Abstract of Thomas Schulte-Herbrüggen

OPTIMISATION ON RIEMANNIAN MANIFOLDS WITH TENSOR-PRODUCT STRUCTURE INCL. APPLICATIONS IN QUANTUM CONTROL AND INFORMATION
We give an account on gradient flows on Riemannian manifolds including new applications in quantum control: we extend former results on unitary groups to closed subgroups of
SU(2n)
with tensor-product structure. An example of importance in quantum information is the partitioning into the product
SU(2)⊗n := SU(2) ⊗ SU(2) ⊗ ...
(known as ''local operations''). The framework is kept sufficiently general for setting up gradient flows and higher-order methods on arbitrary Lie (sub-)groups and (reductive) homogeneous spaces. Relevant convergence conditions are discussed. Illustrative examples and new applications are given, such as figures of merit on the subgroup of local unitary action on n qubits relating to distance measures of pure-state entanglement. We establish the correspondence to relative numerical ranges as well as to best rank-1 approximations of higher order tensors and show applications from quantum information, where our gradient flows on the subgroup SU(2)⊗n provide a faster alternative to tensor-svd techniques. We conclude with an outlook on a quasi-Newton method exploiting the underlying structure of the homogeneous spaces.
(joint work with O. Curtef, G. Dirr, and U. Helmke)

Organisers

Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)