14, 527–586 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. RAND Corporation, Santa Monica (1948)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexandru Baltag
    • 1
  • Jort M. Bergfeld
    • 1
  • Kohei Kishida
    • 1
  • Joshua Sack
    • 1
  • Sonja J. L. Smets
    • 1
  • Shengyang Zhong
    • 1
  1. 1.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands

Personalised recommendations