Relative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information

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dc.creatorDirr, G.
dc.creatorHelmke, U.
dc.creatorKleinsteuber, M.
dc.creatorSchulte-Herbrueggen, T.
dc.date2007-02-01
dc.date.accessioned2026-07-07T12:20:38Z
dc.date.available2026-07-07T12:20:38Z
dc.descriptionMotivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.
dc.descriptionaccompanying paper to math-ph/0701035
dc.identifierhttps://arxiv.org/abs/math-ph/0702005
dc.identifierhttp://arxiv.org/abs/math-ph/0702005
dc.identifierLin. Multilin. Alg. 56 (2008) 27--51
dc.identifierdoi:10.1080/03081080701535898
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/213120
dc.subjectMathematical Physics
dc.subjectQuantum Physics
dc.titleRelative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information
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