一维量子行走的指标理论与元胞自动机
Index theory of one dimensional quantum walks and cellular automata
摘要：
如果一个一维量子晶格系统受到可逆离散时间动力学的一个步骤的影响，那么直观地说，从左边进入任何给定单元块的“量子信息“都必须从右边离开该块。对于两种类型的系统--即量子行走和元胞自动机--我们通过定义一个指数来精确地表达这种直觉，这个指数是一个衡量系统中“量子信息净流量“的量。该指数提供了离散动力学的两个属性的完整表征。第一，两个系统S_1、S_2可以被拼接在一起，在这个意义上，存在一个系统S，它在一个区域中局部地像S_1那样起作用，而在另一个区域中像S_2那样起作用，当且仅当S_1和S_2具有相同的指数。其次，指数标记了这类系统的连通分支：指数相等是S_1到S_2的连续变形存在的充分必要条件。在量子行走的情况下，指数是整数值的，而对于元胞自动机，它取正有理数群中的值。在这两种情况下，映射S -> ind S是一个群同态，如果离散动力学的合成被视为量子系统的群律。平凡指标系统是指那些可以用分块么正实现的系统，而非平凡指标系统的原型是移位。
英文摘要：
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.
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