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arXiv:quant-ph/0203019

Unpredictability of wave function's evolution in nonintegrable quantum systems

Ivan B. Ivanov

Published Mar 5, 2002• 1 citation
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Abstract

It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations $\mathcal O(T)\sim T^α$. On the other hand, the evolution of wave functions in integrable systems can be predicted by the fast algorithms $\mathcal O(T)\sim (log_2 T)^β$ for logarithmically short time and thus there is an algorithmic "compressibility" of their dynamics. The difference between integrable and nonintegrable systems in our approach looks identically for classical and quantum systems. Therefore the minimal number of bit operations $\mathcal O(T)$ needed to predict a state of system for time interval T can be used as universal sign of chaos.

DOI
10.48550/arxiv.quant-ph/0203019

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It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations O(T)∼Tα\mathcal O(T)\sim T^αO(T)∼Tα. On the other hand, the evolution of wave functions in integrable systems can be predicted by the fast algorithms O(T)∼(log2T)β\mathcal O(T)\sim (log_2 T)^βO(T)∼(log2​T)β for logarithmically short time and thus there is an algorithmic "compressibility" of their dynamics. The difference between integrable and nonintegrable systems in our approach looks identically for classical and quantum systems. Therefore the minimal number of bit operations O(T)\mathcal O(T)O(T) needed to predict a state of system for time interval T can be used as universal sign of chaos.
A concept of computational complexity is applied for analysis of classical and quantum dynamical systems. It is argued that evolution of wave functions in nonintegrable quatnum systems lies in complexity class EXP because of rapid growth of number of elementary computational operations needed to predict their future. On the other hand, evolution of wave functions in integrable systems can be predicted by the fast algorithms and thus it belongs to P class. This difference between integrable and nonintegrable systems in our approahc looks identically for classical and quantum systems. In this paper an informational approach is applied for analysis of dynamics in classical and quantum systems to find a universal different between integrable and nonintegrable motion. As a basic tool to analyze compleixty of dynamics we use a number of elementary computational operations O(T) (computational complexity) needed to determine a state of a sytem for time interval T.
A novel concept for quantum chaos in spin systems is proposed which differs distinctively from concepts currently in use. It is argued (a) that the definition of quantum chaos cannot be based on the correspondence principle and (b) that quantum chaos is not equivalent to statistical behavior. Quantum chaos in spin systems implies the existence of a new type of spectrum consisting of excitations which do not form regular patterns with a multiparameter continuum structure in the thermodynamic limit, thus removing the constraint imposed by van Hove singularities that time-dependent correlation functions for pure quantum states cannot decay more rapidly than as powers of t. The irregular spectrum is caused by strong level repulsion resulting from the lack of a sufficient number of conservation laws in nonintegrable quantum many-body systems. The similarities and differences between quantum and classical chaos and, more generally, between quantum (non)integrability and classical (non)integrability are discussed in detail.
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Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.