  Quantum coherence is a fundamental aspect of quantum physics and plays a
central role in quantum information science. This essential property of the
quantum states could be fragile under the influence of the quantum operations.
The extent to which quantum coherence is diminished depends both on the channel
and the incoherent basis. Motivated by this, we propose a measure of
nonclassicality of a quantum channel as the average quantum coherence of the
state space after the channel acts on, minimized over all orthonormal basis
sets of the state space. Utilizing the squared $l_1$-norm of coherence for the
qubit channels, the minimization can be treated analytically and the proposed
measure takes a closed form of expression. If we allow the channels to act
locally on a maximally entangled state, the quantum correlation is diminished
making the states more classical. We show that the extent to which quantum
correlation is preserved under local action of the channel cannot exceed the
quantumness of the underlying channel. We further apply our measure to the
quantum teleportation protocol and show that a nonzero quantumness for the
underlying channel provides a necessary condition to overcome the best
classical protocols.
