How quantum are non-negative wavefunctions?

We consider wavefunctions which are non-negative in some tensor product basis. We study what possible teleportation can occur in such wavefunctions, giving a complete answer in some cases (when one system is a qubit) and partial answers elsewhere. We use this to show that a one-dimensional wavefunction which is non-negative and has zero correlation length can be written in a “coherent Gibbs state” form, as explained later. We conjecture that such holds in higher dimensions. Additionally, some results are provided on possible teleportation in general wavefunctions, explaining how Schmidt coefficients before measurement limit the possible Schmidt coefficients after measurement, and on the absence of a “generalized area law” [D. Aharonov et al., in Proceedings of Foundations of Computer Science (FOCS) (IEEE, 2014), p. 246; e-print arXiv.org:1410.0951] even for Hamiltonians with no sign problem. One of the motivations for this work is an attempt to prove a conjecture about ground state wavefunctions which have an “intrinsic” sign problem that cannot be removed by any quantum circuit. We show a weaker version of this, showing that the sign problem is intrinsic for commuting Hamiltonians in the same phase as the double semion model under the technical assumption that TQO-2 holds [S. Bravyi et al., J. Math. Phys. 51, 093512 (2010)].