An Operator growth hypothesis for open quantum systems

 Under the Hamiltonian evolution, a simple operator turns into a complicated operator. The growth of such an operator is drastically different when the system is connected to a dissipative environment than in a typical closed system. We probe such growth via a recently explored measure of scrambling known as Krylov complexity and aim to propose an operator growth hypothesis in open quantum systems. We review the current progress and challenges of applying this approach to open systems and illustrate some examples from the dissipative SYK model.