Krylov structure in open quantum systems

In recent times, Krylov complexity has emerged as a probe of measuring the operator growth in quantum systems. The growth of a simple operator in the Heisenberg picture is studied and the average position in the newly formed Krylov basis helps one to define a notion of complexity of operators under Liouvillian evolution. The algorithm of forming the Krylov basis and the Krylov complexity are conjectured to capture interesting physics about chaos and phase transitions. I will explain how this notion can be continued in case of an open system where the Liouvillian is replaced by an appropriate Lindbladian. The way of forming a Krylov basis for a non-unitary evolution is also a bit different, which can be done in various ways (we will discuss Arnoldi and Bilanczos, unlike Lanczos for closed unitary evolution). Based on https://arxiv.org/abs/2207.05347 and ongoing work.