Motivated by applications in quantum computing and quantum control, we consider the task of maximizing the trace function Re(tr (CUAU)) on the Lie group of N fold tensor products of special unitary (2×2)-transformations. In order to approach this highly non-convex optimization problem, a new object, the local C-numerical range Wloc(C, A), is introduced and its geometry is studied. We first present examples that illustrate the rather complex geometric structure of the local C-numerical range. It is shown that, in contrast to the ordinary C-numerical range, Wloc(C, A) is in general neither star-shaped nor simply connected. The equivalence of finding bounds on the size of Wloc(C, A) and maximizing the above trace function is derived. We then describe and analyse two intrinsic optimization methods to tackle these problems: (a) a gradient flow with an Euler step discretization scheme and (b) a Jacobi-type algorithm. Explicit step size selections are given for which the gradient algorithm converges to the set of critical points. Finally, numerical experiments are presented.
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).
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 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 ®] , [Google Scholar], 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.
Under mild conditions the experimentally accessible Hamiltonians in ensembles of N coupled, nondegenerate spins-½ are shown to generate the entire Lie group SU(2 N ). This is the case e.g., in nuclear magnetic resonance (NMR) spectroscopy [1], where quantum computing gates [2] are implemented by unitary propagators, viz the ‘rf-pulse sequences’. The maximum achievable coherence transfer amplitude under unitary Hamiltonian dynamics of ensembles translates into a minimum Euclidean distance: the minimal distance between the unitary orbit of some given initial state represented by a density operator (or its signal-relevant components collected in a matrix A) and a given final state or observable (or its components C †). This distance relates to the C-numerical range of A, W C(A) := {tr(UAU -1 C)|U ∈ SU(2 N )}, and its largest absolute value, the C-numerical radius of A, rC(A):=Umax∣tr{UAU−1C}∣[3] [3]. The geometry of the C-numerical range includes as limiting cases the notions of ensembles comprising entangled as well as classically mixed components. In the latter, entropy and relative entropy relate to Euclidean distances. Given two arbitrary matrices A,C ∈ Mat n (C), the first gradient-flow-based computer algorithm [4-6] for minimising the Euclidean distance between the unitary orbit of A and C † (or rather C) in the general case is presented.
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.