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bound to follow many-body entanglement monogamy re- lations [20], which lower its amount as E$_{R}$ grows in di- mension. This is also the reason behind the behavior shown in Fig. 2 (b) , where by inverting the order of the ancilla-environment interactions we lose the invariance of the e$_{L}$ − A concurrence, an effect occurring for precisely the same reasons explained above. On the other hand, as before, the second interaction always sets the lowest de- gree of entanglement. It is worth remarking that Fig. 2 is the result of an independent numerical calculation that, although follows the formal recipe highlighted here, does not rely on the analytic result in Eq. (11). 

## B. Entanglement-Structure Transitions 

The explicit introduction of multipartite entanglement in the process described until now directs us towards the main point of this study. Through the proper tuning of the dynamical and kinematic properties of the system, we shall demonstrate the ability to manipulate the shar- ing structure of the typical multipartite entanglement set among A , e$_{L}$ and e$_{R}$ . For the remaining part of our study, we find it computationally more convenient to abandon concurrence for negativity [21, 22]. This is an entangle- ment measure devised from the Peres-Horodecki criterion on positivity of partial transposition (PPT). It is a nec- essary and sufficient condition for 2 × 2, 2 × 3 and ∞ × ∞ systems, and is a sufficient condition for an arbitrary sys- tem. For a bipartite case we define the negativity as [22] 

where λ$_{neg}$ is the single negative eigenvalue arising after the partial transposition of the density matrix. The con- venience in using negativity stems for its straightforward 

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generalization to the multipartite scenario: we simply have to consider all the possible bipartite splits in a given system and the geometric average of the negativity asso- ciated with any of them. This construction ensures that our measure remains an entanglement monotone [23]. As we are mostly interested in at most three qubits, we con- centrate on tripartite negativity , which in our case reads 

where A | e$_{L}$e$_{R}$ , for instance, indicates the bipartition of qubit A against the group of qubits e$_{L}$ and e$_{R}$ . 

In light of the analysis performed in Sec. II A one can clearly conclude that, if tripartite entanglement is set within the e$_{L}$ − A − e$_{R}$ system, this has to be W-class [24]. In fact, we find non-zero entanglement in any biparti- tion obtained by tracing out one of the components of the overall system. In Fig. 2, we have shown the en- tanglement between the ancilla and each environmental qubit involved in the collisions. Moreover, we have also checked that the e$_{L}$ − e$_{R}$ bipartition turns out to be in- separable in each of the situations addressed there. This is a property not shared by the GHZ-class [25], where by tracing one qubit out of a tripartite register, one gets separable reduced states. In order to check that W-class entanglement is indeed in order here, we have utilized the witness [26, 27] 

with | GHZ 〉 = ( | 000 〉 + | 111 〉 ) $_{/}$√$_{2 a three-qubit GHZ}$ state and 3 / 4 the squared maximum overlap between a GHZ state and a W-class state. Eq. (14) is a valid GHZ witness: a negative expectation value of ˆ W suc- cessfully reveals GHZ-class entanglement. On the other hand, this construction guarantees that a positive expec- tation value is achieved for W-class states [26]. By gener- ating a sample of 1000 random environmental states and taking dim( E$_{R}$ ) up to 4 (so as to match the situation depicted in Fig. 2), we have numerically verified that, as far as an isotropic Heisenberg model is used to model the collisions, 〈 ˆ W〉 > 0 for every single preparation of the en- vironments and regardless of dim( E$_{R}$ ). The calculation has been performed by maximizing the overlap between the sample states and | GHZ 〉 over the tensor product of three local rotations, each acting on A , e$_{L}$ and e$_{R}$ re- spectively. Moreover, the tripartite negativity defined in Eq. (13) turns out to be non-zero for any initial prepara- tion of our statistical sample, and so is the typical value obtained by averaging over it, which guarantees genuine tripartite entangled nature of the e$_{L}$ − A − e$_{R}$ states with strong evidence of their W-class nature. 

The task of this Section is to show that the typical entanglement-sharing structure of such tripartite state can be abruptly modified by biasing the coupling Hamil- tonian towards a specific spin non-preserving model. We 

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