Degrees of entanglement in systems of three indistinguishable bosons: Revisiting the Greenberger-Horne-Zeilinger state

While the concept of entanglement for distinguishable particles is well established, defining entanglement and nonlocality in systems of indistinguishable particles, which require the use of the (anti)symmetrization postulate, remains challenging, and multiple approaches have been proposed to address this issue. In this work we study the problem of detecting genuine tripartite entanglement among systems of indistinguishable bosons. A genuine entangled state is one that cannot be separable under any bipartition, where separability in the indistinguishable regime is defined by the existence of single-particle properties within each subsystem, without the possibility of knowing which property belongs to which subsystem. We use an algorithm that allows us to search for these single-particle properties and consequently rank states according to their degree of separability. In particular, we introduce a state of indistinguishable bosons with properties analogous to those of the standard Greenberger-Horne-Zeilinger state.