RelativeC-numerical ranges for applications in quantum control and quantum information
G. Dirr, U. Helmke, M. Kleinsteuber, Th. Schulte-Herbrüggen
Linear and multilinear algebra
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.
... (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).