TY - JOUR
T1 - The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information
AU - Schulte-Herbrüggen, Thomas
AU - Dirr, Gunther
AU - Helmke, Uwe
AU - Glaser, Steffen J.
PY - 2008/1
Y1 - 2008/1
N2 - 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 [image omitted]. In quantum control of n qubits one may be interested (i) in having USU(2n) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e., to the n-fold tensor product [image omitted]. Interestingly, the latter then leads to a novel entity, the local C-numerical range Wloc(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-Herbrggen, T., 2007, 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.
AB - 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 [image omitted]. In quantum control of n qubits one may be interested (i) in having USU(2n) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e., to the n-fold tensor product [image omitted]. Interestingly, the latter then leads to a novel entity, the local C-numerical range Wloc(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-Herbrggen, T., 2007, 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.
KW - Constrained optimisation
KW - Gradient flow
KW - Optimal quantum control
KW - Relative C-numerical range
UR - http://www.scopus.com/inward/record.url?scp=37249069045&partnerID=8YFLogxK
U2 - 10.1080/03081080701544114
DO - 10.1080/03081080701544114
M3 - Article
AN - SCOPUS:37249069045
SN - 0308-1087
VL - 56
SP - 3
EP - 26
JO - Linear and Multilinear Algebra
JF - Linear and Multilinear Algebra
IS - 1-2
ER -