A new upper bound for the quantum capacity of the $d$-dimensional depolarizing channels is presented. Our derivation makes use of a flagged extension of the 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 state gets fully depolarized. By varying the overlap between the flag 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). We find sufficient conditions for degradability of the flagged channel, which let us calculate its quantum capacity in a suitable parameter region. From this last result we get the upper bound for the depolarizing channel, which by a direct comparison appears to be tighter than previous available results for $d>2$, and for $d=2$ it is tighter in an intermediate regime of noise. In particular, in the limit of large $d$ values, our findings present a previously unnoticed $\mathcal{O}(1)$ correction.

• Received 4 December 2019
• Accepted 4 May 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.020503