Separable and Inseparable Quantum Trajectories

Abstract

The dynamical behavior of interacting systems plays a fundamental role for determining quantum correlations, such as entanglement. In this Letter, we describe temporal quantum effects of the inseparable evolution of composite quantum states by comparing the trajectories to their classically correlated counterparts. For this reason, we introduce equations of motions describing the separable propagation of any interacting quantum system, which are derived by requiring separability for all times. The resulting Schrödinger-type equations allow for comparing the trajectories in a separable configuration with the actual behavior of the system and, thereby, identifying inseparable and time-dependent quantum properties. As an example, we study bipartite discrete-and continuous-variable interacting systems. The generalization of our developed technique to multipartite scenarios is also provided.

Classical evolution in quantum systems

“…The classical dynamics resulting from our method is described in terms of Schrödinger-type equations for each subsystem separately. This is consistent with our recent approach [44] and underlines the the general potential of our method introduced here. As we have shown for different forms of separability (or, conversely, partial entanglement), the entanglement-generating evolution can be clearly distinguished from the classical one, even for initially fully separable mixed states.…”

“…In Ref. [44], we derived and characterized those equations based on the Lagrangian and the least action principle. Here, we used the Hamiltonian to obtain the same dynamics for separable states [58].…”

“…To overcome such a restriction, we recently introduced a method to analyze the entanglement dynamics [44]. This was achieved by constraining the quantum trajectories to nonentangled (i.e., separable) ones.…”

Numerical Construction of Multipartite Entanglement Witnesses

“…As a final example let us consider the dynamics of quantum systems, which is described by the Schrödinger equation. To distinguish the entanglement-generating evolution from the separable dynamics, we recently introduced the separability Schrödinger equations [79], which relate to the SEE in the static case. Again, the SPI can be the starting point for the numerical implementation of this approach.…”

Experimental Reconstruction of Entanglement Quasiprobabilities

“…However, identifying separability through entanglement witnesses requires testing a large number of them, in contrast to the direct criterion of the nonnegativity of EQPs. In addition to witnesses, even the entanglement dynamics of a composite system is accessible through a time-dependent version of the presented eigenvalue approach [42].…”