We study topological phases in one-dimensional open Floquet systems driven by chiral symmetric nonunitary time evolution. We derive a procedure to calculate topological numbers from nonunitary time-evolution operators with chiral symmetry. While the procedure has been applied to open Floquet systems described by nonunitary time-evolution operators, we give the microscopic foundation and clarify its validity. We construct a model of chiral symmetric nonunitary quantum walks classified into class BDI† or AIII, which is one of the enlarged symmetry classes for topological phases in open systems based on experiments of discrete-time quantum walks. Then we confirm that the topological numbers obtained from the derived procedure give correct predictions of the emergent edge states. We also show that the model retains PT symmetry in certain cases, and its dynamics is crucially affected by the presence or absence of PT symmetry.
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