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International Journal of Theoretical Physics

, Volume 50, Issue 2, pp 571–580 | Cite as

Product Łukasiewicz Quantum Logic

  • Cesarino Bertini
  • Roberto Leporini
Article

Abstract

The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a density operator (called qumix). In this framework, any sentence α of the language gives rise to a quantum circuit that transforms the qumix associated to the atomic subformulas of α into the qumix associated to α. In this paper we enrich the language by adding a new connective which expresses truncated sum.

Keywords

Quantum computation Quantum logic 

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References

  1. 1.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936) CrossRefGoogle Scholar
  2. 2.
    Dalla Chiara, M.L., Giuntini, R.: Quantum logics. In: Gabbay, G., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. VI, pp. 129–228. Kluwer, Dordrecht (2002) Google Scholar
  3. 3.
    Dalla Chiara, M.L., Giuntini, R., Leporini, R.: Logics from quantum computation. Int. J. Quantum Inf. 3, 293–337 (2005) MATHCrossRefGoogle Scholar
  4. 4.
    Dalla Chiara, M.L., Giuntini, R., Leporini, R.: Reversibility and irreversibility in quantum computation and in quantum computational logics. In: Algebraic and Proof-Theoretic Aspects of Non-classical Logics. LNAI, vol. 4460, pp. 84–106. Springer, Berlin (2006) CrossRefGoogle Scholar
  5. 5.
    Gudder, S.: Quantum computational logic. Int. J. Theor. Phys. 42, 39–47 (2003) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Petri, C.A.: Grundsätzliches zur Beschreibung diskreter Prozesse. In: Proceedings of the 3rd Colloquium über Automatentheorie, Hannover, 1965, pp. 121–140. Birkhäuser, Basel (1967). English version: Fundamentals of the Representation of Discrete Processes. ISF Report 82.04 (1982). Translated by H.J. Genrich and P.S. Thiagarajan Google Scholar
  7. 7.
    Toffoli, T.: Reversible computing. In: de Bakker, J.W., van Leeuwen, J. (eds.) Automata, Languages and Programming, pp. 632–644. Springer, Berlin (1980). Also available as TechnicalMemo MIT/LCS/TM-151, MIT Laboratory for Computer Science, February (1980) CrossRefGoogle Scholar
  8. 8.
    Zawirski, Z.: Relation of many-valued logic to probability calculus (in Polish, original title: Stosunek logiki wielowartościowej do rachunku prawdopodobieństwa), Poznańskie Towarzystwo Przyjaciół Nauk (1934) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica, Statistica, Informatica e ApplicazioniUniversità di BergamoBergamoItaly

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