# Quantum and private capacities of low-noise channels

## TL;DR

This paper calculates the quantum and private capacities of low-noise quantum channels, revealing they are equal to the single-letter coherent information and showing super-additivity and shielding are negligible in this regime.

## Contribution

It provides the first leading-order determination of capacities for low-noise channels like the depolarizing channel, resolving a 20-year open problem.

## Key findings

- Capacities are equal to the single-letter coherent information in the low-noise limit.
- Super-additivity and degenerate codes have negligible benefits in this regime.
- Shielding does not improve private capacity beyond quantum capacity in low-noise channels.

## Abstract

We determine both the quantum and the private capacities of low-noise quantum channels to leading orders in the channel's distance to the perfect channel. It has been an open problem for more than 20 years to determine the capacities of some of these low-noise channels such as the depolarizing channel. We also show that both capacities are equal to the single-letter coherent information of the channel, again to leading orders. We thus find that, in the low noise regime, super-additivity and degenerate codes have negligible benefit for the quantum capacity, and shielding does not improve the private capacity beyond the quantum capacity, in stark contrast to the situation when noisier channels are considered.