Ergodicity and Born's rule in an entangled two-qubit Bohmian system

We study the Bohmian trajectories of a generic entangled two-qubit system, composed of coherent states of two harmonic oscillators with noncommensurable frequencies and focus on the relation between ergodicity and the dynamical approach to Born's rule for arbitrary distributions of initial conditions. We find that most Bohmian trajectories are ergodic and establish the same invariant ergodic limiting distributions of their points for any nonzero amount of entanglement. In the case of strong entanglement the distribution satisfying Born's rule is dominated by chaotic-ergodic trajectories. Therefore, P -> vertical bar Psi vertical bar(2) for an arbitrary P-0. However, when the entanglement is weak the distribution satisfying Born's rule is dominated by ordered trajectories, which are not ergodic. In this case the ergodic trajectories do not, in general, lead to the distribution of Born's rule, therefore P-0 = vertical bar Psi vertical bar(2) is guaranteed only if P-0 = vertical bar Psi(0)vertical bar(2). Consequently, the existence of chaotic and ergodic Bohmian trajectories does not always lead to the dynamical establishment of Born's rule.