Bipartite entanglement in continuous-variable tripartite systems

Abstract

In the field of continuous-variable tripartite entanglement, the systems utilised can be either asymmetric or symmetric. It is therefore of interest to examine the differences in the entanglement properties of these two types of system, using two examples that are known to produce tripartite entanglement. We examine one asymmetric and one fully symmetric Gaussian continuous-variable system in terms of their tripartite and bipartite entanglement properties. We first treat pure states and are able to find analytic solutions using the undepleted pump approximation for the Hamiltonian models. Our symmetric system exhibits perfect tripartite correlations, but only in the unphysical limit of infinite squeezing. For more realistic squeezing parameters, the two systems exhibit both tripartite and bipartite entanglement. Secondly we treat the more experimentally reasonable situation where the interactions take place inside optical cavities and we are dealing with mixed states. In these cases, where the criteria for genuine tripartite entanglement are more stringent, we find that tripartite entanglement is still available, although over smaller bandwidths than three-mode inseparability. In general, the spectral results are consistent with the analytical solutions. We conclude that none of the outputs are completely analogous to either GHZ or W states, but there are parameter regions of the Hamiltonian dynamics where they produce T states as introduced by Adesso et al. [1,2]. In the intracavity cases, both bipartite entanglement and tripartite inseparability are always present, with genuine tripartite entanglement appearing as the pumping rate is increased. The qualitative differences in the output states for different interaction parameters indicate that continuous-variable tripartite quantum information systems offer a versatility not found in two-mode bipartite systems.