  We introduce a 3-parameter class of maps acting on a bipartite system that
are a natural generalisation of the depolarizing channel (and include it as a
special case). Then, we find the exact regions of the parameter space that
alternatively determine a positive, completely positive, entanglement-breaking
or entanglement-annihilating map. This model displays a much richer behaviour
than the one shown by a simple depolarizing channel, yet it stays exactly
solvable. As an example of this richness, PPT but not entanglement-breaking
maps are found. A simple example of a positive yet indecomposable map is
provided. Then, the study of the entanglement-annihilating property is fully
addressed. Finally, we apply our results to solve the problem of the
entanglement annihilation caused in a bipartite system by a tensor product of
local depolarizing channels. In this context, a conjecture recently posed in
the literature is proven, and many related open questions are answered. To
arrive at this result we furthermore show how the Hadamard product between
quantum states can be implemented via local operations.
