Quantum flags, and new bounds on the quantum capacity of the depolarizing channel

A new bound for the quantum capacity of the $d$-dimensional depolarizing channels is presented. Our derivation make use of a flagged extension of the depolarizing map where the receiver obtains a copy of a state $\sigma_0$ whenever the messages are transmitted without errors, and a copy of a state $\sigma_1$ when instead the original they get fully depolarized. By varying the overlap between the flags states, the resulting transformation nicely interpolates between the depolarizing map (when $\sigma_0=\sigma_1$), and the $d$-dimensional erasure channel (when $\sigma_0$ and $\sigma_1$ have orthogonal support). In our analysis we compute the product-state classical capacity, the entanglement assisted capacity and, under degradability conditions, the quantum capacity of the flagged channel. From this last result we get the upper bound for the depolarizing channel, which by a direct comparison appears to be tighter than previous results. In particular, in the limit of large $d$ values, our findings presents a previously unnoticed $\mathcal O(1)$ correction.