Paper detail

Chaos and ergodicity in an entangled two-qubit Bohmian system

We study in detail the onset of chaos and the probability measures formed by individual Bohmian trajectories in entangled states of two-qubit systems for various degrees of entanglement. The qubit systems consist of coherent states of 1-d harmonic oscillators with irrational frequencies. In weakly entangled states chaos is manifested through the sudden jumps of the Bohmian trajectories between successive Lissajous-like figures. These jumps are succesfully interpreted by the `nodal point-X-point complex' mechanism. In strongly entangled states, the chaotic form of the Bohmian trajectories is manifested after a short time. We then study the mixing properties of ensembles of Bohmian trajectories with initial conditions satisfying Born's rule. The trajectory points are initially distributed in two sets $S_1$ and $S_2$ with disjoint supports but they exhibit, over the course of time, abrupt mixing whenever they encounter the nodal points of the wavefunction. Then a substantial fraction of trajectory points is exchanged between $S_1$ and $S_2$, without violating Born's rule. Finally, we provide strong numerical indications that, in this system, the main effect of the entanglement is the establishment of ergodicity in the individual Bohmian trajectories as $t\to\infty$: different initial conditions result to the same limiting distribution of trajectory points.