  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.

Quantum Physics;Mesoscale and Nanoscale Physics;Optics