On the complementary quantum capacity of the depolarizing channel

The qubit depolarizing channel with noise parameter η transmits an input qubit perfectly with probability 1− η, and outputs the completely mixed state with probability η. We show that its complementary channel has positive quantum capacity for all η > 0. Thus, we find that there exists a single parameter family of channels having the peculiar property of having positive quantum capacity even when the outputs of these channels approach a fixed state independent of the input. Comparisons with other related channels, and implications on the difficulty of studying the quantum capacity of the depolarizing channel are discussed.