= Solution
For the distribution and <POVM> from part 2, apply the <data-processing inequality for quantum relative entropy> to the <measurement channel>. The <quantum mutual information> is the <quantum relative entropy> from the joint <density operator> to the product of its marginals, so for $k\geq2$,
$$
\begin{aligned}
I(\widetilde X:Q)_\rho
&=D(\rho_{\widetilde XQ}\|\rho_{\widetilde X}\otimes\bar\rho_Q)\\
&\geq D(\omega[\epsilon]\|\omega[1-1/k])\\
&=(1-\epsilon)\log_2 k-h_2(\epsilon)-\epsilon\log_2(1-1/k)\\
&\geq(1-\epsilon)\log_2 k-1.
\end{aligned}
$$
The last step uses $h_2(\epsilon)\leq1$ for the <binary entropy> and $\log_2(1-1/k)\leq0$. If $0\leq\epsilon<1$, rearrangement and then the supremum over $P_X$ give \b[the <one-shot classical-quantum coding converse>]:
$$
\boxed{\log_2 k\leq\frac{I(\widetilde X:Q)_\rho+1}{1-\epsilon}
\leq\frac{\sup_{P_X}I(\widetilde X:Q)_{\rho_{\widetilde XQ}}+1}{1-\epsilon}.}
$$
For $k=1$, the sole <POVM> effect is $I_Q$, forcing $\epsilon=0$, and the bound is trivial. For $\epsilon=1$, division by $1-\epsilon$ is undefined: the undivided inequality remains valid and gives no size restriction. The displayed coding bound is therefore understood for $\epsilon<1$, or with a vacuous $+\infty$ right-hand side at $\epsilon=1$.
An alternative uses the original decoder directly. Let $M$ be the uniform message and $\widehat M$ its measured estimate. For $k\geq2$, <Fano's inequality> bounds the <conditional entropy> by $H(M|\widehat M)\leq h_2(\epsilon)+\epsilon\log_2(k-1)$. The <Holevo bound> gives
$$
I(\widetilde X:Q)_\rho\geq I(M:\widehat M)
=\log_2 k-H(M|\widehat M)
\geq\log_2 k-h_2(\epsilon)-\epsilon\log_2(k-1)
\geq(1-\epsilon)\log_2 k-1.
$$
Thus the same coding bound follows by bounding the classical <mutual information> obtainable from the <POVM>.
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