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 , for which the reduced dynamics is Hermitian, a tripartite state , which we call the reference state , such that the set is given as the steered states from the reference state . The set of possible initial states of the system is also given as the steered set from a bipartite reference state . The relation between these two reference states is as , where is the identity map on R and is a Hermitian assignment map, from S to SE . As an important consequence of introducing the reference state , 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 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 ( ).
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 , for which the reduced dynamics is Hermitian, a tripartite state , which we call the reference state , such that the set is given as the steered states from the reference state . The set of possible initial states of the system is also given as the steered set from a bipartite reference state . The relation between these two reference states is as , where is the identity map on R and is a Hermitian assignment map, from S to SE . As an important consequence of introducing the reference state , 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 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 ( ).
Consider the set $\mathcal{S}={{\ensuremath{\rho}}_{SE}}$ of possible initial states of the system-environment, steered from a tripartite reference state ${\ensuremath{\omega}}_{RSE}$. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)] showed that the reduced dynamics of the system, for each ${\ensuremath{\rho}}_{S}\ensuremath{\in}{\mathrm{Tr}}_{E}\mathcal{S}$, is always completely positive if and only if ${\ensuremath{\omega}}_{RSE}$ is a Markov state. There, during the proof, it has been assumed that the dimensions of the system and the environment can vary through the evolution. Here, we show that this assumption is necessary: we give an example for which, though ${\ensuremath{\omega}}_{RSE}$ is not a Markov state, the reduced dynamics of the system is completely positive, for any evolution of the system-environment during which the dimensions of the system and the environment remain unchanged. As our next result, we show that the result of Muller-Hermes and Reeb [A. Muller-Hermes and D. Reeb, Ann. Henri Poincare 18, 1777 (2017)], of monotonicity of the quantum relative entropy under positive maps, cannot be generalized to the Hermitian maps, even within their physical domains.
In the simplest approximation, the reduced dynamics of a quantum system S interacting with its environment E is considered to be given by a completely positive map. But, in general, this is not the case. In fact, the reduced dynamics of the system in not even linear, in general. Whether the reduced dynamics is linear or not is determined by two factors: the set of possible initial states of the system-environment $\mathcal{S}=\left\lbrace ρ_{SE} \right\rbrace $, and the joint system-environment unitary evolution U. When U is factorisable as U=US⊗UE, then we can choose S=D, where D is the set of all system-environment density operators. In other words, when U is factorisable, the reduced dynamics of the system S is linear (in fact unitary) for arbitrary initial state of the system-environment ρSE. We show that this result cannot be generalized to any non-factorisable U: For any non-factorisable unitary evolution of the whole system-environment U, the set S must be chosen as a proper subset of D to achieve linear reduced dynamics. As a byproduct, considering a convex set of possible initial states of the system-environment S such that TrES=TrED, we show that when the reduced dynamics of the system, for one system-environment unitary evolution U1, is positive, but not completely positive, this implies that reduced dynamics is not linear for another U2.
Consider an open quantum system S, interacting with its environment E, and also an ancillary Hilbert space R. The reduced dynamics of the system S is given by a completely positive map, if the set S={ρSE} , of possible initial states of the system-environment, can be written as a steered set from a tripartite Markov state ƮRSE. In this paper, we call such steered states as Markovian states ρSE, and study a physical case, in which the reduced dynamics of the system can be completely positive, even when the initial states of the system-environment ρSE are non-Markovian.
Consider an open quantum system S , interacting with its environment E. Whether the reduced dynamics of the system can be given by a linear map, or not, is an important subject, in the theory of open quantum systems. Dominy, Shabani and Lidar have proposed a general framework for linear Hermitian reduced dynamics. They have considered the case that both the system and the environment are finite dimensional. Their framework can be generalized to include the case that the environment is infinite dimensional too. In this paper, after demonstrating this generalization, we discuss the role of the convexity of the set, of possible initial states of the system-environment, in their framework. Next, we give a proof for the existence of the operator sum representation, for arbitrary linear Hermitian map. This proof enables us to prove the Choi-Jamiołkowski and the Jamiołkowski isomorphisms, straightforwardly.
Open quantum dynamics in a tripartite scenario including a system, its environment, and a passive reference is shown to resolve several open questions regarding not completely positive (NCP) dynamical maps as valid descriptions of open quantum evolution. The steering states of the system and the environment with respect to the reference, reduced down to a dense, compact set of states of the system alone, provide a well-defined domain of action for a bonafide dynamical map describing the open evolution of the system. The map is not restricted to being completely positive but it preserves the positivity of all states in its domain. NCP open dynamics corresponding to different initial configurations of the tripartite system are explored.
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.