Disentanglement of a bipartite system portrayed in a (3+1)D compact Minkowski manifold; quadridistances and quadrispeeds

Abstract

In special relativity, trajectories of particles, whether massive or massless, in 4D, can be displayed in the 3+1 Minkowski space-time manifold. On the other hand, in quantum mechanics, trajectories in phase space are not strictly defined because coordinate and linear momentum cannot be measured simultaneously with arbitrary precision, as these variables do not commute with each other. They are not sharply defined within Hilbert space formalism. Nonetheless, out of the density matrix representing a quantum system the extracted information still yields an enhanced description of its properties, and by arranging adequately the matrix one can acquire additional information from its content. Following these lines of conduct, this paper focuses on a closely related issue, the definition and meaning of velocity and speed of a typical quantum phenomenon, the disentanglement for a bipartite system when its evolution is displayed in a 4D pseudo-space-time, whose coordinates are combinations of the density matrix entries. Formalism is based on the definition of a compact Minkowski manifold, where trajectories are defined using the same reasoning of special relativity in the Minkowski manifold. The space-like and time-like regions acquire different meanings, termed entangled-like and separable-like, respectively. The definition and meaning of velocity and speed of disentanglement follow naturally from the formalism. Depending on the dynamics of the physical system state, trajectories may go forth and back from entanglement to separability regions of the compact Minkowski manifold. When the physical time t is introduced as an intrinsic variable into the formalism, a phenomenon commonly known as sudden death occurs during irreversible evolution when a state that is initially entangled becomes separable.