A Classical Analogue of Entanglement for a Kicked Top

Abstract

It is widely believed that quantum mechanics cannot exhibit chaos, since unitarity of time evolution ensures that distances between quantum states are preserved. However, a parallel argument can be constructed in classical mechanics that would seem to deny the existence of classical chaos too. The argument works by describing classical states as probability distributions in phase space and showing that the inner product between distributions on phase space is preserved under Liouvillian dynamics. Thus, the more faithful classical analogy of a quantum state is not a single phase space trajectory but is instead a phase space distribution, and chaos in such states must be identified by some statistical signatures instead of exponential separation of nearby states. The search for these signatures is the primary goal in quantum chaos research. However, this perspective also naturally motivates the search for classical analogues of these signatures, to reveal the inner machinery of chaos in quantum systems. One widely recognized signature of chaos in quantum systems is the dynamical generation of entanglement. Chaos in the classical system is correlated with a greater entanglement production in the corresponding quantum system. One of the most well-studied examples of this is the kicked top model. In this paper, we construct a classical analogue of bipartite entanglement in terms of the mutual information between phase space distributions of subsystems and find completely analogous signatures of chaos as those found in entanglement for the kicked top Hamiltonian.