Computation in Sofic Quantum Dynamical Systems

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.

[1]  James P. Crutchfield,et al.  Quantum automata and quantum grammars , 2000, Theor. Comput. Sci..

[2]  B. Weiss.  Subshifts of finite type and sofic systems , 1973 .

[3]  James P. Crutchfield,et al.  Computation in Finitary Stochastic and Quantum Processes , 2006 .

[4]  Linda E Reichl,et al.  The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations , 2004 .

[5]  Rusins Freivalds,et al.  Quantum Finite State Transducers , 2001, SOFSEM.

[6]  J. Crutchfield,et al.  Regularities unseen, randomness observed: levels of entropy convergence. , 2001, Chaos.

[7]  C. Beck,et al.  Symbolic dynamics of successive quantum-mechanical measurements. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[8]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[9]  John Watrous,et al.  On the power of quantum finite state automata , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[10]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[11]  James P. Crutchfield,et al.  Computation in Finitary Quantum Processes , 2006 .

[12]  Gustav A. Hedlund.  A New Proof for a Metrically Transitive System , 1940 .

[13]  B. Kitchens.  Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts , 1997 .

[14]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[15]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[16]  Noam Nisan,et al.  Quantum circuits with mixed states , 1998, STOC '98.

[17]  L. Ballentine,et al.  Quantum Theory: Concepts and Methods , 1994 .

[18]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[19]  Symbolic dynamics , 2008, Scholarpedia.

[20]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .

[21]  Mark Fannes,et al.  Quantum Dynamical Systems , 2001 .

[22]  David Z. Albert,et al.  On quantum-mechanical automata , 1983 .

[23]  Kurt Jacobs,et al.  Emergence of chaos in quantum systems far from the classical limit. , 2006, Physical review letters.

[24]  Andris Ambainis,et al.  Two-way finite automata with quantum and classical state , 1999, Theor. Comput. Sci..

[25]  Christopher M. Bishop,et al.  Pattern Recognition and Machine Learning (Information Science and Statistics) , 2006 .

[26]  James P. Crutchfield,et al.  Intrinsic Quantum Computation , 2008 .

[27]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[28]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[29]  Asher Peres,et al.  On quantum-mechanical automata , 1984 .

[30]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[31]  Michael Shub,et al.  Neighborhoods of hyperbolic sets , 1970 .