The Significance of the $C$-Numerical Range and the Local $C$-Numerical Range in Quantum Control and Quantum Information

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dc.creatorSchulte-Herbrueggen, Thomas
dc.creatorDirr, Gunther
dc.creatorHelmke, Uwe
dc.creatorGlaser, Steffen J.
dc.date2007-01-11
dc.date2007-02-02
dc.date.accessioned2026-07-07T12:20:38Z
dc.date.available2026-07-07T12:20:38Z
dc.descriptionThis paper 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 |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii) in restricting the dynamics to {\em local} operations on each qubit, i.e. to the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2). Interestingly, the latter then leads to a novel entity, the {\em local} C-numerical range W_{\rm 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 paper (math-ph/0702005). 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. We conclude by relating the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient flow algorithms.
dc.descriptionupdate relating to math-ph/0702005
dc.identifierhttps://arxiv.org/abs/math-ph/0701035
dc.identifierhttp://arxiv.org/abs/math-ph/0701035
dc.identifierLin. Multilin. Alg. 56 (2008) 3--26
dc.identifierdoi:10.1080/03081080701544114
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/213119
dc.subjectMathematical Physics
dc.subjectQuantum Physics
dc.titleThe Significance of the $C$-Numerical Range and the Local $C$-Numerical Range in Quantum Control and Quantum Information
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