Abstract In this paper we examine the evolution of Bohmian trajectories in the presence of quantum entanglement (QE). We study a simple two-qubit system composed of two coherent states and investigate the impact of QE on chaotic and ordered trajectories via both numerical and analytical calculations.
Abstract 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 → ∞ : different initial conditions result to the same limiting distribution of trajectory points.
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→|Ψ|^{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=|Ψ|^{2} is guaranteed only if P_{0}=|Ψ_{0}|^{2}. Consequently, the existence of chaotic and ergodic Bohmian trajectories does not always lead to the dynamical establishment of Born's rule.
We study in detail the critical points of Bohmian flow, both in the inertial frame of reference (Y-points) and in the frames centered at the moving nodal points of the guiding wavefunction (X-points), and analyze their role in the onset of chaos in a system of two entangled qubits. We find the distances between these critical points and a moving Bohmian particle at varying levels of entanglement, with particular emphasis on the times at which chaos arises. Then, we find why some trajectories are ordered, without any chaos. Finally, we examine numerically how the Lyapunov Characteristic Number (LCN) depends on the degree of quantum entanglement. Our results indicate that increasing entanglement reduces the convergence time of the finite-time LCN of the chaotic trajectories toward its final positive value.
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.
This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born’s rule unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely, whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution of Bohmian particles.
We study in detail the interplay between chaos and entanglement in the Bohmian trajectories of three entangled qubits, made of coherent states of the quantum harmonic oscillator. We find that all the three-dimensional (3D) chaotic trajectories are ergodic; namely, they have a common long time distribution of points regardless of the initial conditions, and for any nonzero entanglement, their number is much larger than in the corresponding two-qubit system. Furthermore, the range of entanglements for which practically all the trajectories are chaotic and ergodic is much larger than in the two-qubit case. Thus, as the dimensionality of the system increases, Born's rule becomes accessible to a wider range of arbitrary initial distributions than in the 2D case. Our numerical results lead to the conjecture that, for multiqubit systems, Born's rule is the limit of almost all initial distributions of particles.