Construction of channels which in every dimension anti-degrade the depolarizing channel

Abstract

We consider the depolarizing channel in $d$ dimension defined as $D_x(ρ)=(1-x)ρ+x\: \textit{tr}(ρ) \frac{I}{d}$, and explicitly find a quantum channel ${\cal N}_x$ which anti-degrades this, when $x\geq\frac{1}{2}$. This proves that the depolarizing channel $D_x$ has zero capacity when $x\geq\frac{1}{2}$. 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-extendibiliy of the Choi matrix, it is known that the channel is anti-degradable when $x\geq \frac{d}{2(d+1)}$, 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 complenetary channel ${\cal D}_x^c$ in the region $x\geq \frac{1}{2}$. This adds to the existing list of quantum channels for which the quantum capacity has been calculated in closed form.