Quantum entanglement growth is commonly summarized by a single entropy, obscuring where correlations reside in the exponentially large Schmidt spectrum and how they form. Here, we introduce Schmidt-scale concentration and dominant Schmidt scale, two coordinates that locate the probability maximum across logarithmic windows in ordered Schmidt-rank space. Applied to quenches of a random-field spin chain, these coordinates distinguish rapid transport of the dominant scale to higher Schmidt rank at weak disorder from strongly suppressed transport despite continued logarithmic entropy growth at strong disorder. The disorder-averaged dynamics exhibit an ordered hierarchy: entropy production peaks first, spectral roughness and exact nonlocal magic peak next, and dominant-Schmidt-scale transport becomes typical only after a substantial delay. Moreover, a solvable head--tail model and controlled numerical experiments reveal the physical origin of this hierarchy: the spectral path determines the order of events, local dynamics on active exchange bonds set their early timing, and intra-subsystem many-body dressing further delays dominant-Schmidt-scale transport. These results establish the Schmidt-scale coordinates as powerful dynamical probes for uncovering fine-grained entanglement structures distinguishing entanglement production, entanglement-spectrum reorganization, and dominant-Schmidt-scale transport beyond entropy alone.
Symmetries shape the quantum-information dynamics of many-body systems, but their effect on nonstabilizerness, the resource complementary to entanglement, is less understood. We compute the stabilizer Rényi entropy, a measure of nonstabilizerness, in
U(1)-symmetric one-dimensional random circuits. The disorder-averaged dynamics is captured by a four-replica tensor network, which we evaluate by
S4-adapted infinite time-evolving block decimation (iTEBD) directly in the thermodynamic limit. Together with a hydrodynamic argument, our results identify a diffusive universality class for the late-time approach of nonstabilizerness to its random-state value, with the stabilizer Rényi entropy gap closing as
1/t. The same scaling is verified in an energy-conserving nonintegrable Ising chain. More broadly, our framework provides a hydrodynamic perspective on nonstabilizerness generation and offers insight into the design of approximate Haar-random states in Hamiltonian dynamics.
The scrambling of quantum information in closed many-body systems, as measured by out-of-timeordered correlation functions (OTOCs), has received considerable attention lately.Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits.Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, "generate their own noise" and look effectively random on sufficient length scales and timescales.In this paper, we extend this approach to systems with locally conserved quantities (e.g., energy).We do this by considering local random unitary circuits with a conserved U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time-ordered and out-of-time-ordered correlation functions; both can have a diffusively relaxing component or "hydrodynamic tail" at late times.We verify the presence of such tails also in a deterministic, periodically driven system.We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific asymmetric shape, decaying as a power law behind the front.These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation.Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential μ, and apply perturbative arguments to show that for μ ≫ 1 the ballistic front of information spreading can only develop at times exponentially large in μ-with the information traveling diffusively at earlier times.We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.
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
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.
We study how conservation laws shape the spreading of quantum coherence in many-body dynamics. Focusing on
U(1)-symmetric random circuits, charge-and-dipole conserving circuits, as well as ergodic Hamiltonian dynamics, we probe coherences both globally, via the participation entropy, and locally, via the relative entropy of coherence. Combining exact vector evolution, matrix product state simulations, and replica tensor networks methods, we find that conservation laws replace the logarithmic saturation of unconstrained circuits with slow hydrodynamic relaxation of the global coherence measures. Locally, symmetry-constrained circuits show a clean rise-peak-fall structure whose peak time grows algebraically with subsystem size. In contrast, ergodic Hamiltonians broaden the peak into an extended plateau at larger subsystems, highlighting a qualitatively distinct mechanism. Coherence thus emerges as a sensitive probe of symmetry-constrained thermalization, linking quantum resource dynamics to many-body transport.