Noisy dynamics of Gaussian entanglement: A transient bound entangled phase before separability

We describe Gaussian bound entangled states---entangled states with a positive partial transpose (PPT)---in four-mode continuous-variable systems. These states appear as a transient phase when certain entangled states with a negative partial transpose (NPT), also known as free entangled Gaussian states, are evolved under a noisy environment. Local thermal baths composed of harmonic oscillators are allowed to interact with one or more modes of the system and a wide variety of initial Gaussian entangled (NPT as well as PPT) states are studied. The robustness of entanglement is defined as the time duration for which the entanglement of the initial state is preserved under the noisy dynamics. We access the separability by utilizing standard semidefinite programming techniques. While most states lose their entanglement after a certain time across all bipartitions, an exception is observed for a three-parameter family of states which we call the generalized four-mode squeezed vacuum states, which transition to bound entangled states and remain so for a finite window of time. This dynamical onset of bound entanglement in continuous-variable systems is the central observation of our work. We carry out the analysis for Haar-random four-mode states (both pure and mixed) to scan the state space for a transient bound entangled phase.