Abstract
A new logic is proposed for reasoning about quantum systems. The logic embodies the postulates of quantum physics and it was designed from the semantics upwards by identifying quantum models with superpositions of classical models. This novel approach to quantum logic is completely different from the traditional approach of Birkhoff and von Neumann. It has the advantage of making quantum logic an extension of classical logic. The key new ingredient of the language of the proposed logic is a rather general modal operator. The logic incorporates probabilistic reasoning (in the style of Nilsson) in order to deal with uncertainty on the outcome of measurements. The logic also incorporates dynamic reasoning (in the style of Hoare) in order to cope with the evolution of quantum systems. A Hilbert calculus for the logic is sketched. A quantum key distribution protocol is specified and analyzed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Foulis, D.J.: A half-century of quantum logic. What have we learned? In: Quantum Structures and the Nature of Reality. Einstein Meets Magritte, vol. 7, pp. 1–36. Kluwer Acad. Publ., Dordrecht (1999)
Chiara, M.L.D., Giuntini, R., Greechie, R.: Reasoning in Quantum Theory. Kluwer Academic Publishers, Dordrecht (2004)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1936)
Kripke, S.A.: Semantical analysis of modal logic. I. Normal modal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9, 67–96 (1963)
Carnielli, W.A., Lima-Marques, M.: Society semantics and multiple-valued logics. In: Advances in Contemporary Logic and Computer Science, Salvador, 1996. Contemporary Mathematics, vol. 235, pp. 33–52. AMS (1999)
Carnielli, W.A.: Possible-translations semantics for paraconsistent logics. In: Frontiers of Paraconsistent Logic, Ghent, 1997. Studies in Logic and Computation, vol. 8, pp. 149–163. Research Studies Press, Hertfordshire (2000)
Nilsson, N.J.: Probabilistic logic. Artificial Intelligence 28, 71–87 (1986)
Nilsson, N.J.: Probabilistic logic revisited. Artificial Intelligence 59, 39–42 (1993)
Bacchus, F.: Representing and Reasoning with Probabilistic Knowledge. MIT Press Series in Artificial Intelligence. MIT Press, Cambridge (1990)
Bacchus, F.: On probability distributions over possible worlds. In: Uncertainty in Artificial Intelligence, 4. Machine Intelligence and Pattern Recognition, vol. 9, pp. 217–226. North-Holland, Amsterdam (1990)
Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Information and Computation 87, 78–128 (1990)
Dishkant, H.: Semantics of the minimal logic of quantum mechanics. Studia Logica 30, 23–32 (1972)
Hoare, C.: An axiomatic basis for computer programming. Communications of the ACM 12, 576–583 (1969)
DiVincenzo, D.P.: Two-bit gates are universal for quantum computation. Physics Reviews A 51, 1015–1022 (1995)
Naur, P.: Revised report on the algorithmic language Algol 60. The Computer Journal 5, 349–367 (1963)
Bennett, C.H., 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. IEEE, Los Alamitos (1984)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mateus, P., Sernadas, A. (2004). Reasoning About Quantum Systems. In: Alferes, J.J., Leite, J. (eds) Logics in Artificial Intelligence. JELIA 2004. Lecture Notes in Computer Science(), vol 3229. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30227-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-540-30227-8_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23242-1
Online ISBN: 978-3-540-30227-8
eBook Packages: Springer Book Archive