Abstract
Performing quantum measurements produces not only the expectation value of a physical observable O but also the probability distribution P(o) of all possible outcomes o. The full counting statistics (FCS) Z(φ, O)≡ Σ_o e^iφ oP(o), a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of Q_A, the charge operator in subsystem A, for 1D systems described by non-Hermitian SYK-like models, which are solvable in the large-N limit. In both the volume-law entangled phase for interacting systems and the critical phase for non-interacting systems, the conformal symmetry emerges, which gives F(φ, Q_A)≡ log Z(φ, Q_A)∼ φ²log |A|. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as F(φ, Q_A)∼ (1-cosφ) |∂ A| for ζ ≫ J, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequence of discrete symmetry, long-range hopping, and generalizations to higher dimensions.