values of $Q$. The plot corresponds to rank $n=7$ and $\eta_{\mathrm{max}}= 0.996\pm 0.001$. []{data-label="fig:max-prob-max"}](fig7.eps)

The numerical studies, involving the generation of 1000 sets of uniformly randomly generated transmission coefficients for each of the systems of up to sizes $n=10$, indicate that with the help of Eq. (\[eq:lowerb-second\]) the first two terms of the expansion Eq. (\[eq:pmax-Taylor\]) can be used to obtain a general lower bound: $$p_n^{\textrm{max}}(\{\eta\}) \ge p_n^{\mathrm{max}}
(\left<\eta\right>) + 2^{-n}Q^2.$$ The inequality implies that the overall contribution from higher order terms is positive, or always balanced by the increase of $B$. The appeal of this lower bound is that it depends only on the size of the system $N=2^n$, and the elementary statistical properties of the noise ($\left<\eta\right>$, $Q$). Therefore, together with the formula (\[eq:maxprob-approx\]) for uniform loss, a straight-forward estimation of success probability is possible before carrying out an experiment.

Up to now, we concentrated on comparing the performance of the search algorithm suffering non-uniform losses with those suffering uniform loss with coefficient equal to the average of the non-uniform distribution. Another physically interesting question is how attenuation alone affects search efficiency. We can formulate this question using the notations above as follows. Consider a randomly generated distribution 