The dynamics of a closed quantum system, under a unitary time evolution
U, is obviously linear. But, the reduced dynamics of an open quantum system
S, interacting with an environment
E, is not linear, in general. Dominy et al. [Quantum Inf. Process. 15, 465 (2016)] considered the case that the set $\mathcal{S}={{\ensuremath{\rho}}_{SE}}$ of possible initial states of the system-environment is convex, and also possesses another property, which they called
U-consistency. They have shown that, under such circumstances, the reduced dynamics of the system
S is linear. Whether the Dominy-Shabani-Lidar framework is the most general one is the subject of this paper. We assume that the reduced dynamics is linear and show that this leads us to their framework. In other words, the reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework.
Consider an open quantum, system interacting with its environment. Whether the reduced dynamics of the system can be given by a linear map or not is an important question in the theory of open quantum systems. Dominy, Shabani and Lidar have proposed a general framework for linear Hermitian reduced dynamics. In addition, it has been shown that their framework is, in fact, the most general one: The reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework. This result has been given in a rather abstract way. Here, we want to give another proof for it, in a more illustrative manner.