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Evaluating the quantum capacity of quantum channels is an important butdifficult problem, even for channels of low input and output dimension. Smithand Smolin showed that the quantum capacity of the Clifford-twirl of a qubitamplitude damping channel (a qubit depolarizing channel) has a quantum capacitythat is at most the coherent information of the qubit amplitude damping channelevaluated on the maximally mixed input state. We restrict our attention toobtaining upper bounds on the quantum capacity using a generalization of Smithand Smolin's degradable extension technique. Given a degradable channel$\mathcal N$ and a finite projective group of unitaries $\mathcal V$, we showthat the $\mathcal V$-twirl of $\mathcal N$ has a quantum capacity at most thecoherent information of $\mathcal N$ maximized over a $\mathcal V$-contractedspace of input states. As a consequence, degradable channels that are covariantwith respect to diagonal Pauli matrices have quantum capacities that are theircoherent information maximized over just the diagonal input states. As anapplication of our main result, we supply new upper bounds on the quantumcapacity of some unital and non-unital channels -- $d$-dimensional depolarizingchannels, two-qubit locally symmetric Pauli channels, and shifted qubitdepolarizing channels.

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