Abstract
Quantum computation has suggested new forms of quantum logic, called quantum computational logics ([2]). The basic semantic idea is the following: the meaning of a sentence is identified with a quregister, a system of qubits, representing a possible pure state of a compound quantum system. The generalization to mixed states, which might be useful to analyse entanglement-phenomena, is due to Gudder ([7]). Quantum computational logics represent non standard examples of unsharp quantum logic, where the non-contradiction principle is violated, while conjunctions and disjunctions are strongly non-idempotent. In this framework, any sentence a of the language gives rise to a quantum tree: a kind of quantum circuit that transforms the quregister associated to the atomic subforrnulas of α into the quregister associated to α.
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
Birkhoff, G., and J. Von Neumann, The logic of quantum mechanics, Annals of Mathematics 37 (1936), 823–843.
Cattaneo, G., M. L. Dalla Chiara, R. Giuntiniand R. Leporini, An unsharp logic from quantum computation,e-print: http://arxiv.org/abs/quant-ph/0201013.
Cattaneo, G., M. L. Dalla Chiara, R. Giuntini and R. Leporini, Quantum computational structures,preprint.
Dalla chiara, M. L., and R. Giuntini, Quantum logics, in G. Gabbayand F. Guenthner (eds.), Handbook of Philosophical Logic, vol. VI, Kluwer, Dordrecht, 2002, pp. 129–228.
Dalla Chiara, M. L., R. Giuntini, A. Leporati and R. Leporini, Qubit semantics and quantum trees,e-print:http://arxiv.org /abs/quant-ph/0211190.
Deutsch, D., A. Ekert, and R. Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic, 3, 2000, pp. 265 283.
Gudder, S., Quantum computational logic,preprint.
Petri, C. A., Gründsatzliches zur Beschreibung diskreter Prozesse,in Proceedings of the 3`d Colloquium über Automatentheorie (Hannover, 1965), Birkhäuser Verlag, Basel, 1967, pp. 121–140. English version: Fundamentals of the Representation of Discrete Processes,ISF Report 82.04 (1982), translated by H. J. Genrich and P. S. Thiagarajan.
Toffoli, T., Reversible Computing, in J. W. DE Bakker, J. Van Leeuwen (eds.), Automata, Languages and Programming, Springer, 1980, pp. 632–644. Also available as TechnicalMemo MIT/LCS/TM-151, MIT Laboratory for Computer Science, February 1980.
Zawirski, Z., Relation of many—valued logic to probability calculus, (in Polish, original title: Stosunek logiki wielowartosciowej do rachunku prawdopodobienstwa), Poznanskie Towarzystwo Przyjacidl Nauk, 1934.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Chiara, M.L.D., Giuntini, R., Leporini, R. (2003). Quantum Computational Logics: A Survey. In: Hendricks, V.F., Malinowski, J. (eds) Trends in Logic. Trends in Logic, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3598-8_9
Download citation
DOI: https://doi.org/10.1007/978-94-017-3598-8_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6414-1
Online ISBN: 978-94-017-3598-8
eBook Packages: Springer Book Archive