higher moments of $\{\eta\}$ are not restricted, therefore we see multiple points for certain $(\left<\eta\right>, Q)$ pairs. (Rank of the hypercube $n=9$.)[]{data-label="fig:max-prob-Q35"}](fig6.eps)

To plot the success probabilities corresponding to arbitrary random coefficients we used the pair of variables $\left<\eta\right>$ and $Q$. On these plots, the higher order terms cause a “spread” of the appearing curves. A sample plot is displayed on Fig. \[fig:max-prob-Q35\] where the relative improvement is compared to the uniform case, in percentages. We observe a general increase of efficiency as compared to the uniform case with the same average loss rate. A general tendency is that for smaller values of $\left<\eta\right>$ the improvement is larger, interrupted, however, by discontinuities. These discontinuities closely follow those of the second order coefficient $B$.

![The difference of the maximum success probabilities, in the presence of direction dependent loss with coefficients $\{\eta\}$, and in the presence of uniform loss with coefficient $\eta_{\mathrm{max}}=\max\{\eta\}$. Physically, the non-uniform case can be obtained from the uniform case by introducing proper attenuation. We use the second moment $Q$ as a measure of deviation from the non-attenuated case. The vertical axis shows $[p_n^{\mathrm{max}}(\{\eta\}) -
  p_n^{\mathrm{max}}(\eta_{\mathrm{max}})] / p_n^{\mathrm{max}}
  (\eta_{\mathrm{max}})$ as a percentage. We observe a systematic improvement for higher 