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