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In this paper, we provide a complete analysis of the Lie algebra
structure of a system of n interacting spin ½ particles with
different gyromagnetic ratios in an electro-magnetic field. We relate
the structure of this Lie algebra to the properties of a graph whose
nodes represent the particles and an edge connects two nodes if and only
if the interaction between the two corresponding particles is active. We
prove that for these systems all the controllability notions, including
the possibility of driving the state or the evolution operator of the
system, are equivalent. We also give a necessary and sufficient
condition for controllability in terms of the properties of the above
described graph. We provide extensions to the case of possibly equal
gyromagnetic ratios

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The control of the state of a system to a given state has important applications in quantum
engineering and control. In this paper, we study a control scheme that controls the
localization of state in a two-level system via homodyne feedback, and analyse the
condition for the steady state and stability of the localization. We also give the parameters
of the feedback and the driving field for control of the system state to localization at any
desired state on the Bloch sphere.

One of the main theoretical challenges in quantum computing is the design of explicit schemes that enable one to effectively factorize a given final unitary operator into a product of basic unitary operators. As this is equivalent to a constructive controllability task on a Lie group of special unitary operators, one faces interesting classes of bilinear optimal control problems for which efficient numerical solution algorithms are sought for. In this paper we give a review on recent Lie-theoretical developments in finite-dimensional quantum control that play a key role for solving such factorization problems on a compact Lie group. After a brief introduction to basic terms and concepts from quantum mechanics, we address the fundamental control theoretic issues for bilinear control systems and survey standard techniques fromLie theory relevant for quantum control. Questions of controllability, accessibility and time optimal control of spin systems are in the center of our interest. Some remarks on computational aspects are included as well. The idea is to enable the potential reader to understand the problems in clear mathematical terms, to assess the current state of the art and get an overview on recent developments in quantum control-an emerging interdisciplinary field between physics, control and computation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Controllability and observability of multi-spin systems in architectures of various symmetries of coupling type and topology are investigated. We complement recent work of explicitly determining the respective dynamic system Lie algebras and thereby also precise reachability sets under symmetry constraints. Here the focus is on the converse: under which conditions can the absence of symmetry be taken as an indicator of universality of the hardware architecture? More precisely, the absence of symmetry implies irreducibility and provides a convenient necessary condition for full controllability. Though much easier to assess than the well-established Lie-algebra rank condition, this is not sufficient unless in an n-qubit system with connected coupling topology the candidate dynamic simple Lie algebra can be identified uniquely as the full unitary algebra su(2<sup>n</sup>). - Here we discuss simple tests confined to solving homogeneous linear equations in order to filter irreducible unitary representations of other candidate algebras of classical type (orthogonal ones and unitary symplectic ones). Finally, we give an outlook under which conditions algebras of exceptional type can also be ruled out.

This article shows how C-numerical-range related new strucures may arise from practical problems in quantum control – and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function . In quantum control of n qubits one may be interested (i) in having U∈SU(2 n ) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e., to the n-fold tensor product . Interestingly, the latter then leads to a novel entity, the local C-numerical range W loc(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying article on Relative C-Numerical Ranges for Application in Quantum Control and Quantum Information [Dirr, G., Helmke, U., Kleinsteuber, M. and Schulte-Herbrüggen, T., 20081.
Dirr , G ,
Helmke , U ,
Kleinsteuber , M and
Schulte-Herbrüggen , T . 2008. Relative C-numerical ranges for application in quantum control and quantum information. Linear and Multilinear Algebra, 56: 27–51. [Taylor & Francis Online], [Web of Science ®]View all references, Linear and Multilinear Algebra, 56, 27–51]. We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. (3) We conclude by connecting the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient-flow algorithms.

Complete controllability is a fundamental issue in the field of control of quantum systems, not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. In this paper we investigate the question of complete controllability for finite-level quantum systems subject to a single control field, for which the interaction is of dipole form. Sufficient criteria for complete controllability of a wide range of finite-level quantum systems are established and the question of limits of complete controllability is addressed. Finally, the results are applied to give a classification of complete controllability for four-level systems. Comment: 14 pages, IoP-LaTeX

Sufficient conditions for complete controllability of $N$-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems. Comment: 8 pages, expanded and revised version

We show that quantum computation is possible with mixed states instead of pure states as inputs. This is performed by embeddin within the mixed state a subspace that transforms like a pure state and that can be identified by labelling it based on logica (spin), temporal, or spatial degrees of freedom. This permits quantum computation to be realized with bulk ensembles far fro the ground state. Experimental results are presented for quantum gates and circuits implemented with liquid nuclear magneti resonance techniques and verified by quantum state tomography.

We consider the problem of controlling the state of a two-level quantum system (quantum bit) via an externally applied electro-magnetic field. The describing model is a bilinear right-invariant system whose state varies on the Lie group of 2×2 special unitary matrices. We study the topological structure of the reachable sets. If two or more independent controls are used, then every state can be achieved in arbitrary time. However, this is no longer true if only one control is available and, in this case, we give an exact characterization of states reachable in arbitrary time. We prove small time local controllability for any state and the existence of a critical time which is the smallest time after which every transfer of state is possible. We provide upper and lower bounds for such a time. The mathematical development is motivated by the problem of manipulating the state of a quantum bit. Every transfer of state may be interpreted as a quantum logic operation and not every logic operation can be obtained in arbitrary time. The analysis we present provides information about the feasibility of a given operation as well as estimates for the speed of a quantum computer.

The effect of restricted control of unitary quantum evolution is investigated with specific attention to NMR spectroscopy. It is demonstrated that in cases where the Hamiltonian through commutation fails to span the entire Lie algebra su(n) for an n-level quantum system, the maximum transfer efficiency may be reduced significantly relative to previously known unitary bounds on spin dynamics. The paper describes methods to determine the degree of controllability and the conditional unitary bounds induced by restricted control. These features are exemplified in relation to heteronuclear coherence transfer by planar and isotropic mixing in liquid state NMR.

Lloyd [Phys. Rev. Lett. 75, 346 (1995)] showed that almost every quantum logic gate is universal in the sense that it can be used to approximate any unitary transformation. The argument relied on a more general fact whose proof was not given in detail. We give a complete proof of this more general fact. © 2000 American Institute of Physics.

We study the manipulation of two-level quantum systems. This
research is motivated by the design of quantum mechanical logic gates
which perform prescribed logic operations on a two-level quantum system,
a quantum bit. We consider the problem of driving the evolution operator
to a desired state, while minimizing an energy-type cost.
Mathematically, this problem translates into an optimal control problem
for systems varying on the Lie group of special unitary matrices of
dimension two, with cost that is quadratic in the control. We develop a
comprehensive theory of optimal control for two-level quantum systems.
This includes, in particular, a classification of normal and abnormal
extremals and a proof of regularity of the optimal control functions.
The impact of the results of the paper on nuclear magnetic resonance
experiments and quantum computation is discussed

The reachable set for a finite dimensional quantum system is shown to be the orbit of the group corresponding to the internal and control Hamiltonians, even if this group is not compact.

In this paper, we study the design of pulse sequences for NMR spectroscopy as a problem of time optimal control of the unitary propagator. Radio frequency pulses are used in coherent spectroscopy to implement a unitary transfer of state. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. Moreover, the general proofs are no more difficult than the specific problems of current interest. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group, what is the minimum time required to steer the system from some initial point to a specified final point? In NMR spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two spin system. Comment: 20 Pages, 3 figures