We introduce a simplified version of Connes-Narnhofer-Thirring’s quantum dynamical entropy for quantum systems. It quantifies the amount of information gained about the initial condition from continuously monitoring an observable. A nonzero entropy growth rate can be obtained by monitoring the thermal fluctuation of an extensive observable in a generic many-body system, away from classical or large $N$ limits. We explicitly compute the entropy rate in the thermodynamic and long-time limit in terms of the two-point correlation functions. We conjecture a universal Planckian bound for the entropy rate. Related results on the purification rate are also obtained.

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• Entanglement entropy
• Quantum chaos
• Quantum statistical mechanics
• Quantum many-body systems