title: Entropy for quantum pure states and quantum H theorem.
abstract: We construct a complete set of Wannier functions that are localized at both given positions and momenta. This allows us to introduce the quantum phase space, onto which a quantum pure state can be mapped unitarily. Using its probability distribution in quantum phase space, we define an entropy for a quantum pure state. We prove an inequality regarding the long-time behavior of our entropy's fluctuation. For a typical initial state, this inequality indicates that our entropy can relax dynamically to a maximized value and stay there most of time with small fluctuations. This result echoes the quantum H theorem proved by von Neumann [Zeitschrift für Physik 57, 30 (1929)]. Our entropy is different from the standard von Neumann entropy, which is always zero for quantum pure states. According to our definition, a system always has bigger entropy than its subsystem even when the system is described by a pure state. As the construction of the Wannier basis can be implemented numerically, the dynamical evolution of our entropy is illustrated with an example.
authors.affiliation: International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
authors.forename: Xizhi
authors.initials: X
authors.lastname: Han
authors.affiliation: International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.|Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.|Wilczek Quantum Center, College of Science, Zhejiang University of Technology, Hangzhou 310014, China.
authors.forename: Biao
authors.initials: B
authors.lastname: Wu