Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that
U(1) conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving
ΔS(t)∼t−1/2 in the hydrodynamic regime and crossing over to
∼exp[−O(t/L2)] when
t≥L2. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum
U(1) conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.
Quantum many-body dynamics generate nonclassical correlations naturally described by quantum resource theories. Quantum magic resources (or nonstabilizerness) capture deviation from classically simulable stabilizer states, while coherence and fermionic non-Gaussianity measure departure from the computational basis and from fermionic Gaussian states, respectively. We track these resources in a subsystem of a one-dimensional qubit chain evolved by random brickwall circuits. For resource-generating gates, evolution from low-resource states exhibits a universal rise-peak-fall behavior, with the peak time scaling logarithmically with subsystem size and the resource eventually decaying as the subsystem approaches a maximally mixed state. Circuits whose gates do not create the resource but entangle neighboring qubits, give rise to a ballistic spreading of quantum resource initially confined to a region of the initial state. Our results give a unified picture of spatiotemporal resource dynamics in local circuits and a baseline for more structured quantum many-body systems.
Quantum state complexity metrics, such as anticoncentration and nonstabilizerness, offer key insights into many-body physics, information scrambling, and quantum computing. Anticoncentration and equilibration of magic resources under dynamics of random quantum circuits occur at times scaling logarithmically with system size. Here, we examine these phenomena in one-dimensional ergodic Floquet models and thermalizing Hamiltonian systems. Using participation and stabilizer entropies to probe anticoncentration and magic resources, we reveal significant differences between the two settings. Floquet systems align with random circuit predictions, exhibiting anticoncentration and magic saturation at timescales logarithmic in system size. In contrast, Hamiltonian dynamics deviate from the random circuit predictions and require times scaling approximately linearly with system size to achieve saturation of participation and stabilizer entropies, which remain smaller than that of the typical quantum states even in the long-time limit. Our findings establish the phenomenology of participation and stabilizer entropy growth in generic many-body systems and emphasize the role of conservation laws in constraining anticoncentration and magic dynamics.
A central question in quantum many-body physics asks how computational complexity — specifically the nonstabilizerness, or "magic," resource that distinguishes genuinely quantum computation from Clifford-circuit simulation — emerges, propagates, and saturates in physical systems governed by conservation laws. This paper synthesizes five recent arXiv preprints spanning quant-ph and cond-mat.stat-mech to argue for a candidate structural pattern: **conservation laws impose hydrodynamic memory on magic-resource dynamics, and this memory manifests as diffusive relaxation signatures that are measurable, resource-measure-dependent, and sensitive to encoding geometry**. The argument is a heuristic reading, not a derivation; the sources share formal structures (replica tensor networks, hydrodynamic arguments, stabilizer Rényi entropy) but are not unified by a single theorem. Concretely: (1) nonstabilizerness in U(1)-symmetric random circuits relaxes diffusively as
1/t [corpus:arxiv:2606.13606v1]; (2) participation entropy — a complementary spread measure — exhibits the same
t−1/2 hydrodynamic tail precisely because U(1) conservation law couples it to diffusing density correlations [corpus:arxiv:2606.11561v1]; (3) in monitored circuits, the steady-state magic depends on resource-measure geometry near the stabilizer layer, with mana scaling linearly and stabilizer Rényi entropy quadratically in the injection parameter [corpus:arxiv:2606.13470v1]; (4) non-isometric encodings — unavoidable in finite-energy bosonic codes and in holographic models — impose fundamental accuracy ceilings on logical operations that are quantitatively controlled by the approximate quantum error correction framework [corpus:arxiv:2606.13559v1]; and (5) optimal classical shadow estimation of unitary channels achieves Heisenberg scaling
O(dε−1) using only parallel queries, establishing the query complexity of extracting magic-relevant channel information [corpus:arxiv:2606.13638v1]. The falsification path common to these claims is concrete: if U(1)-symmetric Floquet circuits with tunable symmetry-breaking perturbations show that the
1/t nonstabilizerness gap and the
t−1/2 participation entropy tail decouple as the symmetry is explicitly broken, the proposed hydrodynamic-memory mechanism is refuted. Sources are drawn from approximately 80 recent preprints in quant-ph, math-ph, and cond-mat.stat-mech. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.11561v1, 2606.11885v1, 2606.12269v1, 2606.13470v1, 2606.13525v1, 2606.13559v1, 2606.13606v1, 2606.13638v1
Quantum resource theories characterize distinct forms of nonclassicality in many-body quantum states, raising the question of whether these resources evolve independently under generic ergodic dynamics. Considering diagnostics quadratic in the state, we show that the dynamics of different resource measures become mutually interdependent and are governed by a few common degrees of freedom. For Haar-random circuits, the purity together with a single resource witness suffices to reconstruct the remaining resource measures, as we demonstrate for coherence, various asymmetries, and number entropies. The same relations hold, to a good approximation, under chaotic Floquet and continuous-time Hamiltonian dynamics, establishing that this dynamical resource equivalence extends beyond random circuits. We further derive an exact coherence--imaginarity relation, remarkably accurate also beyond Haar-random circuits. Our results reveal an emergent simplification of many-body dynamics, in which sufficiently strong scrambling reduces seemingly distinct quantum resources to a few common dynamical degrees of freedom.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.