  It is a fundamental problem how the universal concept of classical chaos
emerges from the microscopic description of quantum mechanics. We here study
standard classical chaos in a framework of quantum mechanics. In particular, we
design a quantum lattice system that exactly simulates classical chaos after an
appropriate continuum limit, which is called the "Hamiltonian equation limit".
The key concept of our analysis is an entanglement entropy defined by dividing
the lattice into many blocks of equal size and tracing out the degrees of
freedom within each block. We refer to this entropy as the "interscale
entanglement entropy" because it measures the amount of entanglement between
the microscopic degrees of freedom within each block and the macroscopic
degrees of freedom that define the large-scale structure of the wavefunction.
By numerically simulating a quantum lattice system corresponding to the
Hamiltonian of the kicked rotor, we find that the long-time average of the
interscale entanglement entropy becomes positive only when chaos emerges in the
Hamiltonian equation limit, and the growth rate of the entropy in the initial
stage is proportional to that of the coarse-grained Gibbs entropy of the
corresponding classical system.
