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arXiv:2609.18691

Universal entanglement embezzlement and divergent nonlocal magic from generic local chaotic quantum evolution

Matias Karjula, Teemu Ojanen, Kim Pöyhönen, Tapio Ala-Nissila, Moein N. Ivaki

quant-phcond-mat.str-elSubmitted Sep 16, 2026• Announced Sep 17, 2026• 0 citations
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Abstract

We show that, starting from a product state, local unitary quantum evolutions generate intermediate states which exhibit a multiscale entanglement-spectrum structure required for universal entanglement embezzlement. This facilitates entanglement extraction from a catalyst many-body state while leaving it asymptotically unchanged. Remarkably, these atypical structures emerge generically at intermediate stages, well before reaching maximum entropy where thermalization has flattened out the spectral hierarchy. The resulting state is accompanied by nonlocal nonstabilizerness that diverges with the system size, consistent with a recently established equivalence between universal embezzlement and divergent nonlocal magic. Thus, without any fine tuning, a chaotic quantum evolution generates intermediate states which form a universal family of catalytic reservoirs.

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We investigate the time dependence of antiflatness in the entanglement spectrum, a measure for nonstabilizerness and lower bound for nonlocal quantum magic resource, on a subsystem of a linear SU(2) plaquette chain during thermalization. Tracing the time evolution of a large number of initial states, we find that the antiflatness exhibits a barrierlike maximum during the time period when the entanglement entropy of the subsystem grows rapidly from the initial value to the microcanonical entropy. The location of the peak is strongly correlated with the time when the entanglement exhibits the strongest growth. This behavior is found for generic highly excited initial computational basis states and persists for coupling constants across the ergodic regime, revealing a universal structure of the entanglement spectrum during thermalization. We conclude that quantitative simulations of thermalization for non-Abelian gauge theories require quantum computing. We speculate that this property generalizes to other quantum chaotic systems, a conjecture supported by analogous behavior observed in real-time simulations of the mixed-field Ising model.

