studied the impact of higher order terms.

The second order Taylor coefficients were determined by fitting over the numerically obtained success probabilities at data points where the higher order moments of the loss distributions were small. An example plot of $B$ is provided on Fig. \[fig:fit-taylor\], for a system $n=8$. The higher order effects were suppressed by selecting the lowest values of $W$ from several repeatedly generated random distributions $\{\eta\}$. A general feature exhibited by all studied cases is that the second order coefficients satisfy the inequality $$B\ge 2^{-n}.
\label{eq:lowerb-second}$$ It is remarkable that this tight lower bound depends only on the size of the system. The dependence of $B$ on $\left<\eta\right>$ is monotonous with discontinuities. We found the number of discontinuities to be proportional to the rank $n$. Our numerical studies have shown that the value of $B$ before the first discontinuity is always a constant, and equal to the empirical lower bound (\[eq:lowerb-second\]).

![The relative improvement of maximum success probability comparing direction dependent loss to uniform loss with identical average loss rates. The difference is measured as $[p_n^{\mathrm{max}}(\{\eta\}) -
  p_n^{\mathrm{max}}(\left<\eta\right>)] /
  p_n^{\mathrm{max}}(\left<\eta\right>)$. All points were generated at $Q=0.35$ and the entire available domain for $\left<\eta\right>$. The values of 