Relative C -numerical ranges for applications in quantum control and quantum information

Abstract

Motivated by applications in quantum information and quantum control, a new type of C-numerical range, the relative C-numerical range denoted as WK (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 ⊂ U(N). The geometric properties of the relative C-numerical range are analyzed in detail. Counterexamples prove that 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 WK (C, A), as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup , 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 WK (C, A) being a circular disc centered at the origin of the complex plane. Finally, the previous results are illustrated in detail for SU(2) ⊗ SU(2).