Tzemos Athanasios C, Contopoulos George
Research Center for Astronomy and Applied Mathematics of the Academy of Athens, Soranou Efesiou 4, Athens GR-11527, Greece.
Phys Rev E. 2020 Oct;102(4-1):042205. doi: 10.1103/PhysRevE.102.042205.
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.
我们研究了一个由两个频率不可通约的谐振子的相干态组成的一般纠缠两量子比特系统的玻姆轨迹，并关注遍历性与任意初始条件分布下玻恩规则的动力学方法之间的关系。我们发现，大多数玻姆轨迹是遍历的，并且对于任何非零纠缠量，其点都建立了相同的不变遍历极限分布。在强纠缠的情况下，满足玻恩规则的分布由混沌遍历轨迹主导。因此，对于任意的(P_0)，有(P→|Ψ|^{2})。然而，当纠缠较弱时，满足玻恩规则的分布由非遍历的有序轨迹主导。在这种情况下，遍历轨迹通常不会导致玻恩规则的分布，因此只有当(P_0 = |Ψ_0|^{2})时，才有(P = |Ψ|^{2})。因此，混沌和遍历玻姆轨迹的存在并不总是导致玻恩规则的动力学确立。
