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

TL;DR

This paper explicitly constructs channels that anti-degrade the depolarizing channel in any dimension for certain noise levels, proving zero capacity and calculating the complementary channel capacity in these regimes.

Contribution

It provides an explicit form of anti-degrading channels for the depolarizing channel when noise exceeds a threshold, establishing zero capacity in this regime.

Findings

Depolarizing channel has zero capacity for x ≥ 1/2.

Explicit anti-degrading channels are constructed for x ≥ 1/2.

Capacity of the complementary channel is calculated for x ≥ 1/2.

Abstract

We consider the depolarizing channel in $d$ dimension defined as $D_x(\rho)=(1-x)\rho+x\: \textit{tr}({\rho}) \frac{I}{d}$, 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.