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Theorem III.4. Given a pure translation-invariant sta- bilizer state ω$_{ξ}$ of stabilizer generator length 2 n + 1 , a region of length L shares 2 n maximally entangled qubit pairs with the rest of the chain if 2 n ≤ L and L qubits pairs if 2 n > L . 

Proof. The proof works exactly as in the bipartite case. In the case 2 n ≤ L the cut stabilizers are only cut on one side. But all stabilizers that are cut on the left-hand side commute with those cut on the right-hand side. Thus, we have two independent cuts of the bipartite case and therefore 2 n pairs of maximally entangled qubits. In the case 2 n > L some stabilizers are cut on both sides. We use the same technique to produce the mutually com- muting anti-commuting pairs which encode the qubits as in Theorem III.3. We always find L pairs of maximally entangled qubits. 

## IV. ENTANGLEMENT GENERATION 

Now we come to the generation of entanglement through CQCA action. As we have seen in the previous section, the bipartite entanglement of a pure translation- invariant stabilizer state depends linearly on the length of the generators of the stabilizer group. So, it suggests itself to study the evolution of the length of the stabilizer generators under CQCA action. First, let us define the asymptotic entanglement generation rate from stabilizer states. 

Definition IV.1. The asymptotic entanglement genera- 

tion rate from stabilizer states for CQCAs is defined as 

where E ( ω$_{ξ}$, t ) is the bipartite entanglement at time t . 

Lemma IV.2. The length 2 n + 1 of the minimal stabi- lizer generators grows asymptotically with 

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for any centered CQCA T and any translation-invariant pure stabilizer state ω$_{ξ}$ . 

Proof. We know that CQCAs map pure translation-in- variant stabilizer states onto pure translation-invariant stabilizer states. The image of a state with stabilizer generators S = { w ( τ$_{x}$ξ ) , x ∈ $^{Z}$} under the action of t steps of a CQCA T is a state with stabilizer gen- erators S t = { w ( τ$_{x}$ t $^{t}$ξ ) , x ∈ $^{Z}$} . Furthermore, we know that any stabilizer state can be generated from the “all-spins-up” state by a CQCA b . So we have S t = { w ( τ$_{x}$ t $^{t}$b (0 , 1)) , x ∈ $^{Z}$} . The length of the sta- bilizer generator is determined by the highest order of the stabilizer generator polynomials, dg( ξ ). Namely, the stabilizer generator is of length 2 · dg( ξ ) + 1. So, we have to calculate dg( t $^{t}$ξ ) = dg( t $^{t}$b 0 1 ). 

( ) For an arbitrary product of CQCAs ∏ k i $_{=1}$t$_{i}$ we can define the series ( a$_{l}$ )$_{1}$$_{≤}$$_{l}$$_{≤}$$_{k}$ = dg( ∏ l i $_{=1}$t$_{i}$ ). It is subaddi- tive, i.e. a$_{n}$$_{+}$$_{m}$ ≤ a$_{n}$ + a$_{m}$ , because the concatenation of CQCAs is essentially the multiplication and addition of polynomials, which is subadditive in the exponents. For subadditive series a$_{n}$ Fekete’s lemma [11] states that the limit lim$_{n}$$_{→∞}$ a n n exists. In our case the series is always positive, so the limit is positive and finite. An easy way to determine the limit is to take a subseries, which of course has the same limit. The subseries of the t = 2 $^{k}$th ( k ∈ $^{N}$) steps is a good candidate, because we can make use of the Cayley-Hamilton theorem to obtain 

Furthermore, we have 

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