Topological phases and the associated multiple edge states are studied for parity and time-reversal ($\mathcal{PT}$)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.

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Dissipative dynamics of quantum systems can be classified topologically based on the correspondence between the Lindbladian in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation and the non-Hermitian Hamiltonian in the Schrödinger equation. While general non-Hermitian Hamiltonians are classified into 38 symmetry classes, the previous work shows that the Lindbladians are classified into 10 symmetry classes due to physical constraints. In this work, however, we unveil a topological classification of Lindbladians based on sublattice symmetry (SLS), which is not previously considered and can increase the number of symmetry classes for the Lindbladians. We introduce shifted SLS so that the Lindbladian can retain this symmetry and take on the same role of SLS for the topological classification. For verification, we construct a model of the dissipative quantum system retaining shifted SLS and confirm the presence of edge states protected by shifted SLS. Moreover, the relationship between the presence of shifted SLS protected edge states and dynamics of an observable quantity is also discussed.

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