Hypersensitivity to perturbations in the quantum baker's map.

We analyze a randomly perturbed quantum version of the baker’s transformation, a prototype of an area-conserving chaotic map. By numerically simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this information grows very rapidly and becomes much larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker’s map thus displays a hypersensitivity to perturbations that is analogous to behavior found earlier in the classical case. This hypersensitivity characterizes “quantum chaos” in a way that is directly relevant to statistical physics. Great progress has been made in studying manifestations of chaos in quantum systems [1], yet there still is controversy as to whether quantum chaos exists at all [2, 3]. A chief reason for this is that the most important characteristic of classical chaotic systems—exponential divergence of trajectories starting at arbitrarily close initial points in phase space—is absent from quantum systems simply because the existence of a quantum scale makes meaningless the concept of two arbitrarily close points in phase space. Any attempt to find exponential divergence of trajectories of Hilbert-space vectors founders immediately, because the linear Schrödinger equation, with its unitary evolution, preserves Hilbert-space inner products. Yet the unitary linear evolution of the Schrödinger equation must be irrelevant to the issue of quantum chaos [2], since any Hamiltonian classical chaotic system can be described by an analogous area-conserving linear Liouville ∗Submitted to Physical Review Letters