Entanglement-assisted capacity of a quantum erasure channel ($C_E({\mathcal{E}}_p)=2(1-p){\log }_{2}D$)

For input dimension $D$, an input of entropy $S(\rho )$ gives output entropy $h_2(p)+(1-p)S(\rho )$ and reference-output entropy $h_2(p)+pS(\rho )$. These follow from entropy of an orthogonal quantum mixture and purity of the input purification. Thus the quantum mutual information is $2(1-p)S(\rho )$, maximized at $\rho =I_D/D$. The entanglement-assisted classical capacity is consequently $2(1-p){\log }_{2}D$ bits per use; for a binary input it is $2(1-p)$.