  We present general numerical methods to construct witness operators for
entanglement detection and estimation of the fidelity. Our methods are applied
to detecting entanglement in the vicinity of a six-qubit Dicke state with three
excitations and also to further entangled symmetric states. All our witnesses
are designed to keep the measurement effort small. We present also general
results on the efficient local decomposition of permutationally invariant
operators, which makes it possible to measure projectors to symmetric states
efficiently
