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 this paper we return to the problem of reduced-state dynamics in the presence of an interacting environment. The question we investigate is how to appropriately model a particular system evolution given some knowledge of the system-environment interaction. When the experimenter takes into account certain known features of the interaction such as its invariant subspaces or its non-local content, it may not be possible to consistently model the system evolution over a certain time interval using a standard Stinespring dilation, which assumes the system and environment to be initially uncorrelated. Simple examples demonstrating how restrictions can emerge are presented below. When the system and environment are qubits, we completely characterize the set of unitaries that always generate reduced dynamics capable of being modeled using a consistent Stinespring dilation. Finally, we show how any initial correlations between the system and environment can be certified by observing the system transformation alone during certain joint evolutions.
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.
The title may be a bit misleading. Perhaps, "On the Complete Positivity of Reduced Quantum Dynamics," would be a more fitting title. Determining whether or not completely positive (CP) maps are required to describe open system quantum dynamics is an extremely important issue concerning the fundamental mathematical foundations of QM, as well as many other areas of physics. it had been typically believed that only CP maps actually describe the dynamical evolution of open quantum systems, however there has been speculation as to whether this is a strict constraint on the mathematical and physical structure of stochastic quantum dynamical maps. The objective of this thesis is to demonstrate that given a particular unitary operator, an initial system state, a final system state, and the dimension of the environment state, there exists no CP map with a composite system-environment, product initial state that is compatible with the given constraints on the reduced quantum dynamics of the system under investigation.
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.
It is now increasingly realized in the study of open system dynamics that initial correlations do not pose a conceptual difficulty as traditionally believed. A similar methodology as used to describe initial product states can be adopted, with the only difference being that the reduced dynamics is possibly not completely positive, entailing that only a restricted set of initial reduced states lead to a physically valid dynamics. Here we study the interplay between the initial correlations, especially in the form of highly entangled states, and the system–environment unitary. In particular, for almost any initial entangled state, one can furnish infinitely many joint unitaries that generate CP dynamics on the system. Restricting to the case of initial, pure entangled states, we obtain the scaling of the dimension of the set of these unitaries and show that it is of zero measure in the set of all possible interaction unitaries.
We study the role of correlations with the environment as the source of non-Markovian quantum evolutions. We first focus on the impact that correlations with the environment can have on the dynamical map that evolve the system. We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We discuss the implications and limitations of the Markov approximation necessary to derive the Kossakowski-Lindblad master equation. A generalized non-Markovian master equation is derived from the dynamical map of systems correlated with their environment. The physical meaning of not completely positive maps is studied to obtain a consistent theory of non-Markovian quantum dynamics. These are associated to inverse maps necessary to establish correlations and they give rise to a canonical embedding map that is local in time. This master equation goes beyond the Kossakowski-Lindblad master equation. Non-equilibrium quantum thermodynamics can be be studied within this theory. Through out this discussion, the general dynamics of two interacting qubits is used as an example for illustrations.