Freeness Reined in by a Single Qubit

Free probability provides a framework for describing correlations between noncommuting 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}≫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 nonuniformly distributed stationary quantum states, which we characterize analytically and confirm numerically.