TY - JOUR

T1 - Index theory of one dimensional quantum walks and cellular automata

AU - Gross, D.

AU - Nesme, V.

AU - Vogts, H.

AU - Werner, R. F.

PY - 2012

Y1 - 2012

N2 - 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, S can be pieced together, in the sense that there is a system S which locally acts like S in one region and like S in some other region, if and only if S and S 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 into S. 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 and the prototypes of systems with non-trivial index are shifts.

AB - 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, S can be pieced together, in the sense that there is a system S which locally acts like S in one region and like S in some other region, if and only if S and S 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 into S. 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 and the prototypes of systems with non-trivial index are shifts.

U2 - 10.1007/s00220-012-1423-1

DO - 10.1007/s00220-012-1423-1

M3 - Article

VL - 310

SP - 419

EP - 454

JO - Comm. Math. Phys.

JF - Comm. Math. Phys.

SN - 1432-0916

ER -