Shannon Entropy and Diffusion Coeffcient in Parity-Time Symmetric Quantum Walks

TL;DR

This paper investigates how diffusion coefficients and Shannon entropy can serve as bulk indicators of topological phase transitions in PT-symmetric non-Hermitian quantum walks, revealing unique features without the need for boundary inhomogeneity.

Contribution

It introduces the use of diffusion coefficients and Shannon entropy as bulk probes for topological phases in non-Hermitian quantum walks, expanding the tools for studying topological states.

Findings

Diffusion coefficient shows abrupt changes near topological transitions.

In PT-symmetric-broken phase, diffusion peaks at phase transition.

Shannon entropy correlates with diffusion coefficient.

Abstract

Non-Hermitian topological edge states have many intriguing properties, but have so far mainly been discussed in terms of bulk-boundary correspondence. Here we propose to use a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient is found to show unique features with the topological phase transitions driven by paritytime( PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite artificial boundaries are not constructed by inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated to the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented here may open up a new avenue for studying the topological state in Non-Hermitian quantum walk systems.