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PAMM

Time optimal factorizations on compact Lie groups

Gunther Dirr, Uwe Helmke B, Martin Kleinsteuber

Published Dec 1, 2004• 1 citation
Publisher

Abstract

Abstract In this paper we study the relationship between factorization problems on SU(2n) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pamm.200410313

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In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.
Abstract Motivated by considerations of pure state entanglement in quantum information, we consider the problem of finding the best rank‐1 approximation to an arbitrary r ‐th order tensor. Reformulating the problem as an optimization problem on the Lie group SU (n1) ⊗ … ⊗ SU (nr) of so‐called local unitary transformations and exploiting its intrinsic geometry yields a new approach, which finally leads to Riemannian variant of the conjugate gradient algorithm. Numerical simulations support that our method offers an alternative to the higher‐order power method for computing the best rank‐1 approximation to a tensor. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract In this paper we study the accessibility properties of finite dimensional (N‐level) open quantum systems in the presence of dissipation and relaxation described by the Lindblad master equation. We specifically focus on the unital Lindbladian case where general results can be obtained. The theory of transitive Lie‐group actions is used to classify the system Lie‐algebras of the Lindblad equation for which the reachable sets have nonempty interior. For the special case of n ‐coupled spin‐1/2 systems, we obtain a particularly simple characterization. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)