$x$ which is a combination of the rank of the hypercube $n$ and the logarithmic transmission parameter $\varepsilon$.[]{data-label="fig:plot_prob_dimless"}](fig3.eps)

![Maximum success probabilities for different ranks of hypercube ($n$) calculated using the theoretical approximation, and numerical simulations. The theoretical curves are drawn with continuous lines of different patterns, and the numerical results are represented by points interconnected with the same line pattern and colour as the theoretical approximates corresponding to the same logarithmic transmission parameter $\varepsilon$.[]{data-label="fig:plot_max_prob_compare"}](fig4.eps)

Direction dependent loss {#sec:non-uniform}
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In the present section we no longer assume equal loss rates, and consider the schematically depicted loss model on Fig. \[fig:loss-scheme\] with arbitrary $\eta_d$ parameters. Because of the high symmetry of the hypercube graph, and the use of mainly identical multiports, we can neglect the position dependence of the transmission coefficients. The operator $\mD$ describing the decoherence mechanism thus acts on a general term of the density operator as $$\mD({\left|d,x\right>}{\left<d',x'\right|}) = \eta_d \eta_{d'} {\left|d,x\right>}{\left<d',x'\right|} +
\delta_{xx'} \delta_{dd'} \eta_d^2 \vrho.$$ To describe the overall effect of this operator on a pure state, we re-introduce the linear decoherence operator in a more general form, $$D = \sum_{d} \eta_d {\left|d\right>}{\left<d\right|} \otimes \Id,
\label{eq:D-inhomo}$$ and use the notation $\{\eta\}$ to denote the set of coefficients 