  Basing on states and channels isomorphism we point out that semidefinite
programming can be used as a quick test for nonzero one-way quantum channel
capacity. This can be achieved by search of symmetric extensions of states
isomorphic to a given quantum channel. With this method we provide examples of
quantum channels that can lead to high entanglement transmission but still have
zero one-way capacity, in particular, regions of symmetric extendibility for
isotropic states in arbitrary dimensions are presented. Further we derive {\it
a new entanglement parameter} based on (normalised) relative entropy distance
to the set of states that have symmetric extensions and show explicitly the
symmetric extension of isotropic states being the nearest to singlets in the
set of symmetrically extendible states. The suitable regularisation of the
parameter provides a new upper bound on one-way distillable entanglement.
