Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.
Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness \unicodex2013 the resource enabling universal quantum computation \unicodex2013 is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.
Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.
Recent advances in quantum simulators allow direct experimental access to ensembles of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics governed by maximum-entropy principles, known as deep thermalization. At infinite temperature this universality is characterized by Haar-random ensembles. More generally, physical constraints such as finite temperature or conservation laws lead to Scrooge ensembles, which are maximally entropic distributions of pure states consistent with these constraints. Here we introduce Scrooge k-designs, which approximate Scrooge ensembles, and use this framework to sharpen the conditions under which Scrooge-like behavior emerges. We first show that global Scrooge designs arise from long-time chaotic unitary dynamics alone, without measurements. Second, we show that measuring a complementary subsystem of a scrambled global state drawn from a global Scrooge 2k-design induces a local Scrooge k-design. Third, we show that a local Scrooge k-design arises from an arbitrary entangled state when the complementary system is measured in a scrambled basis induced by a unitary drawn from a Haar 2k-design. These results show that the resources required to generate approximate Scrooge ensembles scale only with the desired degree of approximation, enabling efficient implementations. Complementing our analytical results, numerical simulations identify coherence, entanglement, non-stabilizerness, and information scrambling as essential ingredients for the emergence of Scrooge-like behavior. Together, our findings advance theoretical explanations for maximally entropic, information-stingy randomness in quantum many-body systems.
The partition function is statistical mechanics' answer to an exponentially large spectrum, distilling it into a single analytic object whose temperature dependence resolves the full structure of the underlying ensemble. We show that magic, the resource separating universal quantum computation from classically simulable stabilizer dynamics, admits precisely such a description. Mapping the Pauli spectrum of a quantum state onto the energy levels of a fictitious many-body system, the Pauli gas, we construct its canonical partition function, the stabilizer partition function, and from its associated free energy a magic monotone that we call the stabilizer work. Both these objects are analytic functions of an inverse-temperature-like parameter and are efficiently estimable via Bell sampling. The framework is analytically tractable. We derive exact ensemble-averaged partition functions for Haar-random, ν-compressible, and pseudomagic states, together with concentration guarantees. We show that for every value of its parameter, the stabilizer work is a faithful, Subadditive magic monotone, while remaining efficiently accessible on quantum hardware and admitting an operational interpretation. Unlike measures that probe a single moment of the Pauli distribution, the stabilizer work is intrinsically moment-generating. As the temperature is tuned from high to low, it interpolates continuously between the stabilizer 2-Rényi entropy and the stabilizer nullity, revealing two previously disconnected monotones as limiting cases of a single object. We demonstrate the framework on low-rank stabilizer simulation, resource interconversion, molecular ground states, and quantum many-body systems. A thermodynamics of magic is therefore not merely an analogy but a working toolkit, opening a statistical-mechanical route to magic properties of quantum systems that no single measure can access.
We investigate the structure of quantum magic in interacting disordered fermionic systems, quantifying non-stabilizerness via the fermionic stabilizer Rényi entropy (SRE). To resolve the distribution of magic across different scales, we introduce a multipartite non-local magic functional, constructed from an inclusion-exclusion combination of subsystem contributions. This measure serves as a fine-grained diagnostic, isolating genuinely global contributions and revealing nontrivial interactions between local and collective supports of magic. We illustrate the measure on paradigmatic multipartite states and apply these diagnostics to the Sachdev-Ye-Kitaev model and its variants. Crucially, for thermal/typical ensembles, we observe a marked disparity between Thermal Pure Quantum (TPQ) states and the thermal density matrix. This reveals a concealed complexity: the immense computational hardness characterizing the unitary evolution is encoded in the specific microstructure of the black hole microstates, while being washed out in the coarse-grained thermodynamic description. Furthermore, in N=2 supersymmetric SYK, we show that while fortuitous BPS states exhibit intermediate stabilizer complexity, the multipartite measure unveils a rich, sector-dependent pattern of global correlations, distinguishing them from generic chaotic states.
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.
Nonstabilizerness, also known as "magic," quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this Letter, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the ℓ-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.