Pure-state deep thermalization predicts that, after sufficient scrambling, projected ensembles approach universal distributions that contain information beyond the reduced density matrix. In realistic quantum systems, however, noise is unavoidable and the projected states are generally mixed. Whether mixed-state deep thermalization admits an analogous universal limiting distribution has remained unclear. Here we answer this question for quantum dynamics containing local quantum channels, i.e., errors. We show that the late-time mixed-state projected ensemble is completely determined by the normalized Choi state of the error channel and is universal, independent of the microscopic details of the underlying unitary dynamics. We establish this result analytically for dual-unitary circuits, global Haar-random dynamics, and local Haar-random circuits at large local dimension, and verify it numerically in qubit circuits and kicked Ising dynamics. In a solvable setting, we further find exponentially fast convergence to the limiting ensemble. Our results extend deep thermalization to noisy quantum dynamics and establish a direct correspondence between quantum errors and universal random mixed-state ensembles.
Pure-state deep thermalization predicts that, after sufficient scrambling, projected ensembles approach universal distributions that contain information beyond the reduced density matrix. In realistic quantum systems, however, noise is unavoidable and the projected states are generally mixed. Whether mixed-state deep thermalization admits an analogous universal limiting distribution has remained unclear. Here we answer this question for quantum dynamics containing local quantum channels, i.e., errors. We show that the late-time mixed-state projected ensemble is completely determined by the normalized Choi state of the error channel and is universal, independent of the microscopic details of the underlying unitary dynamics. We establish this result analytically for dual-unitary circuits, global Haar-random dynamics, and local Haar-random circuits at large local dimension, and verify it numerically in qubit circuits and kicked Ising dynamics. In a solvable setting, we further find exponentially fast convergence to the limiting ensemble. Our results extend deep thermalization to noisy quantum dynamics and establish a direct correspondence between quantum errors and universal random mixed-state ensembles.
We introduce the notion of the mixed state projected ensemble (MSPE), a collection of mixed states describing a local region of a quantum many-body system, conditioned upon measurements of the complementary region which are incomplete. This constitutes a generalization of the pure state projected ensemble in which measurements are assumed ideal and complete, and which has been shown to tend towards limiting pure state distributions depending only on symmetries of the system, thus representing a new kind of universality in quantum equilibration dubbed deep thermalization. We study the MSPE generated by solvable (1+1)d dual-unitary quantum circuit evolution, and identify the limiting mixed state distributions which emerge at late times depending on the size of the incomplete measurement, which we assume to be lossy, finding that they correspond to certain random density matrix ensembles known in the literature. We also derive the rate of the emergence of such universality. Furthermore, we investigate the quantum information properties of the states composing the ensemble, specifically their capacity to teleport quantum information between the ends of the system. The teleportation fidelity is upper bounded by the quantum conditional entropy, which we find exhibits a sharp transition from zero to maximal when the number of measurements lost matches of that the number of degrees of freedom to be teleported. Our results initiate the first investigation of deep thermalization for mixed state ensembles, relevant for present-day quantum simulation experiments wherein measurements are typically not perfect, and also amount to a physical and natural way of sampling from hitherto abstract random density matrix ensembles.
Recent studies of "deep thermalization" have revealed universal physics in quantum many-body dynamics beyond equilibration towards Gibbs states: maximally random quantum state ensembles can emerge on local subsystems, generated by measurements on their complement. In this work, we further identify a new form of universality exhibited in the "finer fingerprints" of quantum dynamics for local subsystems, induced by global unitary time-evolution. Specifically, we consider the projected Kraus ensemble, an ensemble of Kraus operators obtained by unraveling the quantum channel on a small subsystem with respect to knowledge of the classical configurations of its complement. Our central result is a one-parameter random matrix Ansatz that captures the ensemble's emergent statistical behavior along particular spacetime scalings, valid for generic 1D circuit dynamics without conservation laws: the ensemble is described by the product of a complex Ginibre random matrix and an independent log-normal random real scalar. The former encodes scrambling within the subsystem, captured by rotational invariance of the Ginibre measure, whereas the latter encodes fluctuations in the Born probabilities, arising from locality of the underlying dynamics. Our Ansatz can be established in the special cases of dynamics generated by global Haar random unitaries and dual-unitary circuits, while we motivate it for generic circuits using arguments of spacetime duality and the multiplicative ergodic theorem on long products of spatial transfer matrices. Extensive numerical simulations for random and Floquet circuit models verify these predictions. This universality in Kraus operators also provides a microscopic mechanism for deep thermalization in generic 1D quantum circuit dynamics, and has implications for local quantum information recoverability involving classical side information tied to knowledge of the bath.
