Mathematisches Kolloquium

Zusammenfassung: We consider discrete time quantum evolutions on a 1D lattice, which satisfy a strict locality condition. On the one hand these are quantum walks, i.e., evolutions of a single particle on the lattice and, on the other hand, quantum cellular automata describing either an interacting many-particle walk (lattice gas) or a spin system. Mathematically, a cellular automaton is an automorphism of a quasi-local algebra taking strictly local observables to strictly local observables. Translation invariance is not assumed. For both, walks and cellular automata, we want to characterize the following equivalence classes: (1) equality up to partitioned evolutions, for which the system is decoupled into finite size local subsystems (2) connection by a continuous path of evolutions with uniformly bounded neighbourhood size and (3) equality of all "locally computable invariants", i.e., quantities which can be determined from any sufficiently large region of the system, independently of its location. It turns out that in one-dimension these classes coincide and are completely characterized by an index quantity, which takes integer values for quantum walks and positive rational values for cellular automata. In either case the shifts are prototypes of evolutions with non-trivial index.

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