= Solution
{c}
After the first <Bell-basis measurement> and the announced correction, the shared <Werner state> has visibility $v=p^2q$. Ideal <quantum teleportation> through that resource is the <depolarizing channel>
$$
\mathcal D_v(\rho)=v\rho+(1-v)\frac{I_2}{2}.
$$
One way to see this is to separate the singlet branch, which teleports perfectly with the appropriate known correction convention, from $I_4/4$, which has no correlations and yields $I_2/2$.
The additional <Bell-basis measurement> at Alice's station has its own independent reliable branch of probability $q$. In its random-label branch the correction averages the four <Pauli gates>, again giving $I_2/2$. Hence the actual teleported <density matrix> is
$$
\boxed{\rho_{\rm out}=p^2q^2\rho_{\rm in}+(1-p^2q^2)\frac{I_2}{2}.}
$$
On sifted <BB84> rounds, Bob measures in Alice's preparation <basis>. The preserved component gives the correct bit with certainty; the maximally mixed component gives an error with probability $1/2$. Therefore
$$
\boxed{\mathrm{QBER}=\frac{1-p^2q^2}{2}.}
$$
For $p=q=1$ it vanishes; if either parameter is zero it is $1/2$. The result presumes successful full <Bell-basis measurement> outcomes and independent stated imperfections, without extra loss, polarization drift or dark counts.
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