8 a graph of class (c) to one of class (b) in Fig. 5. This is indeed possible. We consider dim( E$_{L}$ ) = dim( E$_{R}$ ) and investigate the behavior of both tripartite and bipartite negativity against the number of environmental qubits and the number of interactions with the ancilla. A typi- cal result, achieved by considering 18 pairs of collisions, is given in Fig. 7 (a) where it is evident that, by consider- ing larger environments, we are able to keep the tripartite entanglement non-zero using repeated collisions between A and the environments. Any two-qubit quantum cor- relation, on the other hand, disappears. The curves are the best-fits to the discrete numerical values obtained via our simulations. We have found that the functional form A e −B (dim( E$_{j}$ ) − 1) (with A , B ∈ R ) gives an excellent agreement with the numerical results. Fig. (7) (b) shows the number of qubits necessary in E$_{j}$ ( j = R,L ) in order to get negativities smaller than 10 − 6 (arbitrarily chosen) at a fixed number of collisions. It is clearly seen that the environmental dimension required to get entangle- ment lower than the chosen threshold is almost always much larger for the tripartite negativity than for the bi- partite ones, thus strengthening our conclusions: we have been able to dynamically drive the system state towards typical GHZ-class entanglement, therefore realizing yet another sharing-structure transition. Using a random-matrix approach, we have shown that the typical entanglement properties of two remote qubit environments, indirectly communicating via the media- tion operated by a shuttling ancilla, can be effectively manipulated both quantitatively and qualitatively. Bi- partite as well as genuinely multipartite entanglement can be faithfully generated and shown to have intrigu- ing statistically averaged properties of resilience against arbitrary pure preparations of the environments, their dimension and the number of interactions that each has with the shuttle. Interestingly, by means of a simple tuning of the details of the interaction, we can achieve transitions in the entanglement-sharing structure from W to two-way GHZ-class [29]. We have studied the ef- fects that an increasing dimension of the environments have in such transitions. The use of random matrices has emerged, here, as a valuable tool for the agile han- dling of the numerical simulations required for the pur- poses of our study, enlarging the range of applicability of the technique to the context of multipartite entangle- ment generation under unfavorable conditions. This is a central problem in current quantum information and computation problems, where we hope that our results FIG. 7: (Color online). (a) Negativity against the environ- ments’ dimension dim( E j ) ( j = L, R ) for λ = 0, 18 pairs of collisions and Jt = 1. The dashed line shows the tripar- tite negativity, while the solid lines are for the entanglement within each bipartite subsystem involving only one environ- ment. The e R − e L entanglement is obviously null. Each curve is rescaled with respect to its maximum value. (b) : We plot the number of qubits required in the remote environments in order to make the tripartite and bipartite negativity smaller than 10 $_{−}$6 when a fixed number of collisions is taken. The top- most curve is for N tri while the other two curves, which are almost superimposed, show the cases associated with N e$_{j}$A . III. CONCLUSIONS FIG. 6: (Color online). Entanglement obtained for an increasingly-sized E L environment (which comprises of up to eight qubits) and a single-qubit E R . We set λ = 0 in Eq. (16) and the rescaled interaction time Jt = 1. (a) Tripartite negativity. (b) Bipartite e L − A entanglement against the dimension of E L and the number of collisions. (c) Same as panel (b) but for the single-qubit environment E R . (a) (b) (c) (a) (b)