Quantum Fano inequality ($S(\sigma )\le h(F_e)+(1-F_e){\log }_2(d^2-1)$)

For a quantum channel on a $d\ge 2$ dimensional system, purify the input to $|\Psi {⟩}_{QR}$ and set $\sigma =(\mathcal{N}\otimes \mathrm{id})(|\Psi ⟩⟨\Psi |)$. Its entanglement fidelity $F_e=⟨\Psi |\sigma |\Psi ⟩$ satisfies $S(\sigma )\le h(F_e)+(1-F_e){\log }_2(d^2-1)$. This is the entropy bound from overlap with a pure state in dimension $d^2$. For a trivial one-dimensional system the output entropy is zero.