论文标题
手性单位和分阶量子步行的索引理论
Index Theory of Chiral Unitaries and Split-Step Quantum Walks
论文作者
论文摘要
Cedzich等人的工作构建。和Suzuki等人，我们考虑了手性单位的拓扑和索引理论特性，这是手性 - 对称自我预性操作员的时间演变的抽象。拆分步行量子步行提供了丰富的示例。我们使用一对投影的索引和Cayley Transform来定义希尔伯特空间和希尔伯特$ C^*$ - 模块的手性统一的拓扑指数。在类似哈密顿的运营商的离散时间演变的情况下，我们将手性单位的指数与哈密顿式的指数联系起来。我们还证明了在Hilbert $ c^*$ - 模块上的各向异性分式步骤量子步行的双面绕组数字公式，该模块扩展了Matsuzawa的结果。
Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^*$-modules, extending a result by Matsuzawa.
