Interscale entanglement production in a quantum system simulating classical chaos

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 (IEE)' 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 wave function. By numerically simulating a quantum lattice system corresponding to the Hamiltonian of the kicked rotor, we find that the long-time average of the IEE 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.