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We present an entanglement analysis of quantum superpositions correspondingto smooth, differentiable, real-valued (SDR) univariate functions. SDRfunctions are shown to be scalably approximated by low-rank matrix productstates, for large system discretizations. We show that the maximum von-Neumannbipartite entropy of these functions grows logarithmically with the systemsize. This implies that efficient low-rank approximations to these functionsexist in a matrix product state (MPS) for large systems. As a corollary, weshow an upper bound on trace-distance approximation accuracy for a rank-2 MPSas $\Omega(\log N/N)$, implying that these low-rank approximations can scaleaccurately for large quantum systems.

Entanglement Properties of Quantum Superpositions of Smooth, Differentiable Functions