  We prove the second law of thermodynamics and the nonequilibirum fluctuation
theorem for pure quantum states.The entire system obeys reversible unitary
dynamics, where the initial state of the heat bath is not the canonical
distribution but is a single energy-eigenstate that satisfies the
eigenstate-thermalization hypothesis (ETH). Our result is mathematically
rigorous and based on the Lieb-Robinson bound, which gives the upper bound of
the velocity of information propagation in many-body quantum systems. The
entanglement entropy of a subsystem is shown connected to thermodynamic heat,
highlighting the foundation of the information-thermodynamics link. We
confirmed our theory by numerical simulation of hard-core bosons, and observed
dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our
result reveals a universal scenario that the second law emerges from quantum
mechanics, and can experimentally be tested by artificial isolated quantum
systems such as ultracold atoms.
