Since $x{\mathop{\mathrm{acot}}\nolimits}x$ is bounded in $x$, the accuracy of the term in brackets is bounded by $O(1/n)$. The most notable consequence of the second $O(1/n)$ contribution is that while in the ideal case the probability $1/2$ is an upper bound, in the lossy case deviations from the leading term, $$p^{\mathrm{max}}(x) = \frac12 \exp(-2x{\mathop{\mathrm{acot}}\nolimits}x) \frac1{1+x^2},
\label{eq:maxprob-approx}$$ can be expected in either direction. The functional form of Eq. (\[eq:maxprob-approx\]), plotted on Fig. \[fig:plot\_prob\_dimless\], allows for a universal interpretation of the dependence of success probability on the transmission rate and the size of the search space through the combined variable $x$. For small losses we can use the approximation (\[eq:x-approx\]) and conclude that the search efficiency depends only on the difference $n/2-\varepsilon$. The approximation is compared with the results of numerical calculations on Fig. \[fig:plot\_max\_prob\_compare\]. We can observe the $O(1/n)$ accuracy of the theoretical curves as expected, hence producing poorer fits at smaller ranks. The positive deviations from the theoretical curves observable at low transmission rates are due to the second $O(1/n)$ term of Eq. (\[eq:maxprob-homo\]).

![Probability of measuring the target state after the optimal number of iterations according to the approximation in Eq. (\[eq:maxprob-approx\]). The probability is plotted against the logarithm of 