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CONTROLLABILITY OF OPEN QUANTUM SYSTEMS

Andreea Grigoriu, HERSCHEL A. RABITZ, Gabriel Turinici

Published May 11, 2012• 1 citation
OpenAlex

Abstract

The problem of controllability of open quantum systems (i.e., quantum systems interacting with an environment), whose dynamics is described by a non-Markowian master equation is addressed. The manipulations of the dynamics is realized with both a laser eld and a tailored nonequilibrium, and generally time-dependent, state of the surrounding environment. Lie algebra theory is used to characterize the structures of the reachable states sets and to prove controllability. The theoretical results are supported by examples.

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The problem of controllability of open quantum systems (i.e., quantum systems interacting with an environment), whose dynamics is described by a non-Markowian master equation is addressed. The manipulations of the dynamics is realized with both a laser eld and a tailored nonequilibrium, and generally time-dependent, state of the surrounding environment. Lie algebra theory is used to characterize the structures of the reachable states sets and to prove controllability. The theoretical results are supported by examples.
In this thesis, we study the controllability and accessibility properties of quantum systems. First, we provide a new proof of the fact that, in general, open quantum systems are not controllable. By controllability we mean that for every pair of initial and target states of the system there exists a finite time T>0 and a control u such that by adding u into the system’s evolution the target state can be reached from the initial state in time T. We then consider a bipartite open system A+B consisting of two interacting qubits A and B under the assumption that only A is directly affected by control u and only A interacts with the environment. We investigate the controllability properties of B. We show that target pure states of B are not reachable in finite time from initial mixed states of B when the initial state of A+B lies in the interior of the set of states. We then examine whether it is possible to maintain the state of B pure. For three specific couplings between A and B we find that the only admissible trajectories are either unaffected by the control u or constant. These results constitute a first negative answer regarding the indirect manipulation of B. Finally, we study the accessibility of B, that is, whether for every initial state of B the attainable set contains an open subset of the set of states of B. For this property, we provide a sufficient condition that can also be applied to more general settings, such as those arising in geometric control theory.
Precisely characterizing and controlling realistic quantum systems under noises is a challenging frontier in quantum sciences and technologies. In developing reliable controls for open quantum systems, one is often confronted with the problem of the lack of knowledge on the system controllability. The purpose of this paper is to give a numerical approach to this problem, that is, to approximately compute the reachable set of states for coherently controlled quantum Markovian systems. The approximation consists of setting both upper and lower bounds for system's reachable region of states. Furthermore, we apply our reachability analysis to the control of the relaxation dynamics of a two-qubit nuclear magnetic resonance spin system. We implement some experimental tasks of quantum state engineering in this open system at a near optimal performance in view of purity: e.g., increasing polarization and preparing pseudopure states. These results demonstrate the usefulness of our theory and show interesting and promising applications of environment-assisted quantum dynamics.
Quantum mechanics has both deepened our understanding of the physical reality and allowed development of quantum technologies beating their classical counterparts in efficiency, security, and sensitivity. As a major downside, the quantum properties behind these technologies are very sensitive to the effects the system’s interaction with its environment. The theory of open quantum systems describes how such interaction influences the system, and as a consequence, how the quantum features are lost, controlled, and possibly returned by manipulating the environment. This thesis begins with a short description of the most relevant features and results of the open quantum system theory. We continue by presenting some commonly used measures to quantify the information encoded in quantum systems and the correlations between spatially or temporally separated parties. The literature review ends with discussion of non-Markovianity and memory effects which combine the fields of open quantum systems and quantum information in a natural way. The rest of the thesis presents the main results of Publications I – VII. We study the memory effects and physicality conditions of open system dynamics described by the vast family of phase-covariant master equations. We show how the open quantum system picture can be exploited to design probing protocols to extract information without knowing how the probe and system interact. Such protocols are considered also from the point-of view of efficiently obtaining the information required in the probing, both in theory and by experimental implementation. The experiment serves also as evidence for applicability of recently developed geometric tools to design informationally incomplete measurements for solving so-called membership problems. We present and experimentally realize a protocol where control over intial correlations between polarization and frequency gives full freedom to implement any dephasing dynamics in the open system polarization, thus dictating how its information content changes in time. We study the two-qubit memory effects of dephasing dynamics and compare them to the effect of such noise in the transmission efficiency in superdense coding. Optimal noise configurations in different situations are found. Finally, we characterize the families of initial states for which the decay of quantum correlations depends or does not depend on the noise location.
A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S+A, extending previous results, and prove that complete controllability of S+A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states.
A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S + A, extending previous results, and prove that complete controllability of S + A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states.