Bounding the quantum capacity with flagged extensions

TL;DR

This paper introduces flagged extensions of quantum channels to derive new upper bounds on their quantum and private capacities, improving bounds for key channels like depolarizing and amplitude damping.

Contribution

It develops general conditions for degradability of flagged extensions and applies them to improve capacity bounds for specific quantum channels.

Findings

Established new upper bounds for depolarizing channel capacities.

Derived bounds for BB84 and generalized amplitude damping channels.

Suggested potential for tighter bounds using tensor powers of channels.

Abstract

In this article we consider flagged extensions of convex combination of quantum channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum $Q$ and private $P$ capacities of any channel being a mixture of a unitary map and another channel, with the probability associated to the unitary component being larger than $1/2$. We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.