Reference state for arbitrary U-consistent subspace

Abstract

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 $\mathcal{S}$, for which the reduced dynamics is Hermitian, a tripartite state ${\omega }_{RSE}$, which we call it the reference state, such that the set $\mathcal{S}$ is given as the steered states from the reference state ${\omega }_{RSE}$,. The set of possible initial states of the system is also given as the steered set from a bipartite reference state ${\omega }_{RS}$. The relation between these two reference states is as ${\omega }_{RSE}=i{d}_R\otimes {\Lambda }_S({\omega }_{RS})$, where $i{d}_R$ is the identity map on $R$ and ${\Lambda }_S$ is a Hermitian assignment map, from $S$ to $SE$. As an important consequence of introducing the reference state ${\omega }_{RSE}$, 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 ${\omega }_{RSE}$ 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 ${\omega }_{RSE}$ (${\omega }_{RS}$).

9 pages, Theorems 3, 4 and 5, on the CP-ness of the reduced dynamics, are added

References in corpus (6)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Vanishing quantum discord is necessary and sufficient for completely positive maps
• Vanishing quantum discord is not necessary for completely-positive maps
• Completely positive maps within the framework of direct-sum decomposition of state space
• Structure of correlated initial states that guarantee completely positive reduced dynamics
• The reference system and not completely positive open quantum dynamics

Cited by in corpus (5)

• Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear
• Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy
• When the assignment map is completely positive
• Positivity of the assignment map implies complete positivity of the reduced dynamics
• Initial correlations in open quantum systems are always detectable