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

G. Dirr·U. Helmke·M. Kleinsteuber·T. Schulte-Herbrueggen
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Abstract

Motivated 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 $SU(2) \otimes SU(2)$.

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G. Dirr, U. Helmke, M. Kleinsteuber, T. Schulte-Herbrueggen. 2007-02-01. Relative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information. https://doi.org/10.1080/03081080701535898

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