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Entropy 2012, 14(7), 1259-1273; doi:10.3390/e14071259

Quantum Dynamical Entropies and Gács Algorithmic Entropy

1 Department of Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy 2 INFN, Trieste, Strada Costiera 11, I-34151 Trieste, Italy
Received: 13 April 2012 / Revised: 8 June 2012 / Accepted: 3 July 2012 / Published: 12 July 2012
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
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Several quantum dynamical entropies have been proposed that extend the classical Kolmogorov–Sinai (dynamical) entropy. The same scenario appears in relation to the extension of algorithmic complexity theory to the quantum realm. A theorem of Brudno establishes that the complexity per unit time step along typical trajectories of a classical ergodic system equals the KS-entropy. In the following, we establish a similar relation between the Connes–Narnhofer–Thirring quantum dynamical entropy for the shift on quantum spin chains and the Gács algorithmic entropy. We further provide, for the same system, a weaker linkage between the latter algorithmic complexity and a different quantum dynamical entropy proposed by Alicki and Fannes.
Keywords: quantum spin chains; algorithmic complexity; dynamical entropy quantum spin chains; algorithmic complexity; dynamical entropy
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Benatti, F. Quantum Dynamical Entropies and Gács Algorithmic Entropy. Entropy 2012, 14, 1259-1273.

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