@article{GrossNesmeVogtsetal.2012,
  author    = {Gross, David and Nesme, V. and Vogts, H. and Werner, Reinhard F.},
  title     = {Index theory of one dimensional quantum walks and cellular automata},
  series = {Communications in mathematical physics},
  volume    = {310},
  journal   = {Communications in mathematical physics},
  number    = {2},
  publisher = {Springer},
  address   = {New York},
  issn      = {0010-3616},
  doi       = {10.1007/s00220-012-1423-1},
  pages     = {419 -- 454},
  year      = {2012},
  abstract  = {If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S-1, S-2 can be "pieced together", in the sense that there is a system S which acts like S-1 in one region and like S-2 in some other region, if and only if S-1 and S-2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S-1 into S-2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S bar right arrow. ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.},
  language  = {en}
}
@article{GuetschowNesmeWerner2012,
  author    = {Guetschow, Johannes and Nesme, Vincent and Werner, Reinhard F.},
  title     = {Self-similarity of cellular automata on abelian groups},
  series = {Journal of cellular automata},
  volume    = {7},
  journal   = {Journal of cellular automata},
  number    = {2},
  publisher = {Old City Publishing Science},
  address   = {Philadelphia},
  issn      = {1557-5969},
  pages     = {83 -- 113},
  year      = {2012},
  abstract  = {It is well known that the spacetime diagrams of some cellular automata have a self-similar fractal structure: for instance Wolfram's rule 90 generates a Sierpinski triangle. Explaining the self-similarity of the spacetime diagrams of cellular automata is a well-explored topic, but virtually all of the results revolve around a special class of automata, whose typical features include irreversibility, an alphabet with a ring structure, a global evolution that is a ring homomorphism, and a property known as (weakly) p-Fermat. The class of automata that we study in this article has none of these properties. Their cell structure is weaker, as it does not come with a multiplication, and they are far from being p-Fermat, even weakly. However, they do produce self-similar spacetime diagrams, and we explain why and how.},
  language  = {en}
}