= Solution
Write $q_j=\langle z_j|\rho|z_j\rangle$, $r_j=\langle z_j|Y(\rho)|z_j\rangle$, and $c=\max_{i,j}|\langle y_i|z_j\rangle|^2$. Then
$$
r_j=\sum_ip_i|\langle z_j|y_i\rangle|^2\leq c,
$$
and therefore
$$
D(Z(\rho)\|Z(Y(\rho)))
=\sum_jq_j\log\frac{q_j}{r_j}
\geq-S(Z(\rho))-\log c.
$$
The <data-processing inequality for quantum relative entropy> applied to $Z$ and part c give
$$
S(Y(\rho))-S(\rho)
\geq D(Z(\rho)\|Z(Y(\rho))).
$$
Combining them proves
$$
\boxed{S(Y(\rho))+S(Z(\rho))\geq-\log c+S(\rho)}.
$$
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