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.
We find that measuring sign-free states in a sign-free basis cannot generate significantly more entanglement than the correlations existing before measurement. In particular, we show that for a sign-free stabilizer state, its MIE is upper-bounded by its mutual information, while for a more general sign-free qubit state, its MIE is upper-bounded by a correlation function. We also verify the MIE diagnostic in various critical systems, including the recently discovered “measurement induced phase transitions" and the ground states of several one-dimensional critical spin chains. Indeed, we find the numerical results from the above sign-free critical states support our finding, while the non-sign-free critical states can generate unbounded MIE.
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