5 Example III.1. The stabilizer group with the gener- ators S = { X ⊗ X, Z ⊗ Z } stabilizes the Bell state ψ = 1 $_{/}$√$_{2(}$ | 1 , 1 〉 + | 0 , 0 〉 ) . To check this, we just have to apply the stabilizer generators to the state (we omit the normalization): ( X ⊗ X ) ψ = ( X ⊗ X )( | 1 〉 ⊗ | 1 〉 ) + ( X ⊗ X )( | 0 〉 ⊗ | 0 〉 ) = | 0 , 0 〉 + | 1 , 1 〉 = ψ ( Z ⊗ Z ) ψ = ( Z ⊗ Z )( | 1 〉 ⊗ | 1 〉 ) + ( Z ⊗ Z )( | 0 〉 ⊗ | 0 〉 ) = ( − | 1 〉 ) ⊗ ( − | 1 〉 ) + | 0 〉 ⊗ | 0 〉 = ( − 1) $^{2}$| 1 , 1 〉 + | 0 , 0 〉 = ψ ♦ For infinitely many qubits we can’t just write down the state, so we use an abstract definition. A translation- invariant stabilizer state is defined by a translation-in- variant set of operators S = { w ( τ$_{x}$ξ ) , x ∈ $^{Z}$} , where τ$_{x}$ is the lattice translation by x sites. In [1] it was proven that such a set defines a pure translation-invariant sta- bilizer state if and only if ξ is reflection invariant and the Laurent polynomials of ξ have no common divisors: gcd( ξ$_{+}$, ξ$_{−}$ ) = 1. For the polynomials, this means that 1. ξ is of odd length, because reflection-invariant Lau- rent polynomials of even length are always divisible by 1 + u . We will write l = 2 n + 1. 2. ξ (0) = ( 0 0 ) ; thus, the center element is not 1 . Else ξ would have the divisor u − 1 + u . An example is S$_{i}$ = X$_{i}$$_{−}$$_{1}$ ⊗ Z$_{i}$ ⊗ X$_{i}$$_{+1}$ . The condi- tion that the polynomials of ξ are coprime is also a condition for the column vectors of a CQCA matrix. This indicates a close connection between CQCAs and transla-tion-invariant stabilizer states. Indeed, CQCAs map pure translation-invariant stabilizer states onto each other. Furthermore, every translation-invariant stabilizer state can be generated by a single step of a CQCA from the “all-spins-up” state, which is a stabilizer state with stabilizer generators S = { w ( τ$_{x}$ (0 , 1)) = Z$_{x}$, x ∈ $^{Z}$} . Thus, we can study the entanglement generation proper- ties of CQCAs acting on pure translation-invariant sta- bilizer states by just studying the entanglement in these states. 3. At least two different types of Pauli matrices (both different from the identity) have to occur (e.g. X and Y ). Otherwise, ξ$_{+}$ = 0 or ξ$_{−}$ = 0 or ξ$_{+}$ = ξ$_{−}$ , each case implying common divisors. Its trace is tr t$_{F}$ = u − 1 + 1 + u . The time evolution that is shown in Figure 3 is very similar for all CQCAs with the same trace. This is observed also for other classes and subclasses of CQCAs. Mathematically, however, in general there is no notion of equivalence known for CQ- CAs with the same trace. An exception are CQCAs with trace u − 1 + u , which can be shown to be all equivalent in the sense that they can all be transformed into the standard glider CQCA (7) via space-time evolutions as shown in Figure 3. The class of a CQCA is determined by the trace of its matrix. If the trace is a constant tr t = c, c ∈ $^{Z$_{2}$}$, the automaton is periodic. If it is of the form tr t = u − n + u $^{n}$, n ∈ $^{N}$, the CQCA has gliders that move n steps on the lattice every time step. All other cases show fractal behavior [2]. We have already seen an example of a glider CQCA. The following CQCA exhibits fractal behavior: Figure 3: Time evolution of the fractal CQCA T F with tr t F = u − 1 + 1 + u . Time increases upwards. The differ- ent colors mark the different Pauli matrices. t$_{F}$ = ( u − 1 + 1 + u 1 1 0 ) . (9) t = bt$_{G}$b − 1 (10) where b is a CQCA [2]. III. PURE TRANSLATION-INVARIANT STABILIZER STATES A. Definition B. Entanglement A stabilizer state ω is a common eigenstate to a group of commuting operators S = 〈{ S$_{i}$ }〉 . This means that ω ◦ S = ω, ∀ S ∈ S . The group is generated by the set of generators $^{S}$= { S$_{i}$ } by multiplication. It therefore suffices to check the stabilizer condition ω ◦ S = ω for the generators S$_{i}$ . For finitely many qubits, this can be easily understood with the following example: Entanglement in the stabilizer formalism was first studied in [10]. It was found that the entropy of en- tanglement with respect to a bipartite cut of some sta- bilizer state is exactly the number of generators of the correlation subgroup. The correlation subgroup S AB for