Abstract
We analyze how measured quantum dynamical systems store and process information, introducing sofic quantum dynamical systems. Using recently introduced information-theoretic measures for quantum processes, we quantify their information storage and processing in terms of entropy rate and excess entropy, giving closed-form expressions where possible. To illustrate the impact of measurement on information storage in quantum processes, we analyze two spin-1 sofic quantum systems that differ only in how they are measured.
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References
Kitchens, B.P.: Symbolic dynamics: one-sides, two-sided, and countable state Markov shifts. Springer, Heidelberg (1998)
Hedlund, G.H., Morse, M.: Symbolic dynamics i. Amer. J. Math. 60, 815–866 (1938)
Hedlund, G.H., Morse, M.: Symbolic dynamics ii. Amer. J. Math. 62, 1–42 (1940)
Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Champaign-Urbana (1962)
Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995)
Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, Heidelberg (1990)
Reichl, L.E.: The transition to chaos: Conservative classical systems and quantum manifestations. Springer, New York (2004)
Habib, S., Jacobs, K., Shizume, K.: Emergence of chaos in quantum systems far from the classical limit. Phys. Rev. Lett. 96, 10403–10406 (2006)
Alicki, R., Fannes, M.: Quantum dynamical systems. Oxford University Press, Oxford (2001)
Beck, C., Graudenz, D.: Symbolic dynamics of successive quantum-mechanical measurements. Phys. Rev. A 46(10), 6265–6276 (1992)
Wiesner, K., Crutchfield, J.P.: Computation in finitary quantum processes. e-print arxiv/quant-ph/0608206, 2006 (submitted)
Cover, T., Thomas, J.: Elements of Information Theory. Wiley-Interscience, Chichester (1991)
Albert, D.Z.: On quantum-mechanical automata. Physics Letters 98A, 249–251 (1983)
Peres, A.: On quantum-mechanical automata. Physics Letters 101A, 249–250 (1984)
Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theor. Comp. Sci. 237, 275–306 (2000)
Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: 38th IEEE Conference on Foundations of Computer Science, pp. 66–75. IEEE Computer Society Press, Los Alamitos (1997)
Aharonov, D., Kitaev, A., Nisan, N.: Quantum circiuts with mixed states. In: 30th Annual ACM Symposium on the Theory of Computing, pp. 20–30. ACM Press, New York (1998)
Ambainis, A., Watrous, J.: Two-way finite automata with quantum and classical states. Theoretical Computer Science 287, 299–311 (2002)
Freivalds, R., Winter, A.: Quantum finite state transducers. Lect. Notes Comp. Sci. vol. 2234, pp. 233–242 (2001)
Peres, A.: Quantum theory: concepts and methods. Kluwer Academic Publishers, Dordrecht (1993)
Hirsch, M., Palis, J., Pugh, C., Shu, M.: Neighborhoods of hyperbolic sets. Inventiones Math. 9, 121–134 (1970)
Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (2001)
Bishop, C.M.: Pattern recognition and machine learning. Springer Verlag, Singapore (2006)
Weiss, B.: Subshifts of finite type and sofic systems. Monatshefte für Mathematik 77, 462–474 (1973)
Crutchfield, J.P., Feldman, D.P.: Regularities unseen, randomness observed: Levels of entropy convergence. Chaos 13, 25–54 (2003)
Crutchfield, J.P., Wiesner, K.: Intrinsic quantum computation. , e-print arxiv/quant-ph/0611202, 2006 (submitted)
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Wiesner, K., Crutchfield, J.P. (2007). Computation in Sofic Quantum Dynamical Systems. In: Akl, S.G., Calude, C.S., Dinneen, M.J., Rozenberg, G., Wareham, H.T. (eds) Unconventional Computation. UC 2007. Lecture Notes in Computer Science, vol 4618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73554-0_20
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DOI: https://doi.org/10.1007/978-3-540-73554-0_20
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