Quantum flags, and new bounds on the quantum capacity of the depolarizing channel

TL;DR

This paper introduces a new method using flagged extensions to derive tighter bounds on the quantum capacity of depolarizing channels, improving previous results especially for higher dimensions and revealing a novel correction for large d.

Contribution

The authors develop a flagged extension approach to bound quantum capacity, providing tighter bounds for depolarizing channels across various dimensions and noise regimes.

Findings

New upper bounds on quantum capacity for d>2

Tighter bounds in an intermediate noise regime for d=2

Discovery of an O(1) correction for large d

Abstract

A new 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 flags 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). In our analysis we compute the product-state classical capacity, the entanglement assisted capacity and, under degradability conditions, the quantum capacity of the flagged channel. 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 presents a previously unnoticed $\mathcal O(1)$ correction.