Julia code for time-dependent two-point correlation functions and finite-temperature quantum Fisher information (QFI) of Gibbs states in the 1D transverse-field Ising chain.

We consider the 1D transverse-field Ising model

$$ \hat H_0 = \sin(\gamma)\sum_{i=1}^{N}\sigma_i^z - \cos(\gamma)\sum_{i=1}^{N-1}\sigma_i^x\sigma_{i+1}^x, $$

and introduce a classical intensive parameter $\theta$ via

$$ \hat H(\theta) = \hat H_0 + \theta \sum_i \sigma_i^x, \qquad \hat \rho_\theta = \frac{e^{-\beta \hat H(\theta)}}{\mathrm{Tr}\big[e^{-\beta \hat H(\theta)}\big]}. $$

Here $\theta$ plays the role of a longitudinal field thermodynamically conjugate to the total magnetization $\sum_i \sigma_i^x$, in the sense discussed in arXiv:2511.05042.

This repository provides:

• Exact two-point correlation functions
  (computed via the Jordan–Wigner transformation and the resulting free-fermion representation), such as $\langle \sigma_i^x(t)\sigma_j^x\rangle$, $\langle \sigma_i^z(t)\sigma_j^z\rangle$, and their site-averaged sums.

• Exact finite-temperature QFI with respect to the parameter $\theta$, evaluated around $\theta=0$, using the integral representation

$$ \mathcal{F}_\theta = \frac{2}{\pi}\int_{-\infty}^{+\infty} d\omega \tanh^2 \Bigl(\frac{\beta\omega}{2}\Bigr) \frac{1}{\omega^2}S(\omega), $$

where $S(\omega)$ is the symmetrized autocorrelation spectrum of the conjugate operator $\hat O = \sum_i \sigma_i^x$.
In practice, $S(\omega)$ is obtained from the real-time correlation functions computed in correlator.jl.

• The thermodynamic bounds on QFI discussed in arXiv:2511.05042, including both the variance bound and the susceptibility-based bound.

• correlator.jl
  Implements the Jordan–Wigner mapping and free-fermion solution, and evaluates real-time two-point correlation functions of the transverse-field Ising chain (following Phys. Rev. B 56, 11659).

• qfi_vs_bounds.jl
  Uses these correlation functions to compute the QFI of the thermal state via the time-integral representation of QFI, and evaluates the corresponding thermodynamic bounds.

1. Install Julia.

2. Clone this repository:

   git clone https://github.com/YemingMeng/IsingQFI cd IsingQFI

3. Instantiate the project environment (this will download all Julia dependencies):

   julia --project=./julia_env -e 'using Pkg; Pkg.instantiate(); Pkg.precompile()'

After this, you can either run the example commands below from your shell,
or start a Julia REPL with

julia --project=./julia_env

and include the scripts manually.

Generate the transverse-field Ising xx–correlator curves (a Plotly HTML file correlator_xx.html will be written to the current directory):

julia --project=./julia_env -e 'include("correlator.jl"); correlator_xx_example()'

Similarly, for the zz–correlator:

julia --project=./julia_env -e 'include("correlator.jl"); correlator_zz_example()'

The output is an interactive Plotly HTML file which you can open in any modern browser.

To compute the finite-temperature QFI and its thermodynamic bounds (per site) and generate the corresponding plot (e.g. Fig. 2(a) of arXiv:2511.05042):

julia --project=./julia_env qfi_vs_bounds.jl

• By default, the script uses a relatively small system size (e.g. n=15) for quick previews.
• To reproduce the figure in the paper more faithfully, open qfi_vs_bounds.jl and increase n to 100.
  This substantially increases the computational cost, so expect a much longer runtime.

If you use this code or build on it in your work, please cite the accompanying paper:

@Article{IsingQFI2025, author = {Meng, Ye-Ming and Shi, Zhe-Yu}, title = {Bounds on quantum Fisher information and uncertainty relations for thermodynamically conjugate variables}, journal = {arXiv preprint arXiv:2511.05042}, year = {2025}, doi = {10.48550/arXiv.2511.05042}, }