Abstract：Measuring complexity has been a major and long quest in both quantum and classical dynamics. While there exists a direct connection between chaos and algorithmic complexity of trajectories in classical physics, the problem is particularly elusive for quantum mechanics, where the notion of trajectory is forbidden by the Heisenberg uncertainty principle and complexity can be attributed not only to the lack of integrability but also to the tensor-product structure of the Hilbert space, that is, to entanglement.
The phase space representation of quantum mechanics is a very convenient framework to investigate quantum complexity, in that one can compare classical and quantum dynamical evolutions of distributions in phase space.
We measure complexity by means of either the number of harmonics or the separability entropy of the Wigner distribution. Both quantities, as well as scrambling, characterized by the growth rate of the square commutator between two observables, in the semiclassical limit are determined by the Lyapunov exponent of the underlying classical dynamics. Our results are illustrated in a model of two coupled nonlinear oscillators, both in the chaotic and in the integrable regime. In particular, we show that it is possible to use the above quantities to detect in the time domain the integrability to chaos crossover in many-body quantum systems.
