  We present a framework for investigating wave function spreading in
$\mathcal{PT}$-symmetric quantum systems using spread complexity and spread
entropy. We consider a tight-binding chain with complex on-site potentials at
the boundary sites. In the $\mathcal{PT}$-unbroken phase, the wave function is
delocalized. We find that in the $\mathcal{PT}$-broken phase, it becomes
localized on one edge of the tight-binding lattice. This localization is a
realization of the non-Hermitian skin effect. Localization in the
$\mathcal{PT}$-broken phase is observed both in the lattice chain basis and the
Krylov basis. Spread entropy, entropic complexity, and a further measure that
we term the Krylov inverse participation ratio probe the dynamics of wave
function spreading and quantify the strength of localization probed in the
Krylov basis. The number of Krylov basis vectors required to store the
information of the state reduces with the strength of localization. Our results
demonstrate how measures in Krylov space can be used to characterize the
non-hermitian skin effect and its localization phase transition.
