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

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