$\mathcal{PT}$ symmetry, that is, a combined parity and time-reversal symmetry, is a key milestone for non-Hermitian systems exhibiting entirely real eigenenergy. In the present work, motivated by a recent experiment, we study $\mathcal{PT}$ symmetry of the time-evolution operator of nonunitary quantum walks. We present the explicit definition of $\mathcal{PT}$ symmetry by employing a concept of symmetry time frames. We provide a necessary and sufficient condition so that the time-evolution operator of the nonunitary quantum walk retains $\mathcal{PT}$ symmetry even when parameters of the model depend on position. It is also shown that there exist extra symmetries embedded in the time-evolution operator. Applying these results, we clarify that the nonunitary quantum walk in the experiment does have $\mathcal{PT}$ symmetry.

• Received 30 March 2016

DOI:https://doi.org/10.1103/PhysRevA.93.062116