functions $p^{(m)}(x)$ exactly for all sampling sizes, constructing the GR states for up to $m=14$ qubits[^1], analyzing their structure and complexity.

In Fig. \[fig:entropies\]a we plot the entanglement required to enlarge the quantum register by one bit, from $m$ to $m+1.$ This is the entropy $S[\rho^{(m,1)}]$ in the notation above. As shown in Fig. \[fig:entropies\]b, the bipartition never exceeds one *e-bit* of entanglement, and exhibits an exponential decay at large sizes that goes as $2^{-\gamma m}$ with $\gamma$ between 1.73 and 1.84, depending on the simulation and probability distribution. The behavior is therefore more favorable than the bound from Eq. , which overestimates the entanglement.

Fig. \[fig:entropies\]c shows that just like the entropy increase of adding one more qubit is small, the cumulative entropy obtained by studying all bipartitions of $m+k$ qubits also remains small, and with a similar tendency. Moreover, for a detailed enough sampling with $m=14,$ there is no difference between the GR states and the simpler discretizations from Sect. \[sec:other-discretizations\], shown here with dash-dot lines.

To test whether these favorable dependencies are artifacts of our choice of distributions, we have varied the parameters of the distributions and also tested situations where one of them  acquires more features. 