Reference state for arbitrary U-consistent subspace

TL;DR

This paper introduces a reference state framework for analyzing the conditions under which the reduced dynamics of a system interacting with an environment is Hermitian and completely positive, generalizing previous results with new state constructions.

Contribution

It proposes a tripartite reference state approach to characterize initial system-environment states and extends prior work by linking Markov states to completely positive dynamics.

Findings

The reference state $ ext{ω}_{RSE}$ is key to determining Hermitian and CP reduced dynamics.

The set of initial states can be represented as steered states from the reference state.

Complete positivity of dynamics is equivalent to the reference state being a Markov state.

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 $\omega_{RSE}=id_{R}\otimes \Lambda_{S}(\omega_{RS})$, where $id_{R}$ 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 $\omega_{RSE}$ ($\omega_{RS}$).