Abstract
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.
Highlights
Several proposals have been put forward with the aim of extending the dynamical entropy of Kolmogorov and Sinai (KS-entropy) [1] to the quantum realm [2,3,4,5,6]
While the KS-entropy is related to the rate at which information is produced by the dynamics with respect to an equilibrium state, algorithmic complexity theory has been developed by Kolmogorov, Entropy 2012, 14
Probability theory cannot sort out regular from random binary strings; for instance, in the case of a fair coin tossing, all strings of N zeroes and ones have the same probability 2−N. Algorithmic complexity measures their randomness based upon the difficulty of description by means of programs that run by Universal Turing Machines (UTM) to reproduce the target string
Summary
Several proposals have been put forward with the aim of extending the dynamical entropy of Kolmogorov and Sinai (KS-entropy) [1] to the quantum realm [2,3,4,5,6]. Since in quantum mechanics there are neither phase-space nor trajectories and, observations perturb the observed system, many different non-commutative extensions can be envisaged, all of them reducing to the KS-invariant in the case of classical, that is, commutative systems. While the KS-entropy is related to the rate at which information is produced by the dynamics with respect to an equilibrium state, algorithmic complexity theory has been developed by Kolmogorov, Entropy 2012, 14.
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