Separable and Inseparable Quantum Trajectories

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.

Figure 1

(a) The relation between the different equations. The time-independent equation for the joint system is the EE [Eq. (2)]. For product states, $|a⟩\otimes |b⟩$, the static equations are the SEEs [Eq. (10)]. The joint evolution of the system is described via the SE [Eq. (1)]. Here, we derived the “missing link”—the SSEs [Eqs. (6a) and (6b)]—which describe the propagation of product states. (b) Trajectories on the Bloch sphere. The solutions of the SSE (dashed) are compared to the solutions of the SE (solid) for one half period. All curves are on the depicted plane. The initial conditions are $|a_0⟩=|\uparrow ⟩$ and $|b_0⟩=(|\uparrow ⟩+|\downarrow ⟩)/\sqrt{2}$. (c) Trajectories in phase space for one half period. The solutions of the SSE and SE are shown as dashed and solid curves, respectively. The initial states are the coherent states $|e^{i\pi /4}/2⟩$ and $|-e^{i\pi /4}/2⟩$ for the subsystems $A$ and $B$, respectively.