fast, making the representation  efficient.

It is known that MPS’s admit an efficient, sequential construct [@schoen2005; @schoen2007]. This algorithm consists of $m$ steps. On each step, an ancilla with $\log_2\chi_i$ qubits is correlated with a fresh new qubit using a unitary operation of dimension $(d\max\{\chi_i,\chi_{i+1}\})^2.$ This unitary operation is potentially smaller and more efficient than the GR unitary, because the whole process has bounded time-cost $T_\text{GR}\sim \mathcal{O}(m\chi^2).$

Numerical study of 1D distributions {#sec:numerics-1d}
-----------------------------------

![(a) Entropy of entanglement between the $m$ first qubits and an additional qubit that is added to a GR state. We use $(\sigma,\mu)=(1,0),(1,1),(1,0),(1,0)$ for the Gaussian, Lognormal, Lorentzian and non-convex distributions, choosing the intervals $[a,b]=[\mu-6\sigma,\mu+6\sigma],[10^{-16},\mu+50\sigma],\; [\mu-10\sigma,mu+10\sigma]$ and $[0,7\sigma].$ (c-d) Similar as (a), but we plot the entanglement of a bipartition with $(k,M-k)$ qubits, for $M=14.$[]{data-label="fig:entropies"}](fig-entropies.pdf){width="\linewidth"}

Let us put these ideas to the test using three paradigmatic distributions $$\begin{aligned}
  \label{eq:Gaussian}
  \mbox{Gaussian:}
  & \quad p_G(x;\sigma,\mu) = \frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2/\sigma^2}, \\
  \label{eq:log-normal}
  \mbox{Log-normal:}
  & \quad p_{ln}(x;\sigma,\mu) = \frac{1}{x}p_G(\log(x)),\\
  \label{eq:Lorentzian}
  \mbox{Lorentzian:}
  & \quad p_{L}(x;\sigma,\mu) = \frac{\sigma}{2\pi}\frac{1}{(x-\mu)^2+\sigma^2}.\end{aligned}$$ We will also consider an unconventional function that also has a finite bandwidth, but which is not log-convex $\sigma$ $$\label{eq:non-convex}
  p_{nc}(x;\sigma) \propto e^{-x}\sin(2\sigma x)^2\cos(3\sigma x)^2.$$ We have chosen all these distributions because we can compute the 