arXiv:2510.11681•Physical Review Letters•Mar 25, 2026•2 citations
We show that, starting from a product state, local unitary quantum evolutions generate intermediate states which exhibit a multiscale entanglement-spectrum structure required for universal entanglement embezzlement. This facilitates entanglement extraction from a catalyst many-body state while leaving it asymptotically unchanged. Remarkably, these atypical structures emerge generically at intermediate stages, well before reaching maximum entropy where thermalization has flattened out the spectral hierarchy. The resulting state is accompanied by nonlocal nonstabilizerness that diverges with the system size, consistent with a recently established equivalence between universal embezzlement and divergent nonlocal magic. Thus, without any fine tuning, a chaotic quantum evolution generates intermediate states which form a universal family of catalytic reservoirs.
Entanglement and magic are resources that reveal complementary aspects of quantum many-body systems. Their interplay is captured by nonlocal magic, the magic that survives arbitrary local changes of basis. Yet their markedly different dynamical behavior leaves open how this irreducible component of magic spreads. Here we connect nonlocal magic to the capacity of entanglement, a tractable quantity measuring fluctuations of the entanglement Hamiltonian. This connection enables analytically controlled predictions across a wide range of many-body dynamics, from chaotic to integrable systems, which we investigate also using large-scale numerical simulations. In chaotic systems, nonlocal magic exhibits a transient buildup, with logarithmic growth in time followed by decay to a size-independent value. For integrable systems, we develop a quasiparticle picture in quantitative agreement with numerics, showing that the same initial growth instead leads to saturation at a value logarithmic in subsystem size. These contrasting behaviors have an operational consequence for entanglement embezzlement: the strongest scramblers embezzle only transiently, whereas free-fermionic dynamics can sustain universal embezzlement in the steady state.
We show that the emergence of a ``magic barrier'', characterized by a transient maximum of nonstabilizerness during rapid entanglement growth, is a universal feature of thermalization within quantum ergodic theories, which is intimately connected to random matrix spectral properties. Starting from a pure product state, we analytically determine the time evolution of entanglement entropy, antiflatness, capacity of entanglement, and nonlocal trace distance magic under unitary dynamics generated by random matrix Hamiltonians drawn from all three Gaussian ensembles. We find that the transient maxima of antiflatness, capacity of entanglement, and nonlocal magic occur on the same characteristic timescale as the maximum rate of entanglement growth. These results establish that the magic barrier is a universal feature of RMT thermalization, suggesting universality across quantum chaotic systems. We conclude that quantitative simulations of thermalization in ergodic quantum theories inherently require quantum computation.
Magic, or nonstabilizerness, is the resource that lifts Clifford circuits to universal quantum computation and has become a standard diagnostic of many-body states. For a state shared between two parties, however, a basic question has remained open: how much of the magic resides in the correlations between the parties rather than in their local bases? Isolating this nonlocal magic requires minimizing over all local bases, an optimization that has so far resisted exact solution. Here we solve it for the stabilizer fidelity: the nonlocal magic of every pure multiqubit state is the distance of its entanglement spectrum from the closest spectrum of Bell pairs. The same quantity governs an apparently unrelated task: a family of states universally embezzles entanglement under local operations and classical communication if and only if its nonlocal magic diverges. The deciding property is not the amount of entanglement but the way the entanglement spectrum spreads its weight across factor-of-two windows of rank, so that critical chains and random-singlet states, with identical logarithmic entanglement scaling, carry unbounded and vanishing nonlocal magic, respectively. Nonlocal magic thereby becomes an operationally meaningful property of quantum correlations, directly accessible to tensor-network simulations and, through entanglement spectroscopy, to experiments.
Embezzlement of entanglement is the counterintuitive process in which entanglement is extracted from a resource system using local unitary operations, with almost no detectable change in the resource's state. It has recently been argued that any state of a relativistic quantum field theory can serve as a resource for perfect embezzlement. We study the circuit complexity of embezzlement, using sequences of states that enable arbitrary precision for the process, commonly called universal embezzling families. In addition, we argue that this approach provides a well-defined model for the complexity of embezzlement from quantum field theories. Under fairly general assumptions, we establish a generic lower bound on the complexity, which increases with the precision of the process or embezzled entanglement, and diverges as these become infinite. As an example, we consider a 1d1d1d critical system as the resource and derive an exponentially growing lower bound on the complexity. Consequently, the findings imply that circuit complexity acts as a physical obstruction to perfect embezzlement. Supplementary to the main results, we derive lower bounds for common models of circuit complexity for state preparation, based on the difference between the Schatten norms of the initial and final states.
Entanglement and nonstabilizerness capture distinct aspects of quantum complexity, yet their relation through the entanglement spectrum remains poorly understood. Here we develop a spectral framework for bipartite nonlocal nonstabilizerness. We introduce a generalized anti-flatness and derive upper and lower bounds on the nonlocal stabilizer Rényi entropy (SRE) in terms of Rényi entanglement entropy and spectral non-uniformity. We further reduce the local-unitary optimization to a single-unitary problem and prove that the ordered Schmidt reference state is a local minimum of the SRE for integer α≥2α\geq2α≥2. Applying these results to exponentially and algebraically decaying spectra reveals parametrically distinct relations between entanglement and nonlocal nonstabilizerness. For broad spectra, we introduce a dyadic-shell sandwich construction that determines the asymptotic SRE scaling. At one-dimensional conformal critical points, it yields a universal hierarchy of double-logarithmic scaling laws for integer α≥2α\geq2α≥2. Our spectral bounds and dyadic-shell sandwich construction provide general tools for analyzing nonlocal SRE, offering a flexible framework that can be applied to a wide range of entanglement spectra in quantum many-body systems.
We investigate the time dependence of antiflatness in the entanglement spectrum, a measure for nonstabilizerness and lower bound for nonlocal quantum magic resource, on a subsystem of a linear SU(2) plaquette chain during thermalization. Tracing the time evolution of a large number of initial states, we find that the antiflatness exhibits a barrierlike maximum during the time period when the entanglement entropy of the subsystem grows rapidly from the initial value to the microcanonical entropy. The location of the peak is strongly correlated with the time when the entanglement exhibits the strongest growth. This behavior is found for generic highly excited initial computational basis states and persists for coupling constants across the ergodic regime, revealing a universal structure of the entanglement spectrum during thermalization. We conclude that quantitative simulations of thermalization for non-Abelian gauge theories require quantum computing. We speculate that this property generalizes to other quantum chaotic systems, a conjecture supported by analogous behavior observed in real-time simulations of the mixed-field Ising model.