Quantum thermalization in a many-body system is defined by the approach of local subsystems toward a universal form, describable as an ensemble of quantum states wherein observables acquire thermal expectation values.Recently, it has been demonstrated that the distribution of these quantum states can also exhibit universal statistics, upon associating each state with the outcome of a local projective measurement of the complementary subsystem.Specifically, this collection of pure quantum states-called the projected ensemble-can, under certain conditions, mimic the behavior of a maximally entropic uniformly random ensemble, i.e., form a quantum state design, representing a "deeper" form of quantum thermalization.In this work, we investigate the dynamical process underlying this novel emergent universality.Leveraging a space-time duality mapping for one-dimensional quantum circuits, we argue that the physics of dynamical purification, which arises in the context of monitored quantum systems, constrains the approach of the projected ensemble toward the uniform distribution.We prove that the absence of dynamical purification in the space-time dual dynamics (a condition realized in dual-unitary quantum circuits with appropriate initial states and final measurement bases) generically yields exact state designs for all moments k at the same time, extending previous rigorous results [Ho and Choi, Phys.Rev. Lett.128, 060601 (2022)].Conversely, we show that, departing from these conditions, dynamical purification can lead to a separation of time scales between the formation of a quantum state design for moment k = 1 (regular thermalization) and for high moments k 1 (deep thermalization).Our results suggest that the projected ensemble can probe nuanced features of quantum dynamics inaccessible to regular thermalization, such as quantum information scrambling.
Deep thermalisation -- where ensembles of pure states on a local subsystem, conditioned on measurement outcomes on its complement, approach universal maximum-entropy ensembles -- represents a stronger form of ergodicity than conventional thermalisation. We show that this framework fails dramatically for mixed initial states, evolved unitarily, even with infinitesimal initial mixedness. To address this, we introduce a new paradigm of deep thermalisation for mixed states, fundamentally distinct from that for pure-state ensembles. In our formulation, the deep thermal ensemble arises by tracing out auxiliary degrees of freedom from a maximum-entropy ensemble defined on an augmented system, with the ensemble structure depending explicitly on the entropy of the initial state. We demonstrate that such ensembles emerge dynamically in generic, locally interacting chaotic systems. For the self-dual kicked Ising chain, which we show to be exactly solvable for a class of mixed initial states, we find exact emergence of the so-defined mixed-state deep thermal ensemble at finite times. Our results therefore lead to fundamental insights into how maximum entropy principles and deep thermalisation manifest themselves in unitary dynamics of states with finite entropy.
Can measuring part of a quantum system make the remainder more random than the whole? Deep thermalization is a phenomenon in which measurement on part of a thermalizing quantum many-body system produces a subsystem ensemble of states whose higher moments approximate those of Haar-random states, i.e. a quantum state design. In this work, we tightly limit how much randomness from the global state can be transferred to the subsystem in this way: without further assumptions, a global k-design degrades to at most a k/2-design on the unmeasured subsystem. We show this optimality bound by constructing a global k-design for which each sampled state hides a sign preference in its projected distribution that can always be discovered by analyzing moments of its projected ensemble greater than k/2. Strikingly, we show this design loss can be reversed when the global design is generated by chaotic dynamics. Under the chaotic constant-time evolution of a Gaussian Unitary Ensemble (GUE) Hamiltonian, we find that the projected ensemble becomes exactly Haar-random in the thermodynamic limit --- even if the global state seems to be only a O(1)-design. Moreover, we show this phenomenon does not require the full GUE structure: it persists under substantially weaker assumptions on the spectrum and eigenbasis, and admits finite-size versions with increasingly accurate projected designs as the measured subsystem grows. Our results identify global chaotic dynamics as a mechanism by which measurements can ``concentrate'' quantum randomness and place tight limits on when such amplification is possible.
Measuring part of a many-body wave function generates a projected ensemble of pure quantum states on the unmeasured subsystem. Recent advances in deep thermalization have shown that, in chaotic systems, this ensemble universally converges to the maximally random ensemble compatible with its average density matrix, known as the Scrooge ensemble. By contrast, free-fermion systems are nonchaotic, and their quantum states are constrained to remain Gaussian. Motivated by this distinction, we introduce the fermionic Gaussian Scrooge ensemble to describe deep thermalization in generic free-fermion systems. This ensemble is defined as a distortion of the Gaussian Haar ensemble by the average density matrix, and its moments reveal an enlarged symmetry of Gaussian states in the replicated Hilbert space. We demonstrate the emergence of the fermionic Gaussian Scrooge ensemble in two complementary settings: (1) we analytically prove that the projected ensemble of the SYK2 model follows the fermionic Gaussian Scrooge ensemble at arbitrary evolution times; (2) we provide numerical evidence that it emerges at sufficiently long times in random Gaussian circuits with charge conservation. Our results establish the fermionic Gaussian Scrooge ensemble as a universal description of deep thermalization in free-fermion systems.