  We show that there is genuine quantum chaos despite that quantum dynamics is
linear. This is revealed by introducing a physical distance between two quantum
states. Qualitatively different from existing distances for quantum states, for
example, the Fubini-Study distance, the physical distance between two mutually
orthogonal quantum states can be very small. As a result, two quantum states,
which are initially very close by physical distance, can diverge from each
other during the ensuing quantum dynamical evolution. We are able to use
physical distance to define quantum Lyaponov exponent and quantum chaos
measure. The latter leads to quantum analogue of the classical Poincar\'e
section, which maps out the regions where quantum dynamics is regular and the
regions where quantum dynamics is chaotic. Three different systems, kicked
rotor, three-site Bose-Hubbard model, and spin-1/2 XXZ model, are used to
illustrate our results.
