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arXiv:2512.13803

Freeness Reined in by a Single Qubit

Alexander Altland, Francisco Divi, Tobias Micklitz, Maedeh Rezaei

quant-phcond-mat.dis-nnSubmitted Dec 15, 2025• Announced Dec 17, 2025• 1 citation
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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.

Comments
5 pages, 5 figures. 4 pages of Supplemental Material

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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/.

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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≫1D0 \gg 1D0≫1. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order O(1)O(1)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.
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≫1D_0\gg 1D0​≫1. We find that, even in this setting, the correlation functions predicted by free probability theory receive corrections of order O(1)O(1)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 ∣⟨Ut⟩∣2‾\overline{|⟨U^t⟩|²}∣⟨Ut⟩∣2​ for discrete times t=1,…,300t = 1,\dots,300t=1,…,300. Two-point correlator ("CBA") data: values of ⟨AB(t)⟩‾\overline{⟨A B(t)⟩}⟨AB(t)⟩​ stored as a complex array (the plots use the real part). Results are provided for multiple ancilla-environment coupling strengths g∈{0.5, 0.6, 0.8, 1.0}g \in \{0.5,\,0.6,\,0.8,\,1.0\}g∈{0.5,0.6,0.8,1.0}, internally mapped to the rate γ=g2/2\gamma = g^{2}/2γ=g2/2. Default model size: an environment register of 666 qubits plus one ancilla qubit, yielding a total Hilbert-space dimension D=128D = 128D=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/.
A bstract A classical dynamical system can be viewed as a probability space equipped with a measure-preserving time evolution map, admitting a purely algebraic formulation in terms of the algebra of bounded functions on the phase space. Similarly, a quantum dynamical system can be formulated using an algebra of bounded operators in a non-commutative probability space equipped with a time evolution map. Chaos, in either setting, can be characterized by statistical independence between observables at t = 0 and t → ∞, leading to the vanishing of cumulants involving these observables. In the quantum case, the notion of independence is replaced by free independence, which only emerges in the thermodynamic limit (asymptotic freeness). In this work, we propose a definition of quantum chaos based on asymptotic freeness and investigate its emergence in quantum many-body systems including the mixed-field Ising model with a random magnetic field, a higher spin version of the same model, and the SYK model. The hallmark of asymptotic freeness is the emergence of the free convolution prediction for the spectrum of operators of the form A (0) + B ( t ), implying the vanishing of all free cumulants between A (0) and B ( t ) in the thermodynamic limit for an infinite-temperature thermal state. We systematically investigate the spectral properties of A (0) + B ( t ) in the above-mentioned models, show that fluctuations on top of the free convolution prediction follow universal Wigner-Dyson statistics, and discuss the connection with quantum chaos. Finally, we argue that free probability theory provides a rigorous framework for understanding quantum chaos, offering a unifying perspective that connects many different manifestations of it.
The local-operator entanglement (LOE) measures the classical simulability of a Heisenberg operator and is conjectured to witness many-body chaos in locally interacting systems. Using tools from free probability, we analytically compute its asymptotic value for Haar random dynamics for all Rényi indices. We find that to the first two orders in large dimension, for traceless operators it exactly reproduces the Page curve for random states. In contrast to higher-order out-of-time-ordered correlators (OTOCs), which depend on more complex operator correlations via free cumulants, the leading-order LOE is independent of the initial operator. \add{We detail exactly how these higher-order correlations are encoded in the coefficients of the finite-size corrections. Therefore, guided by our Haar result, we argue that the long-time value of the LOE entropies in chaotic systems is robust: they depend only on a simple sub-sector of the hierarchy of correlations encoded by the full Eigenstate Thermalization Hypothesis. Our results therefore provide a conceptual boundary between operator simulability and higher-order scrambling diagnostics.
Recent experimental and theoretical developments in many-body quantum systems motivate the study of their out-of-equilibrium properties through multitime correlation functions. We consider the dynamics of higher-order out-of-time-order correlators (OTOCs) in a minimal circuit model for quantum dynamics. This model mimics the dynamics of a structured subsystem locally coupled to a maximally random environment. We prove the exponential decay of all higher-order OTOCs and fully characterize the relevant timescales, showing how local operators approach free independence at late times. We show that the effects of the environment on the local subsystem can be captured in a higher-order influence matrix, which allows for a Markovian description of the dynamics provided an auxiliary degree of freedom is introduced. This degree of freedom directly yields a dynamical picture for the OTOCs in terms of free cumulants from free probability, consistent with recent predictions from the full eigenstate thermalization hypothesis. This approach and the relevant influence matrix are expected to be applicable in more general settings and present a first step to characterizing quantum memory in higher-order OTOCs.
Unitary randomness underpins both fundamental tasks in quantum information and the modern theory of quantum chaos. On one side, a central concept is that of approximate unitary designs: circuits that look random according to small moments and for forward-in-time protocols. In a distinct setting, out-of-time-ordered correlators (OTOCs), intensely studied as a measure of information scrambling, have recently been shown to probe freeness between Heisenberg operators, the noncommutative generalization of statistical independence. Bridging these two concepts, we study the emergence of freeness in a random matrix product unitary ensemble. We prove that, with only polynomial bond dimension, these unitaries reproduce Haar values of higher-order OTOCs for local, finite-trace observables, while traceless observables instead require exponential resources. Indeed, local observables are precisely those predicted to thermalize in chaotic many-body systems according to the eigenstate thermalization hypothesis. Moreover, adding to previous literature, we show how random matrix product unitaries constitute approximate designs: we exactly compute the frame potential of the ensemble, showing convergence to the Haar value with polynomial deviations and so indicating that global observables are freely independent on-average. Our results highlight the need to refine previous notions of unitary design in the context of operator dynamics, guiding us towards protocols for quantum advantage and shedding light on the emergent complexity of chaotic many-body systems.
We study a so-called semi-ergodic brickwork dual-unitary circuits where, in the infinite volume limit, the two-point correlation functions of single-site operators exhibit ergodic behavior along one light ray and non-ergodic behavior along the other light ray. Here, however, we study intermediate and long-time dynamics of a system in a finite, large volume. Under such dynamics, the Heisenberg evolution of a single traceless single-site operator lies within a restricted subspace, and this time evolution can be mapped to a simpler problem of a single qutrit scattering with a bunch of qubits sequentially. Despite the model being non-integrable and free from any quenched disorder, the operator entanglement grows at most logarithmic in time, contrary to prior expectations. The auto-correlation function can be written in terms of a sum of products of SO(3)SO(3)SO(3) matrices, allowing for a random matrix prediction for the auto-correlation function at late times. The operator size distribution also becomes bimodal at certain times, displaying intermediate behavior between chaotic and free systems.