Index theory of one dimensional quantum walks and cellular automata

TL;DR

This paper introduces an index theory for one-dimensional quantum walks and cellular automata, providing a complete classification of their dynamics based on quantum information flow, with implications for system composition and deformation.

Contribution

It defines a novel index that characterizes quantum information flow, classifies systems by index, and links index values to system properties like shifts and partitioned unitaries.

Findings

The index is integer-valued for quantum walks.

The index takes positive rational values for cellular automata.

The index map is a group homomorphism under composition.

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 locally 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 -> 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.