FIG. 3: (Color online). Bipartite and tripartite negativity re- sulting from a bilocal interaction model ruled by Eq. (15), for single-qubit environments and | 000 〉 e$_{L}$Ae$_{R}$ initial state. The full disappearance of entanglement at λ = 1 occurs in virtue of this specific choice of initial state. At λ = 0 we have a point of entanglement-sharing change. Other three such points are visibile within the shown range of λ ’s. thus recast Eq. (2) into ( j = L, R ) where we have clearly taken J$_{2}$$_{,}$$_{3}$ = λJ ( λ ∈ R ) and J$_{1}$ = J . The reasons behind this choice are best explained by means of the following analysis. For the sake of argument we focus on single-qubit environments and take qubits A, e$_{L}$ and e$_{R}$ as prepared in their respective ground state. By considering an e$_{L}$ − A collision followed by an e$_{R}$ − A one, both ruled by Eq. (15), we get Despite its innocence, the entanglement-sharing prop- erties of Eq. (16) are quite interesting. In Fig. 3 we show the bipartite and tripartite negativity against the anisotropy parameter λ , for Jt = 1 (arbitrary choice). The behavior of the bipartite entanglement is strongly dependent on the choice of λ . In particular, at λ = 0, which leaves us with ˆ H$_{b,j}$ = Jσ$_{1}$$_{,A}$ ⊗ σ$_{1}$$_{,e}$$_{j}$ , only two out of three possible bipartitions are inseparable: the e$_{L}$ − e$_{R}$ subsystem remains separable regardless of the choice for Jt . This is easily checked by studying the partial trans- posed of the state resulting from | ψ 〉 e$_{L}$Ae$_{R}$ upon trace over A . And yet, the tripartite negativity N tri is rather large at λ = 0 (cfr. Fig. 3, dashed line), demonstrating that we have a tripartite entangled state that, in evident contrast with the results of the previous Section, is out of the W-class. [28]. This qualitative feature is typical as it survives to a statistical average over random ini- tial preparations of the environments, their dimensions and the number of A − e$_{j}$ interactions ( j = L, R ). This As anticipated, we also allow the ancilla to interact multiple times, in sequence, with each environment (each time with the very same e$_{L,R}$ qubit). Multiple interac- tions, however, do not appear to have any qualitative relevance on the generated entanglement structure, al- though there is an evident quantitative effect on the amount of entanglement shared both in bipartite and tri- partite sense. In Fig. 4 (a) and (b) we plot the typical tripartite and bipartite entanglement of the A − e$_{L}$ system and it is easily verified that the same behavior is found for the A − e$_{R}$ system. For the whole considered range of val- ues for λ the bipartite negativity is always non zero, sig- where ϱ$_{e}$$_{L,R}$ denotes the reduced density matrix of qubit e$_{L,R}$ after (random) preparation, unitary evolution and appropriate partial traces [30]. We can thus rightfully claim for the anticipated entanglement transition induced by the tuning of the effective order parameter embod- ied by λ . The naturally generated W-class state arising from the use of an unbiased Heisenberg coupling is non- trivially changed into a GHZ-type entanglement charac- terized by a two-way sharing structure. The transition is typical in the sense explained so far: given any pure initial environmental state, we are able to create either type of tripartite entanglement structures simply by tun- ing the interaction properties. latter feature is shown in Figs. 4, where we plot the aver- age bipartite and tripartite negativity against λ ∈ [ − 5 , 5] and the number of collisions. Fig. 4 (c) shows that at λ = 0 there is no quantum correlations shared by the two environments, even in this statistically typical scenario. 6 How can we understand this result and relate it to the claimed change in multipartite entanglement struc- ture? We refer to the studies conducted by Plesch and Buˇ zek [29] on the classification of multiparticle quan- tum correlations via entangled graphs. Following their lines, we represent a multi-qubit system with an inherent structure of shared entanglement as a connected graph. Each vertex embodies a qubit while a bond connecting two vertices represents bipartite entanglement shared by the components of the corresponding reduced state. For three qubits, four possible classes are identified, as shown in Fig. 5. In particular, Eq. (16), as well as any of the states resulting from the application of random unitaries over the chosen reference state | 000 〉 e$_{L}$$_{Ae$_{R}$}$, have a sharing structure corresponding to the graph in Fig. 5 (c) , panel (a) being representative of W-class. As it is argued in Ref. [29], by following this graph-based classification of multipartite entanglement, a state showing tripartite cor- relations with the two-way residual entanglement struc- ture of panel (c) is GHZ-class. Moreover, a remark is due: such a two-way entanglement-sharing structure is possible, for pure states, only if there exist classical cor- relation between the separable qubits ( e$_{L}$ and e$_{R}$ in our case). This property is readily confirmed in Eq. (16) and any random element of our statistical sample. In fact, we find that ˆ H$_{b,j}$ = J [ σ$_{1}$$_{,A}$ ⊗ σ$_{1}$$_{,e}$$_{j}$ + λ 3 ∑ i =2 σ$_{i,A}$ ⊗ σ$_{i,e}$$_{j}$ ] , (15) ϱ$_{e}$$_{R}$$_{e}$$_{L}$ = ϱ$_{e}$$_{L}$ ⊗ ϱ$_{e}$$_{R}$ (17) | ψ 〉 e$_{L}$Ae$_{R}$ = e − 2 $^{iλJt}$cos( Jt − λJt ) | 0 〉 $_{e$_{L}$}$[cos( Jt − λJt ) | 00 〉 − i sin( Jt − λJt ) | 11 〉 ]$_{Ae}$$_{R}$ sin( Jt − λJt ) | 1 〉 e$_{L}$ ⊗ [sin( λJt + Jt ) | 01 〉 − i cos( λJt + Jt ) | 10 〉 ]$_{Ae}$$_{R}$ . (16)