Quantum Flags and New Bounds on the Quantum Capacity of the Depolarizing Channel

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 σ_{0} whenever the messages are transmitted without errors, and a copy of a state σ_{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 σ_{0}=σ_{1}), and the d-dimensional erasure channel (when σ_{0} and σ_{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 O(1) correction.