  We present an entanglement analysis of quantum superpositions corresponding
to smooth, differentiable, real-valued (SDR) univariate functions. SDR
functions are shown to be scalably approximated by low-rank matrix product
states, for large system discretizations. We show that the maximum von-Neumann
bipartite entropy of these functions grows logarithmically with the system
size. This implies that efficient low-rank approximations to these functions
exist in a matrix product state (MPS) for large systems. As a corollary, we
show an upper bound on trace-distance approximation accuracy for a rank-2 MPS
as $\Omega(\log N/N)$, implying that these low-rank approximations can scale
accurately for large quantum systems.
