  We study the ergodicity and mixing of quantum kicked rotor (QKR) with two
distinct approaches. In one approach, we use the definitions of quantum
ergodicity and mixing recently proposed in [Phys. Rev. E 94, 022150 (2016)],
which involve only eigen-energies (Floquet quasi-energies for QKR). In the
other approach, we study ergodicity and mixing with quantum Poincar\`e section,
which is plotted with a method that maps a wave function unitarily onto quantum
phase space composed of Planck cells. Classical Poincar\`e section can be
recovered with the effective Planck constant gradually diminishing. We
demonstrate that the two approaches can capture the quantum and classical
characteristics of ergodicity and mixing of QKR, and give consistent results
with classical model at semiclassical limit. Therefore, we establish a
correspondence between quantum ergodicity (mixing) and classical ergodicity
(mixing).
