Data for "Measurement Induced Entanglement in Conformal Field Theory"

Abstract

Local measurements can radically reshape patterns of many-body entanglement, especially in long-range entangled quantum-critical states. Yet, analytical results addressing the effects of measurements on many-body states remain scarce, and measurements are often approximated as forcing specific measurement outcomes. We study measurement-induced entanglement (MIE) in TomonagaLuttinger liquids, a broad family of 1+1d quantum critical states described at low energies by compact free boson conformal field theories (CFT). Using a replica-trick to address the randomness of the measurement outcomes, we compute exactly the entanglement induced by measuring the local charge operator for Tomonaga-Luttinger liquids, in very good agreement with matrix-product state calculations. We show that the MIE for physical quantum measurements is fundamentally different from the entanglement induced by forcing measurement outcomes, and has a natural interpretation in terms of Born averaging over conformally-invariant boundary conditions.

Data

The archive contains two main data components:

- Theory_data/
  CSV files containing theoretical predictions for different Rényi indices and interaction parameters. File names encode the relevant quantity (MIE or forced MIE), the Rényi index, and the interaction parameter `Delta`.

- Numerical_data/
  Numerical data for MIE and forced MIE in the XX and XXZ chains as a function of the cross-ratio. The folders are:

  - free-fermion-mie/: MIE data at the free-fermion point of the XX chain (`Delta = 0`).
  - free-fermion-forced-mie/: forced-MIE data for the fixed outcome `10101010...` at the free-fermion point of the XX chain (`Delta = 0`).
  - mie-dmrg/: MIE data for the interacting XXZ chain for `Delta = -0.3` and `0.5`.
  - forced-mie-dmrg/: forced-MIE data for the fixed outcome `10101010...` for the interacting XXZ chain for `Delta = -0.3` and `0.5`.

Folder names encode simulation parameters such as system size `L`, Rényi index `n`, interaction parameter `Delta`, and, where relevant, numerical cutoff values.

The data files are intended for direct reuse in plotting, comparison with theory, and further post-processing.

Code

The Code/ directory contains the scripts and notebooks used to generate, analyze, and visualize the results in this record.

- Python scripts
  - ee_free_born_renyi.py
  - ee_free_forced_renyi.py

  These scripts implement free-fermion calculations for MIE and forced MIE in the XX chain.

- Julia scripts
  - mie_dmrg_born_symmetric.jl
  - mie_dmrg_forced_symmetric.jl

  These scripts implement DMRG-based calculations of MIE and forced MIE for the interacting XXZ chain. They rely on Julia packages including ITensors, ITensorMPS and JLD2.

- Jupyter notebooks
  - Fig1_suppmat.ipynb
  - Fig1a_renyi_1_mie_born.ipynb
  - Fig1b_free_fermion_mie_born.ipynb
  - Fig_1c_miedifferenceforcedmie.ipynb

  These notebooks are provided for analysis and figure reproduction.

- Julia environment files
  - Project.toml
  - Manifest.toml

  These files define the Julia package environment used for the DMRG calculations. `Project.toml` lists the direct dependencies, while `Manifest.toml` records the exact resolved package versions needed to reproduce the Julia setup more reliably.

Reuse notes

Theoretical data can be read directly as CSV files using standard tools such as Python, Julia, MATLAB, or spreadsheet software. Numerical data are organized by parameter values to facilitate comparison across methods and system sizes. The provided code can be used as a reference for reproducing the published results. The included Julia environment files (`Project.toml` and `Manifest.toml`) are provided to support reproducibility of the DMRG calculations.