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

TL;DR

This paper analyzes the measurement-induced entanglement in Tomonaga-Luttinger liquids using conformal field theory, deriving exact formulas for cumulants and the entanglement entropy distribution, revealing critical behavior and bimodal distributions.

Contribution

It introduces a novel analytical approach combining replica trick and CFT to compute MIE statistics in quantum critical systems, providing explicit formulas and numerical validation.

Findings

Cumulants of MIE exhibit critical behavior across all orders.

The entanglement entropy distribution is bimodal with fat tails.

Analytical results agree well with numerical simulations.

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.