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

Abstract

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 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.

Keywords:

• Relative C-numerical range
• Constrained optimisation
• Gradient flow
• Optimal quantum control

AMS Subject Classification: :

• 47A12
• 81P68

Acknowledgements

This work was supported in part by the integrated EU-programme QAP. T.S.H. thanks Chi-Kwong Li and Leiba Rodman for their kind hospitality during a visit to the College of William and Mary at Williamsburg.