Freeness Reined in by a Single Qubit

TL;DR

This paper investigates how the correlations predicted by free probability theory are affected by coupling a single qubit to a large quantum system, revealing persistent corrections due to non-uniform stationary states.

Contribution

It demonstrates that even minimal coupling introduces significant corrections to free probability predictions in large quantum systems.

Findings

Corrections to free probability are of order O(1) even at long times.

Non-uniform stationary states cause deviations from free probability.

Analytical and numerical methods confirm persistent modifications.

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 $D0 \gg 1$. 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.