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 it 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 [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)]: 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).

We consider a bipartite quantum system S (including parties A and B), interacting with an environment E through a localized quantum dynamics FSE . We call a quantum dynamics FSE localized if, e.g., the party A is isolated from the environment and only B interacts with the environment: FSE = idA ⊗ FBE, where idA is the identity map on the part A and FBE is a completely positive (CP) map on the both B and E. We will show that the reduced dynamics of the system is also localized as ES = idA ⊗ĒB, whereĒB is a CP map on B, if and only if the initial state of the system-environment is a Markov state. We then generalize this result to the two following cases: when both A and B interact with a same environment, and when each party interacts with its local environment.

We study the dynamics of two lower bounds of concurrence in bipartite quantum systems when one party goes through an arbitrary channel. We show that these lower bounds obey the factorization law similar to that of [Konrad et al., Nat. Phys. 4, 99 (2008)]. We also, discuss the application of this property, in an example.

Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett 106, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurementinduced nonlocality (MIN) for an arbitrary m×n dimensional bipartite density matrix ρ for the case where one of its reduced density matrix, ρ a , is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, ρ a has d degenerate subspaces with dimension mi(mi ≤ m, i = 1, 2, ..., d). We show that according to the degeneracy of ρ a , if we expand ρ in a suitable basis, the evaluation of MIN for an m × n dimensional state ρ, is degraded to finding the MIN in the mi × n dimensional subspaces of state ρ. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary m × n state ρ for which mi ≤ 2, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. In the final, we explain the evaluation of MIN for 3 × n dimensional states in details.

Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary $m\times n$ dimensional bipartite density matrix $\rho$ for the case where one of its reduced density matrix, $\rho^{a}$, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, $\rho^{a}$ has $d$ degenerate subspaces with dimension $m_{i} (m_{i} \leq m , i=1, 2, ..., d)$. We show that according to the degeneracy of $\rho^{a}$, if we expand $\rho$ in a suitable basis, the evaluation of MIN for an $m\times n$ dimensional state $\rho$, is degraded to finding the MIN in the $m_{i}\times n$ dimensional subspaces of state $\rho$. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary $m\times n$ state $\rho$ for which $m_{i}\leq 2$, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. Finally, we explain the evaluation of MIN for $3\times n$ dimensional states in details.