Abstract
Quantum computation deals with projective measurements and unitary transformations in finite dimensional Hilbert spaces. The paper presents a propositional logic designed to describe quantum computation at an operational level by supporting reasoning about the probabilities associated to such measurements: measurement probabilities, and transition probabilities (a quantum analogue of conditional probabilities). We present two axiomatizations, one for the logic as a whole and one for the fragment dealing just with measurement probabilities. These axiomatizations are proved to be sound and complete. The logic is also shown to be decidable, and we provide results characterizing its complexity in a number of cases.
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References
Abadi, M., Halpern, J.Y.: Decidability and expressiveness for first-order logics of probability. Information and Computation 112(1), 1–36 (1994)
Bacchus, F.: Representing and Reasoning with Probabilistic Knowledge. MIT Press, Cambridge (1990)
Ben-Or, M., Kozen, D., Reif, J.H.: The complexity of elementary algebra and geometry. Journal of Computer and System Sciences 32(1), 251–264 (1986)
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1936)
Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: Proc. 20th ACM Symp. on Theory of Computing, pp. 460–467 (1988)
Carnap, R.: Logical Foundations of Probability. University of Chicago Press, Chicago (1950)
Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Information and Computation 87(1/2), 78–128 (1990)
Gudder, S.: Quantum Probability Theory. Academic Press, San Diego (1989)
Nilsson, N.: Probabilistic logic. Artificial Intelligence 28, 71–87 (1986)
Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, Dordrecht (1995)
Pratt, V.R.: Linear logic for generalized quantum mechanics. In: Proc. Of Workshop on Physics and Computation (PhysComp 1992), Dallas, pp. 166–180. IEEE, Los Alamitos (1992)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. Univ. of California Press, Berkeley (1951)
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© 2003 Springer-Verlag Berlin Heidelberg
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van der Meyden, R., Patra, M. (2003). A Logic for Probability in Quantum Systems. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_34
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DOI: https://doi.org/10.1007/978-3-540-45220-1_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40801-7
Online ISBN: 978-3-540-45220-1
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