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Performing quantum measurements produces not only the expectation value of aphysical observable $O$ but also the probability distribution $P(o)$ of allpossible outcomes $o$. The full counting statistics (FCS) $Z(\phi, O)\equiv\sum_o e^{i\phi o}P(o)$, a Fourier transform of this distribution, contains thecomplete information of the measurement outcome. In this work, we study the FCSof $Q_A$, the charge operator in subsystem $A$, for 1D systems described bynon-Hermitian SYK-like models, which are solvable in the large-$N$ limit. Inboth the volume-law entangled phase for interacting systems and the criticalphase for non-interacting systems, the conformal symmetry emerges, which gives$F(\phi, Q_A)\equiv \log Z(\phi, Q_A)\sim \phi^2\log |A|$. In short-rangeentangled phases, the FCS shows area-law behavior which can be approximated as$F(\phi, Q_A)\sim (1-\cos\phi) |\partial A|$ for $\zeta \gg J$, regardless ofthe presence of interactions. Our results suggest the FCS is a universal probeof entanglement phase transitions in non-Hermitian systems with conservedcharges, which does not require the introduction of multiple replicas. We alsodiscuss the consequence of discrete symmetry, long-range hopping, andgeneralizations to higher dimensions.

Full Counting Statistics across the Entanglement Phase Transition of Non-Hermitian Hamiltonians with Charge Conservations