A quantum system which evolves in time while interacting with an external environ- ment is said to be an open quantum system (OQS), and the influence of the environment on the unperturbed unitary evolution of the system generally leads to non-unitary dynamics. This kind of open system dynamical evolution has been typically modeled by a Standard Prescription (SP) which assumes that the state of the OQS is initially uncorrelated with the environment state. It is here shown that when a minimal set of physically motivated assumptions are adopted, not only does there exist constraints on the reduced dynamics of an OQS such that this SP does not always accurately describe the possible initial cor- relations existing between the OQS and environment, but such initial correlations, and even entanglement, can be witnessed when observing a particular class of reduced state transformations termed purity extractions are observed. Furthermore, as part of a more fundamental investigation to better understand the minimal set of assumptions required to formulate well defined reduced dynamical quantum maps, it is demonstrated that there exists a one-to-one correspondence between the set of initial reduced states and the set of admissible initial system-environment composite states when G-consistency is enforced. Given the discussions surrounding the requirement of complete positivity and the reliance on the SP, the results presented here may well be found valuable for determining the ba- sic properties of reduced dynamical maps, and when restrictions on the OQS dynamics naturally emerge.
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 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 an open quantum system is often modeled by introducing a bath initially in a product state with the system, letting the system and bath evolve unitarily together, and then tracing over the bath. However, often this model may not accurately reflect a particular experimental situation. Here, we provide some restrictions on one's ability to model an open quantum system using an initial product state when some information about the system-bath interaction is known. For instance, when certain symmetries exist in the system-bath interaction, we compute limitations on how much the system's purity can be increased if the bath is initially uncorrelated white noise. Furthermore, when the system and bath are qubits, we find that effectively only swap and product unitaries always generate system dynamics capable of being modeled using an initial product state. Finally, we show how any initial correlations between the system and environment are detectable, in principle, by observing the system transformation alone during certain joint evolutions. Our results have application in experimental quantum control and quantum computing where the system and environment are often assumed to be initially uncorrelated.
Standard theoretical descriptions of the dynamics of open quantum systems rely on the assumption that the correlations with the environment can be neglected at some reference (initial) time. While being reasonable in specific instances, such as when the coupling between the system and the environment is weak or when the interaction starts at a distinguished time, the use of initially uncorrelated states is questionable if one wants to deal with general models, taking into account the mutual influence that the open-system and environmental evolutions perform on each other. Here we introduce a perturbative method that can be applied to any microscopic modeling of the system-environment interaction, including fully general initial correlations. Extending the standard technique based on projection operators that single out the relevant part of the global dynamics, we define a family of projections adapted to a convenient decomposition of the initial state, which involves a convex mixture of product operators with proper environmental states. This leads us to characterize the open-system dynamics via an uncoupled system of differential equations, which are homogeneous and whose number is limited by the dimensionality of the open system, for any kind of initial correlations. Our method is further illustrated by means of two case studies, for which it reproduces the expected dynamical behavior in the long-time regime more consistently than the standard projection technique.
In the study of open quantum systems modeled by a unitary evolution of a bipartite Hilbert space, we address the question of which parts of the environment can be said to have a "classical action" on the system, in the sense of acting as a classical stochastic process. Our method relies on the definition of the Environment Algebra, a relevant von Neumann algebra of the environment. With this algebra we define the classical parts of the environment and prove a decomposition between a maximal classical part and a quantum part. Then we investigate what other information can be obtained via this algebra, which leads us to define a more pertinent algebra: the Environment Action Algebra. This second algebra is linked to the minimal Stinespring representations induced by the unitary evolution on the system. Finally in finite dimension we give a characterization of both algebras in terms of the spectrum of a certain completely positive map acting on the states of the environment.