Consider an open quantum, system interacting with its environment. Whether the reduced dynamics of the system can be given by a linear map or not is an important question in the theory of open quantum systems. Dominy, Shabani and Lidar have proposed a general framework for linear Hermitian reduced dynamics. In addition, it has been shown that their framework is, in fact, the most general one: The reduced dynamics of the system is linear if and only if it can be formulated within the Dominy-Shabani-Lidar framework. This result has been given in a rather abstract way. Here, we want to give another proof for it, in a more illustrative manner.
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.
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
TrE S=TrE D, 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.
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.
We study the influence of the preparation of an open quantum system on its reduced time evolution. In contrast to the frequently considered case of an initial preparation where the total density matrix factorizes into a product of a system density matrix and a bath density matrix the time evolution generally is no longer governed by a linear map nor is this map affine. Put differently, the evolution is truly nonlinear and cannot be cast into the form of a linear map plus a term that is independent of the initial density matrix of the open quantum system. As a consequence, the inhomogeneity that emerges in formally exact generalized master equations is in fact a nonlinear term that vanishes for a factorizing initial state. The general results are elucidated with the example of two interacting spins prepared at thermal equilibrium with one spin subjected to an external field. The second spin represents the environment. The field allows the preparation of mixed density matrices of the first spin that can be represented as a convex combination of two limiting pure states, i.e., the preparable reduced density matrices make up a convex set. Moreover, the map from these reduced density matrices onto the corresponding density matrices of the total system is affine only for vanishing coupling between the spins. In general, the set of the accessible total density matrices is nonconvex.
In the study of open quantum systems, one commonly describes the evolution of a system of interest through reduced dynamics, obtained by treating the environment indirectly rather than as a part of the full model. This thesis presents an expository account of an alternative, dilation-based viewpoint in the finite-dimensional setting, where a family of reduced dynamics is represented through unitary evolution on a larger system consisting of the original system together with an ancillary environment. After reviewing the reduced-dynamics perspective and the language of quantum channels, we formulate finite-dimensional quantum dynamics as channel-valued dynamical curves and use this framework to discuss Stinespring dilations of such curves. We then present exact dilation results for analytic dynamical curves, explain the singular behavior that can arise at t=0, and describe approximation results showing that Lipschitz-continuous dynamical curves admit approximate finite-dimensional Stinespring dilations. The thesis therefore provides a mathematically focused introduction to dilation-based modeling of quantum dynamics and argues that a change of perspective can lead to new ways of formulating problems in the theory of open quantum systems.
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).