= Quantum Fano inequality
{title2=$S(\sigma)\leq h(F_e)+(1-F_e)\log_2(d^2-1)$}
For a <quantum channel> on a $d\geq2$ dimensional system, purify the input to $|\Psi\rangle_{QR}$ and set $\sigma=(\mathcal N\otimes\operatorname{id})(|\Psi\rangle\langle\Psi|)$. Its <entanglement fidelity> $F_e=\langle\Psi|\sigma|\Psi\rangle$ satisfies $S(\sigma)\leq h(F_e)+(1-F_e)\log_2(d^2-1)$. 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.
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