The role of chaotic and ordered trajectories in establishing Born's rule

TL;DR

This paper investigates how the interplay of ordered and chaotic trajectories in Bohmian mechanics influences the emergence of Born's rule, showing that entanglement levels determine the proportion of each trajectory type and their role in quantum probability.

Contribution

It provides a detailed analysis of the relationship between entanglement, trajectory types, and the emergence of Born's rule in Bohmian mechanics, highlighting the role of chaos and order.

Findings

Ordered trajectories increase as entanglement decreases.

Chaotic trajectories are ergodic regardless of entanglement.

The ratio of ordered to chaotic trajectories influences Born's rule emergence.

Abstract

We study in detail the trajectories, ordered and chaotic, of two entangled Bohmian qubits when their initial preparation satisfies (or not) Born's rule for various amounts of quantum entanglement. For any non zero value of entanglement ordered and chaotic trajectories coexist and the proportion of ordered trajectories increases with the decrease of the entanglement. In the extreme cases of zero and maximum entanglement we have only ordered and chaotic trajectories correspondingly. The chaotic trajectories of this model are ergodic, for any given value of entanglement, namely the limiting distribution of their points does not depend on their initial conditions. Consequently it is the ratio between ordered and chaotic trajectories which is responsible for the dynamical establishment (or not) of Born's rule.