Channel covariance, twirling, contraction, and some upper bounds on the quantum capacity

TL;DR

This paper develops new upper bounds on quantum channel capacity by generalizing twirling and contraction techniques, simplifying capacity estimation for certain classes of quantum channels.

Contribution

It introduces a generalized degradable extension method using group twirling to derive upper bounds on quantum capacity for covariant channels.

Findings

Upper bounds for $d$-dimensional depolarizing channels.

Upper bounds for two-qubit locally symmetric Pauli channels.

Upper bounds for shifted qubit depolarizing channels.

Abstract

Evaluating the quantum capacity of quantum channels is an important but difficult problem, even for channels of low input and output dimension. Smith and Smolin showed that the quantum capacity of the Clifford-twirl of a qubit amplitude damping channel (a qubit depolarizing channel) has a quantum capacity that is at most the coherent information of the qubit amplitude damping channel evaluated on the maximally mixed input state. We restrict our attention to obtaining upper bounds on the quantum capacity using a generalization of Smith and Smolin's degradable extension technique. Given a degradable channel $\mathcal N$ and a finite projective group of unitaries $\mathcal V$, we show that the $\mathcal V$-twirl of $\mathcal N$ has a quantum capacity at most the coherent information of $\mathcal N$ maximized over a $\mathcal V$-contracted space of input states. As a consequence, degradable channels that are covariant with respect to diagonal Pauli matrices have quantum capacities that are their coherent information maximized over just the diagonal input states. As an application of our main result, we supply new upper bounds on the quantum capacity of some unital and non-unital channels -- $d$-dimensional depolarizing channels, two-qubit locally symmetric Pauli channels, and shifted qubit depolarizing channels.