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on β$_{e}$$_{2}$ . By properly averaging over any possible prepara- tion of the environmental qubit e$_{2}$ , we are led to a typical e − A entanglement that is identical to Eq. (6). These considerations can be extended to dim( E ) = n , although the complexity of an analytical proof scales exponentially with n . However, our numerical calculations are in per- fect agreement with the analytic conclusions, supporting the general validity of the arguments used here and thus demonstrating the claimed invariance. 

The relevance of this result and its significance become evident when one considers that random-matrix theory is widely believed to provide a good effective descrip- tion of the state of a system in thermal equilibrium at a high temperature [19]. In such unfavorable conditions, the achievement of entanglement regardless of the initial state of the environment ( i.e. its effective temperature) and its robustness against the complexity of E ’s structure and the initial (pure) state of A is a remarkable feature. Our study reveals that through the use of a suitable in- teraction we can reliably create entanglement between a clean ancilla qubit and a mixed-state environment qubit. Furthermore, the invariance of the amount of generated entanglement with respect to the size of the environment strongly contrasts with the intuitive belief that C eA would have followed a monogamy constraints similar to those responsible for the bipartite entanglement in an n -qubit system [13, 20]. Besides its intrinsic interest, such an in- variance will be proven crucial for the understanding of the study conducted in Sec. II 

## II. TWO-ENVIRONMENT CASE: ENTANGLEMENT-STRUCTURE TRANSITION 

We now turn our attention to the main scenario of our investigation, which consists of two independent and spa- tially separated environments, E$_{R}$ and E$_{L}$ . This time, the ancilla A shuttles between them, colliding always with the same qubit of each environment. Again, we are interested in typical values of entanglement and, in order to cancel any dependence on the initial prepara- tion of the environments, we rely on a random-matrix approach. We first show that, in line with the results un- veiled in Sec. I, one of the environments is entangled with A with a constant typical degree, dependent solely on Jt . This is accompanied by a few other results related to the details of the environment-ancilla interactions. Most importantly, however, we demonstrate that the genuine tripartite entanglement established among A and the en- vironmental qubits involved in the collisions is subjected to a drastic change in its structure, depending on the shape of the interaction Hamiltonian and the number of collisions. The fact that these results refer to typical ( i.e. statistically averaged) quantities, make them even more intriguing. Not only multipartite entanglement can be generated: through suitable interaction engineering we can actually efficiently dictate its form and nature. 

## A. Quantitative analysis of entanglement 

Let us first consider the case where dim( E$_{L}$ ) = 1 (we label its only element as { e$_{L}$ } ), while E$_{R}$ consists of qubits { e$_{1}$, e$_{2}$, .., e$_{n}$ } . If A collides only once, first with E$_{R}$ and then E$_{L}$ , we find that the degree of entanglement between the single-qubit environment E$_{L}$ and the ancilla remains constant against the number of qubits belonging to E$_{R}$ , as show in Fig. 2 (a) . We can easily understand this in light of the results discussed in the previous Sec- tion, where we have shown that the typical entanglement between A and a single-qubit environment does not de- pend on the input state of the ancilla. As the collision with E$_{R}$ , followed by the trace over its degrees of free- dom, simply prepares A in a mixed state, the invariance of C e$_{L}$A against dim( E$_{R}$ ) is clear: quantitatively, C e$_{L}$A is identical to Eq. (6). On the other hand, by calling e$_{R}$ the E$_{R}$ qubit with which A collides, we have that C e$_{R}$A < C e$_{L}$$_{A}$, regardless of dim( E$_{R}$ ), a behavior that can be rigorously explained as follows. For the sake of simplicity, we consider the simplest case where both the environments consist of a single qubit. By means of the decomposition in Eq. (3), we prepare a general (sepa- rable) initial state of e$_{R}$ and e$_{L}$ by specifying two sets of parameters { φ$_{j}$, ψ$_{j}$, χ$_{j}$, α$_{j}$ } ( j = L, R ). The ancilla is then let interact with e$_{R}$ and e$_{L}$ , in this order. The degrees of freedom of e$_{L}$ are thus traced out so as to as- certain the concurrence between A and e$_{R}$ . This turns out to be 

There is no dependence on the state of e$_{L}$ in this ex- pression, although the second collision has occurred and no average has yet been taken. However, this is quite clear in virtue of the fact that we are considering only a single interaction with each environment. After the col- lision with e$_{L}$ , there is no way for A to convey to e$_{R}$ information on the state of the left qubit. In fact, if we instead trace out e$_{R}$ , we find precisely Eq. (7). Look- ing for typical values, i.e. calculating averages over the tensor product of any possible unitary matrix by using the appropriate Haar measure introduced before, we get C e$_{R}$A = | cos(2 Jt ) sin(4 Jt ) | / 2 < C e$_{L}$A ∀ Jt . Quantita- tively, for Jt = 1 the typical e$_{R}$ − A concurrence is ≃ 0 . 157, which perfectly agrees with the independent numerical estimate provided in Fig. 2 (a) (dashed line). The ordering relation involving C e$_{R}$A and C e$_{L}$A is pre- served as the number of qubits in E$_{R}$ grows, as shown in Fig. 2 (a) , where the difference C e$_{L}$A − C e$_{R}$A appear to increase with dim( E$_{R}$ ). This is a clear effect of the profound differences, stemming from the possibility to achieve multipartite entanglement, between the present situation and the single-environment case addressed in Sec. I. For dim( E$_{R}$ ) > 1, one cannot exclude that the random unitaries taken in order to prepare E$_{R}$ set some genuine multipartite entanglement among its ele- ments. Consequently, the entanglement set by the dy- namics studied here and shared by e$_{R}$ and A would be 

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