Abstract

Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework under a minimal deviation from freeness: the coupling of a single ancilla qubit to a Haar-distributed quantum circuit of dimension $D_0\gg 1$. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order $O(1)$. These modifications persist at long times, when the dynamics of the coupled system is already ergodic. We trace their origin to non-uniformly distributed stationary quantum states, which we characterize analytically and confirm numerically.

Data and code for Freeness Reined in by a Single Qubit

Here, we provide the numerical code and precomputed Monte-Carlo data used to reproduce the spectral form factor (SFF) and two-point correlation function (CBA) figures in the companion manuscript "Freeness Reined in by a Single Qubit."

The repository includes

Precomputed Monte-Carlo data

• Spectral form factor data: estimates of $\overline{|⟨U^t⟩|²}$ for discrete times $t = 1,\dots,300$.
• Two-point correlator ("CBA") data: values of $\overline{⟨A  B(t)⟩}$ stored as a complex array (the plots use the real part).
• Results are provided for multiple ancilla-environment coupling strengths $g \in \{0.5,\,0.6,\,0.8,\,1.0\}$, internally mapped to the rate $\gamma = g^{2}/2$.
• Default model size: an environment register of $6$ qubits plus one ancilla qubit, yielding a total Hilbert-space dimension $D = 128$.

The numerics are performed using Python code included in the repository. A Jupyter notebook loads the data and reproduces all plots in publication-ready form.

Files

• 01\_sff\_cba.py: Monte-Carlo simulation script (generates Data/*.pkl)
• 02\_plots.ipynb: plotting notebook (loads Data/*.pkl and generates figures)
• Data/: precomputed dataset
  • sff\_cba\_multi\_g\_env6\_samples1000000.pkl
• Results/: output figures produced by the notebook:
  • SFF\_multig\_env6\_samples1000000.(pdf|png)
  • CBA\_env6\_samples1000000.(pdf|png)

The Jupyter notebook automatically loads the data stored in the folder Data/ and writes the resulting figures to Results/.