We consider the depolarizing channel in d dimension defined as \(D_x(\rho )=(1-x)\rho +x\text {tr}(\rho ) \frac{I}{d}\) , and explicitly find a quantum channel \(\mathcal{N}_x\) which anti-degrades this, when \(x\ge \frac{1}{2}\) . This proves that the depolarizing channel \(D_x\) has zero capacity when \(x\ge \frac{1}{2}\) . As a corollary, this implies that any quantum channel when contaminated by white noise stronger than this value loses its capacity completely. Although by arguments based on symmetric extendibility of the Choi matrix, it is known that the channel is anti-degradable when \(x\ge \frac{d}{2(d+1)}\) , [1–3], the explicit form of the anti-degrading channel in this larger interval is not known. We also calculate in closed form the capacity of the complementary channel \(\mathcal{D}_x^c\) in the region \(x\ge \frac{1}{2}\) . This adds to the existing list of quantum channels for which the quantum capacity has been calculated in closed form, see [4, 5], and [6].
