{\left\lbrace \left|\delta_d\right| | d=0,\ldots,n-1 \right\rbrace}$. Then we have $${\left< \left. r\vphantom{r} \right| r \right>} \le \sum_{k=1}^t \etamax^{t-k}
\delta_{\mathrm{max}} \eta^{k-1} = \frac{\etamax}{\eta}
\frac{\delta_{\mathrm{max}}}{\etamax-\eta} (\etamax^t - \eta^t).$$ Since $U$ is unitary, its contribution to the above formula is trivial. Our upper bound on $|K|$ hence becomes $|K| \le 1/{\sqrt n}
(\etamax\delta_{\mathrm{max}}/\eta) (\etamax^t - \eta^t) / (\etamax -
\eta)$. Combining the results, we obtain a lower bound on the probability for measuring the target node, $$\begin{gathered}
p_n(\{\eta\},t) \ge \eta^{2t} \left\{ \sqrt{p_n^{(i)}(t)} \right. \\
  \left. - \frac{\etamax}{\eta} \frac{\delta_{\mathrm{max}}}{\etamax-\eta}
  \left[ \left(\frac{\etamax}{\eta}\right)^t -1 
  \right] \right\}^2,
\label{eq:p-lower-bound}\end{gathered}$$ where $p_n^{(i)}(t)$ stands for the corresponding probability of the ideal (lossless) case. We maximize the lower bound with respect to the arbitrary parameter $\eta$. The procedure can be carried out noting that $\delta_{\mathrm{max}} = \max\{ \etamax-\eta,
\eta-\eta_{\mathrm{min}} \}$, thereby we find the maximum at $\eta=\bar{\eta} \equiv (\etamax + \eta_{\mathrm{min}})/2$, yielding the formula $$p_n(\{\eta\},t) \ge \bar{\eta}^{2t}\left\{ \sqrt{p_n^{(i)}(t)} -
   ({\etamax}/{\bar{\eta}}) \left[
   \left({\etamax}/{\bar{\eta}}\right)^t -1 \right] \right\}^2.
\label{eq:opt-lower-bound}$$ To interpret the formula (\[eq:opt-lower-bound\]), we consider the two terms in the curly braces separately. The first term returns the success probability for uniform losses with transmission coefficient $\bar\eta$. The second term may be considered as a correction term that depends not only on some average 