covariance matrix $\Sigma=\mathrm{diag}\{\sigma_1,\sigma_2,\ldots\}$ become products states of one-dimensional distributions, such as those studied in Sects. \[sec:bounds\] and \[sec:numerics-1d\]. We also introduce the order (B), where qubits are first sorted by significance and only then by coordinate. This order is inspired by the renormalization group, and deeply similar to the multi-scale representation of 2D quantum image encodings [@latorre2005]. For a distribution with two random variables, $x_1$ and $x_2,$ using three qubits per variable, the two orders read $$\begin{aligned}
  \mbox{Order (A):}& \ket{s_1s_2} \to \ket{s_1^1}\ket{s_1^2}\ket{s_1^3}\ket{s_2^1}\ket{s_2^2}\ket{s_2^3},
  \;\mbox{and}\label{eq:orders}\\
  \mbox{Order (B):}& \ket{s_1s_2}\to \ket{s_1^1}\ket{s_2^1}\ket{s_1^2}\ket{s_2^2}\ket{s_1^3}\ket{s_2^3}. \nonumber\end{aligned}$$

![(a) Two-mode Gaussian probability distribution with variances $(\sigma_{min},\sigma_{max}),$ rotated an angle $\theta.$ We work with a discretization interval $[-7\sigma_{max},7\sigma_{max}]^{\otimes 2},$ compressing $\sigma_{min}$ down to 0.1, which is a 20dB squeezing of the variance. (b) Maximum entanglement of all bipartitions, for different variances, orientations and orders. In solid lines, we plot the trivial sampling  with $N\times m=28$ qubits in total. We compare those plots with the same simulation using an MPS at a higher sampling (circles). We probe two angles $\theta=0$ and $\pi/4$ in Eq. , and two orders . (c) Entanglement entropies for all bipartitions of $Nn=24$ qubits into $(k,Nm-k),$ for the highly squeezed state $\sigma_{min}=0.1\sigma_{max}$ with $\theta=\pi/4.$ (d) Similar plot but 