Universal Statistics of Measurement-Induced Entanglement in Tomonaga-Luttinger liquids

Abstract: We study the statistics of measurement-induced entanglement (MIE) after partial measurement on a class of one-dimensional quantum critical states described by Tomonaga-Luttinger liquids at low energies. Using a replica trick to average over measurement outcomes in the charge basis and tools from conformal field theory (CFT), we derive closed-form expressions for the cumulants of MIE. We show that exact Born-averaging over microscopic measurement outcomes becomes equivalent at low energy to averaging over conformal boundary conditions weighted by their corresponding partition functions. Our results yield distinctive critical behavior across all cumulants in the regime where the unmeasured parts of the system are maximally separated. We also obtain the full distribution of the post-measurement entanglement entropy, finding that it is generically bimodal and exhibits fat-tails. We corroborate our analytical predictions by numerical calculations and find good agreement between them.