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

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.

💡 Research Summary

This paper investigates the full statistical properties of measurement‑induced entanglement (MIE) in one‑dimensional quantum critical states described at low energies by Tomonaga‑Luttinger liquids (TLLs). The authors consider a spin‑½ XXZ chain in its critical regime (−1 < Δ ≤ 1) and perform projective measurements of the local charge (σᶻ) on a disjoint region B = B₁∪B₂, leaving two unmeasured intervals A and C. For each measurement outcome m, the post‑measurement pure state |ψₘ⟩ is obtained, and the Rényi entanglement entropy Sₙ(A|m) of region A is computed. The central object of interest is the Born‑averaged entanglement MIE = ∑ₘ pₘ Sₙ(A|m), where pₘ is the Born probability of outcome m.

To treat the exponentially large space of outcomes, the authors employ an exact replica trick. They construct a generalized replica partition function Zₙ(k₁,k₂) in which the measurement outcomes are encoded as inhomogeneous Dirichlet boundary conditions on the compact boson field ϕ in the measured intervals. By mapping the replicated geometry to a cylinder with circumference 2π and length h(ζ) (ζ being the conformal cross‑ratio of the four interval endpoints), the problem reduces to evaluating CFT correlators with boundary insertions. Crucially, they demonstrate that at low energies the Born average over microscopic outcomes is equivalent to an average over conformal boundary conditions weighted by their partition functions. This “Born‑averaging = boundary‑averaging” principle allows them to write all cumulants κₗ = ⟨(Sₙ)ˡ⟩_c in closed form.

The analysis yields a universal scaling for every cumulant in the limit of maximal separation between A and C (ζ → 0). For Rényi index n larger than a threshold, all cumulants scale as ζ^{α} with α = 1/(2K), where K is the Luttinger parameter. The scaling exponent is independent of the order of the cumulant, indicating a single universal critical exponent governing the full distribution. For smaller n the scaling acquires subleading corrections, but the overall picture remains robust.

Beyond the average, the authors compute the full probability distribution P(Sₙ) of the post‑measurement entanglement. They find that P(Sₙ) is generically bimodal: one peak at zero entanglement (corresponding to outcomes that completely destroy correlations) and another peak near log 2 (the maximal entanglement for a single EPR pair). The distribution possesses heavy tails, meaning that rare outcomes can generate a substantial amount of entanglement even when the typical outcome yields almost none. In the ζ → 0 limit the log 2 peak diminishes, yet its weight remains finite, implying a non‑zero probability of creating an EPR pair from a critical ground state—a manifestation of the state acting as a quantum “wire”.

The paper also introduces disorder‑induced entanglement (DIE), obtained by averaging over random Dirichlet boundary conditions (modeling quenched impurity‑like disorder). Using the same replica‑CFT machinery, the authors derive universal expressions for DIE and show that its scaling mirrors that of MIE.

Numerical checks are performed on the lattice XXZ model using exact diagonalization and tensor‑network methods. The authors compute the first few cumulants and the full distribution for various system sizes, Luttinger parameters, and measurement configurations. The numerical data collapse onto the analytical predictions, confirming the universality of the scaling exponents and the bimodal shape of the distribution.

In conclusion, the work provides one of the first comprehensive, analytic characterizations of the full statistics of measurement‑induced entanglement in a critical one‑dimensional system. By linking Born‑averaged quantities to averages over conformal boundary conditions, the authors open a pathway to study measurement‑driven phenomena in other CFTs and higher dimensions. The results have direct implications for measurement‑based quantum computation, entanglement distribution protocols, and the fundamental understanding of how projective measurements reshape quantum correlations in many‑body systems.