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.

中文翻译：

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纠缠的双量子比特波姆系统中的混沌和遍历性

我们详细研究了混沌的开始以及由不同纠缠度的双量子位系统纠缠态中的单个波姆轨迹形成的概率度量。量子位系统由具有无理频率的一维谐振子的相干状态组成。在弱纠缠状态中，混沌表现为连续的 Lissajous 图形之间的 Bohmian 轨迹的突然跳跃。这些跳跃被“节点点-X 点复合体”机制成功解释。在强纠缠态中，波姆轨迹的混沌形式在很短的时间内就显现出来。然后，我们研究了初始条件满足波恩规则的玻姆轨迹系综的混合特性。轨迹点最初分布在两组 $S_1$ 和 $S_2$ 中，具有不相交的支撑，但随着时间的推移，它们在遇到波函数的节点时会突然混合。然后大部分轨迹点在 $S_1$ 和 $S_2$ 之间交换，而不会违反 Born 规则。最后，我们提供了强有力的数值指示，在这个系统中，纠缠的主要影响是在各个波姆轨迹中建立遍历性为 $t\to\infty$：不同的初始条件导致轨迹点的相同限制分布. s 规则。最后，我们提供了强有力的数值指示，在这个系统中，纠缠的主要影响是在各个波姆轨迹中建立遍历性为 $t\to\infty$：不同的初始条件导致轨迹点的相同限制分布. s 规则。最后，我们提供了强有力的数值指示，在这个系统中，纠缠的主要影响是在各个波姆轨迹中建立遍历性为 $t\to\infty$：不同的初始条件导致轨迹点的相同限制分布.