The reduced dynamics of the system S, interacting with the environment E, is not given by a linear map, in general. However, if it is given by a linear map, then this map is also Hermitian. In order that the reduced dynamics of the system is given by a linear Hermitian map, there must be some restrictions on the set of possible initial states of the system-environment or on the possible unitary evolutions of the whole SE. In this paper, adding an ancillary reference space R, we assign to each convex set of possible initial states of the system environment S, for which the reduced dynamics is Hermitian, a tripartite state ωRSE, which we call the reference state, such that the set S is given as the steered states from the reference state ωRSE. The set of possible initial states of the system is also given as the steered set from a bipartite reference state ωRS. The relation between these two reference states is as ωRSE = idR ⊗ ΛS(ωRS), where idR is the identity map on R and ΛS is a Hermitian assignment map, from S to SE. As an important consequence of introducing the reference state ωRSE, we generalize the result of Buscemi (2014 Phys. Rev. Lett. 113 140502): we show that, for a U-consistent subspace, the reduced dynamics of the system is completely positive, for arbitrary unitary evolution of the whole system-environment U, if and only if the reference state ωRSE is a Markov state. In addition, we show that the evolution of the set of system-environment (system) states is determined by the evolution of the reference state ωRSE (ωRS).

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