Operator Growth and Spread Complexity in Open Quantum Systems

Commonly, the notion of "quantum chaos” refers to the fast scrambling ofinformation throughout complex quantum systems undergoing unitary evolution.Motivated by the Krylov complexity and the operator growth hypothesis, wedemonstrate that the entropy of the population distribution for an operator intime is a useful way to capture the complexity of the internal informationdynamics of a system when subject to an environment and is, in principle,agnostic to the specific choice of operator basis. We demonstrate itseffectiveness for the Sachdev-Ye-Kitaev (SYK) model, examining the dynamics ofthe system in both its Krylov basis and the basis of operator strings. We provethat the former basis minimises spread complexity while the latter is aneigenbasis for high dissipation. In both cases, we probe the long-time dynamicsof the model and the phenomenological effects of decoherence on the complexityof the dynamics.