Explicit definition of PT symmetry for nonunitary quantum walks with gain and loss

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 PT symmetry of the time-evolution operator of nonunitary quantum walks. We present the explicit definition of 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 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 PT symmetry.