Invariance of quantum correlations under local channel for a bipartite quantum state

We show that the quantum discord in a bipartite quantum state is invariant under the action of a local quantum channel if and only if the channel is invertible. We also show that the geometric measure of discord is preserved under the action of a local channel if the channel is invertible and if the metric used to define the geometric discord is monotonic. In particular, both versions of quantum discord are invariant under a local unitary channel