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

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 ⊂ 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 loc (2 n) := SU (2) ⊗ · · · ⊗ 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) ⊗ SU (2).