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Haar measure dU$_{r}$ = (8 π $^{3}$) − 1 sin(2 φ ) dφdψdχdα [11, 15, 16]. In order to evaluate the entanglement set within the system, we use Wootters’ concurrence [17]. For a general bipartite pure or mixed state described by the density matrix ϱ , concurrence is given by 

<img src='content_image/405717.jpg'>

where λ$_{1}$ ≥ λ$_{j}$ ( j = 2 , 3 , 4) are the eigenvalues of ρ ( σ$_{2}$ ⊗ σ$_{2}$ ) ρ $^{∗}$( σ$_{2}$ ⊗ σ$_{2}$ ). When A is prepared in its ground state and upon evolution of the e − A system as ϱ = ˆ U$_{h}$ ˆ U$_{r}$ρ ˆ U † r ˆ U † $_{h}$, we get the density matrix 

<img src='content_image/405718.jpg'>

with C ( x ) = cos( x ) and S ( x ) = sin( x ). It is then straightforward to see that the concurrence shared by e and A after the application of ˆ U$_{r}$ and ˆ U$_{h}$ results in the elegant expression 

This shows that, given a specific preparation of the en- vironment, the only parameters governing the entangle- ment are φ and Jt . The typical value of C eA is obtained by averaging the above expression over any possible uni- tary matrix U$_{r}$ uniformly drawn according to the proper Haar measure. Explicitly, we have to calculate 

<img src='content_image/405720.jpg'>

which may seem a special result arising from the pure- state preparation of the ancilla. However, this is defi- nitely not the case, as we now demonstrate. By starting with the mixed ancilla state 

and calculating the evolution of the e − A system, one gets a density matrix that is an easy generalization of Eq. (5). By inspecting the eigenvalues of ρ ( σ$_{2}$ ⊗ σ$_{2}$ ) ρ $^{∗}$( σ$_{2}$ ⊗ σ$_{2}$ ), whose explicit form is too lenghty to be reported here, one finds no dependence on χ, ψ or α , in full analogy with the case of Eq. (7). The explicit calculation of con- currence leads us to the expression 

<img src='content_image/405723.jpg'>

As ∫ π/ 2 0 cos(2 φ ) sin(2 φ ) dφ = 0, the average concurrence (calculated using the appropriate Haar measure, as done before) turns out to be identical to Eq. (6). The study 

<img src='content_image/405719.jpg'>

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can be straightforwardly generalised to the case of qubit A being prepared in any coherent-superposition state, the only difficulty being a slightly more complicated expres- sion for the concurrence corresponding to any set prepa- ration of E . The message, however, is rather clear: re- gardless of the state into which the ancilla is prepared, the typical e − A entanglement is simply set by the rescaled interaction time Jt . This is strictly valid only in the statistical sense: if specific instances of preparation of both A and e are taken, such an independence does not hold anymore. However, contrary to a naive expec- tation, the typical entanglement does not vanish. As we see later, this result is the key to understand what occurs in the two-environment case. 

We now approach the second step of our proof by studying the invariance of the e − A entanglement with respect to dim( E ) when A is prepared in a pure state. For this task, we consider a simple extension of the previous case to a two-qubit environment E = { e$_{1}$, e$_{2}$ } , initially prepared in a state described by 

<img src='content_image/405721.jpg'>

with χ the elements of the tensor accounting for the cor- relations between e$_{1}$ and e$_{2}$ and β$_{e}$$_{j}$ ( j = 1 , 2) the Bloch vector of qubit e$_{j}$ ( j = 1 , 2) [9, 18]. This form holds for both entangled and separable two-qubit states and is thus a formal description of an arbitrary preparation of E . By taking e ≡ e$_{2}$ (an arbitrary choice that does not affect the generality of our discussion) we follow the recipe for evolution described above. This time, before calculating the e − A concurrence, we have to trace over e$_{1}$ ’s degrees of freedom. Through a tedious but otherwise straightforward calculation, we see that although the tri- partite E − A density matrix depends on χ and β$_{e}$$_{1}$ , the reduced density matrix of the e − A system only depends 

