  The sign structure of quantum states is closely connected to quantum phases
of matter, yet detecting such fine-grained properties of amplitudes is subtle.
Here we employ as a diagnostic measurement-induced entanglement (MIE): the
average entanglement generated between two parties after measuring the rest of
the system. We propose that for a sign-free state, the MIE upon measuring in
the sign-free basis decays no slower than correlations in the state before
measurement. Concretely, we prove that MIE is upper bounded by mutual
information for sign-free stabilizer states (essentially CSS codes), which
establishes a bound between scaling dimensions of conformal field theories
describing measurement-induced critical points in stabilizer systems. We also
show that for sign-free qubit wavefunctions, MIE between two qubits is upper
bounded by a simple two-point correlation function, and we verify our proposal
in several critical ground states of one-dimensional systems, including the
transverse field and tri-critical Ising models. In contrast, for states with
sign structure, such bounds can be violated, as we illustrate in critical
hybrid circuits involving both Haar or Clifford random unitaries and
measurements, and gapless symmetry-protected topological states.
