Representation theory framework for analysing passive linear optics beyond the dilute regime

This repository contains symbolic and numerical tools to study concentration and anti-concentration phenomena for passive linear optics, together with notebooks that reproduce the main computations and figures.

It corresponds to the numerical framework used for the article "Representation theory framework for analysing passive linear optics beyond the dilute regime", by Hela Mhiri, Hugo Thomas, Léo Monbroussou, Zoë Holmes, and Elham Kashefi.

Requirements

• Python 3.10+ (tested with Python 3.11)
• Jupyter Notebook or JupyterLab
• Python packages:
  • numpy
  • scipy
  • sympy
  • matplotlib
  • tqdm
  • ipykernel (recommended for notebook kernels)

Setup

From the repository root:

python -m venv .venv
source .venv/bin/activate
pip install --upgrade pip
pip install numpy scipy sympy matplotlib tqdm jupyterlab ipykernel

Then launch notebooks:

jupyter lab

How To Run

Open and run notebooks from the repository root (important so src.* imports resolve): Anti-Concentration.ipynb

Generated figures are saved under plots/.

Repository Structure

Representation_Theory_Framework-Passive_Linear_Optics/
├── src/
│   ├── Purity_analysis.py
│   ├── Observable_toolbox.py
│   ├── states_toolbox.py
│   └── anti_concentration_toolbox.py
├── plots/
│   ├── Inverse_P2_per_n_cube.pkl
│   ├── Inverse_P2_per_n_3_2.pkl
│   ├── Concentration_with_bounds.pdf
│   ├── Inverse_P2_per_n_per_coeff_linear.pkl
│   ├── Inverse_P2_per_n_square.pkl
├── Anti-Concentration.ipynb
├── plots/
└── LICENSE

src modules

• Purity_analysis.py:
  • Core purity computations, irrep dimensions, combinatorial helpers, and state-projection related quantities.
• Observable_toolbox.py:
  • Observable-related formulas and norms (single-irrep observables, photon-number products, optimized purity computations).
• states_toolbox.py:
  • Fock-state construction and permutation-invariant state utilities.
• anti_concentration_toolbox.py:
  • Anti-concentration formulas, including P2_calculus(n, m) and its helper functions.

Notes

• The notebooks are the main entry points for reproducing results and plots.