7 FIG. 4: (Color online). (a) Tripartite negativity of the E L − A − E R system against λ ∈ [ − 5 , 5] and the number of collisions, for a dimensionless interaction strength Jt = 1 in an anisotropic Heisenberg coupling involving single-qubit environments. (b) Bipartite negativity for the A − E j qubit-pair, regardless of j = L, R . (c) Bipartite negativity for the E L − E R system. For λ = 0 there is no entanglement in this bipartition, while the ancilla-environment ones are inseparable. naling that this interaction model guarantees the genera- tion of entanglement between the ancilla and an environ- mental qubit even under the “minimum-control” condi- tions where one is unable to stop the ancilla-environment interactions after a desired number of collisions. More- over, as claimed above, at λ = 0 the e$_{L}$ − e$_{R}$ system is indeed separable, regardless of the number of considered collisions (see the N e$_{L}$e$_{R}$ = 0 line in Fig. 4 (c) ). It is very informative to study the effects that increas- ing the size of the environments has on the degree and type of shared entanglement for λ = 0. We first take dim( E$_{L}$ ) ≥ 1 and leave E$_{R}$ with a single qubit. The scheme of interaction in the usual one, with e$_{L}$ being struck first. The features associated with the resulting entanglement sharing are shown in Figs. 6. In panel (a) we plot the tripartite negativity set within the state of the interacting qubits after tracing out the non-interacting elements from the E$_{L}$ environment. We find that N tri decreases with the dimension of E$_{L}$ , which is a physi- cally meaningful result in light of monogamy relations that the system should adhere to [20]. A similar behav- ior is found for the bipartite e$_{L}$ − A entanglement, as shown in Fig 6 (b) , which can be easily interpreted in light of the discussion in Sec. II A. Finally, panel (c) shows the e$_{R}$ − A entanglement, which exhibits a con- stant trend that is consistent with our previous results on the isotropic Heisenberg Hamiltonian. We omit show- ing the e$_{R}$ − e$_{L}$ entanglement as this is exactly null, in these conditions. The entanglement structure remains of the two-way GHZ class [29] although for dim( E$_{L}$ ) ≳ 8, N e$_{L}$A ≃ 0. This implies the disappearance of the leftmost bond in Fig. 5 (c) but no further change, as genuine tri- partite entanglement is still present in the e$_{L}$ − A − e$_{R}$ system (see Fig. 6 (a) ). The above analysis covers only partially the issue of increasing-sized environments, as it addresses an asym- metric situation. Another interesting effect arises if E$_{R}$ is let grow in size. In fact, the rate at which bipartite quan- tum correlations decrease in the case considered above suggests that, by increasing dim( E$_{R}$ ), all residual en- tanglement would die off, eventually, leaving us with a sharing structure that reminds more of a typical GHZ state. In other words, we are looking for a configura- tion allowing an additional entanglement transition from FIG. 5: (Color online). Entanglement graphs for the tripar- tite setting. Each sphere is a qubit, each bond represents a non-separable state. (a) W-class states having no separable bipartition. (b) Fully separable and GHZ-class states, with no residual bipartite entanglement. (c) Entangled-class with bipartition-dependent residual entanglement. If the state con- tains genuine tripartite entanglement, it can be enumerated in the GHZ class. Systems e L and e R share classical correla- tions. (d) Biseparable states encompassing a Bell pair [9]. (a) (b) (c)