Dirr, Gunther (Univ. of Würzburg), Helmke, Uwe (Univ. of Wuerzburg), Kleinsteuber, Martin (National ICT Australia Ltd), GLASER, Steffen (Tech. Univ. Muenchen (TUM)), SCHULTE-HERBRUEGGEN, Thomas (Tech. Univ. of Munich)

The Local C-Numerical Range: Examples, Conjectures, and Numerical Algorithms

Scheduled for presentation during the Mini-Symposium "Geometric Optimisation in Systems and Control I" (WeA07), Wednesday, July 26, 2006, 12:05−12:30, Room H

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 24, 2024

Abstract

Motivated by applications in quantum computing and quantum control, we consider the task of maximizing the trace fuction Re(tr(C*UAU*)) on the Lie group of the N-fold tensor product of special unitary 2-by-2 matrices. In order to approach this highly non-convex optimization problem, a new object, the local C-numerical range of an operator 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, the local one is in general neither star-shaped nor simply connected. The equivalence of finding bounds on its size 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 examples are presented.