We investigate the bound on the speed of state transformation in the quantum systems that are coupled to general environments. We focus on the energy fluctuation of the target system and the coupling to the environment, and clarify the universal constraint on the state transformation deriving the general speed-limit inequality. This speed-limit inequality is applicable to any open systems including physical situations that have not been studied so far. We apply the inequality to the thermalization phenomena in isolated quantum systems, and the quantum cyclic engines. For the thermalization, we provide several inequalities to estimate the equilibration time. In the cyclic heat engines, we derive a rigorous constraint on the state change in the cycle where the Planck constant plays a fundamental role.
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