Explicit definition of $\mathcal{PT}$ symmetry for non-unitary quantum walks with gain and loss

TL;DR

This paper explicitly defines $c0b4$ symmetry for non-unitary quantum walks with gain and loss, providing conditions for symmetry preservation and confirming experimental observations of $c0b4$ symmetry in such systems.

Contribution

It introduces an explicit $c0b4$ symmetry definition for non-unitary quantum walks using symmetry time frames and identifies conditions for symmetry retention with position-dependent parameters.

Findings

Derived necessary and sufficient conditions for $c0b4$ symmetry in non-unitary quantum walks.

Confirmed the presence of $c0b4$ symmetry in recent experimental setups.

Revealed additional embedded symmetries in the time-evolution operator.

Abstract

$\mathcal{PT}$ symmetry, that is, a combined parity and time-reversal symmetry is a key milestone for non-Hermite 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 non-unitary 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 non-unitary 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 non-unitary quantum walk in the experiment does have $\mathcal{PT}$ symmetry.