Unified Framework for Direct and Complete Characterization of an Unknown Kraus Operator and Density Matrix Using a Single Input State

Characterization of quantum measurements and dynamical processes is typically performed using pure state preparations. However, in realistic experimental settings, the preparation of pure states is often infeasible due to noise and system constraints. In this work, we present a unified framework that enables the direct and complete characterization of an unknown Kraus operator using only a single input state. The same framework also supports the characterization of unknown observable, unitary operator, and density matrix. Remarkably, all these tasks are accomplished using a single input state, a set of projector-based unitary evolution operators, and the measurement of a single observable. Importantly, our approach imposes no constraints on the strength of the coupling between the system and a probe.