Gunther Dirr

The significance of the C -numerical range and the local C -numerical range in quantum control and quantum information

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., 2008, 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 Hahns 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.

Spin Dynamics: A Paradigm for Time Optimal Control on Compact Lie Groups

The development of efficient time optimal control strategies for coupled spin systems plays a fundamental role in nuclear magnetic resonance (NMR) spectroscopy. In particular, one of the major challenges lies in steering a given spin system to a maximum of its so-called transfer function. In this paper we study in detail these questions for a system of two weakly coupled spin-½ particles. First, we determine the set of maxima of the transfer function on the special unitary group SU(4). It is shown that the set of maxima decomposes into two connected components and an explicit description of both components is derived. Related characterizations for the restricted optimization task on the special orthogonal group SO(4) are obtained as well. In the second part, some general results on time optimal control on compact Lie groups are re-inspected. As an application of these results it is shown that each maximum of the transfer function can be reached in the same optimal time. Moreover, a global optimization algorithm is presented to explicitly construct time optimal controls for bilinear systems evolving on compact Lie groups. The algorithm is based on Lie-theoretic time optimal control results, established in [15], as well as on a recently proposed hybrid optimization method. Numerical simulations show that the algorithm performs well in the case a two spin-½ system.

GRADIENT FLOWS FOR OPTIMIZATION IN QUANTUM INFORMATION AND QUANTUM DYNAMICS: FOUNDATIONS AND APPLICATIONS

Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is sufficiently general for setting up gradient flows on (sub)manifolds, Lie (sub)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SUloc(2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SUloc(2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.

Relative C -numerical ranges for applications in quantum control and quantum information

Motivated by applications in quantum information and quantum control, a new type of C-numerical range, the relative C-numerical range denoted as WK (C, A), is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical C-numerical range by any of its compact and connected subgroups K ⊂ U(N). The geometric properties of the relative C-numerical range are analyzed in detail. Counterexamples prove that its geometry is more intricate than in the classical case: e.g., W K (C, A) is neither star-shaped nor simply connected. Yet, a well-known result on the rotational symmetry of the classical C-numerical range extends to WK (C, A), as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup , i.e., the n-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for WK (C, A) being a circular disc centered at the origin of the complex plane. Finally, the previous results are illustrated in detail for SU(2) ⊗ SU(2).

Separable Lyapunov functions for monotone systems

Separable Lyapunov functions play vital roles, for example, in stability analysis of large-scale systems. A Lyapunov function is called max-separable if it can be decomposed into a maximum of functions with one-dimensional arguments. Similarly, it is called sum-separable if it is a sum of such functions. In this paper it is shown that for a monotone system on a compact state space, asymptotic stability implies existence of a max-separable Lyapunov function. We also construct two systems on a non-compact state space, for which a max-separable Lyapunov function does not exist. One of them has a sum-separable Lyapunov function. The other does not.

Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping

This paper presents a survey on Riemannian geometry methods for smooth and nonsmooth constrained optimization. Gradient and subgradient descent algorithms on a Riemannian manifold are discussed. We illustrate the methods by applications from robotics and multi antenna communication. Gradient descent algorithms for dextrous hand grasping and for sphere packing problems on Grassmann manifolds are presented respectively.

Controllability Aspects of Quantum Dynamics: A Unified Approach for Closed and Open Systems

Knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in quantum technology. The purpose of this paper is to analyze such controllability aspects for finite dimensional bilinear quantum control systems. We use a unified approach based on Lie-algebraic methods from nonlinear control theory to revisit known and to establish new results for closed and open quantum systems. In particular, we provide a simplified characterization of different notions of controllability for closed quantum systems described by the Liouville-von Neumann equation. We derive new necessary and sufficient conditions for accessibility of open quantum systems modelled by the Lindblad-Kossakowski master equation. To this end, we exploit a well-studied topic of differential geometry, namely the classification of all matrix Lie-groups which act transitively on the Grassmann manifold or the punctured Euclidean space. For the special case of coupled spin-1/2 systems, we obtain a remarkably simple characterization of accessibility. These accessibility results correct and refine previous statements in the quantum control literature.

Illustrating the Geometry of Coherently Controlled Unital Open Quantum Systems

We extend standard Markovian open quantum systems (quantum channels) by allowing for Hamiltonian controls and elucidate their geometry in terms of Lie semigroups. For standard dissipative interactions with the environment and different coherent controls, we particularly specify the tangent cones (Lie wedges) of the respective Lie semigroups of quantum channels. These cones are the counterpart of the infinitesimal generator of a single one-parameter semigroup. They comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled open quantum dynamics. Such a differential characterization is highly valuable for approximating reachable sets of given initial quantum states in a plethora of experimental implementations.

An EnestrÖm-Kakeya Theorem for Hermitian Polynomial Matrices

We extend the Enestroumlm-Kakeya theorem and its refinement by Hurwitz to polynomial matrices with positive semidefinite coefficients. We determine an annular region containing the zeros of . A stability result for systems of linear difference equations is given as an application.

Riemannian Optimization on Tensor Products of Grassmann Manifolds: Applications to Generalized Rayleigh-Quotients

We introduce a generalized Rayleigh-quotient