This work addresses how the growth of invariant operators is influenced by their underlying symmetry structure. For this purpose, we introduce the symmetry-resolved Krylov complexity, which captures the time evolution of each block into which an operator, invariant under a given symmetry, can be decomposed. We find that, at early times, the complexity of the full operator is equal to the average of the symmetry-resolved contributions. At later times, however, the interplay among different charge sectors becomes more intricate. In general, the symmetry-resolved Krylov complexity depends on the charge sector, although in some cases this dependence disappears, leading to a form of Krylov complexity equipartition. Our analysis lays the groundwork for a broader application of symmetry structures in the study of Krylov space complexities with implications for thermalization and universality in many-body quantum systems.
This work addresses how the growth of invariant operators is influenced by their underlying symmetry structure. For this purpose, we introduce the symmetry-resolved Krylov complexity, which captures the time evolution of each block into which an operator, invariant under a given symmetry, can be decomposed. We find that, at early times, the complexity of the full operator is equal to the average of the symmetry-resolved contributions. At later times, however, the interplay among different charge sectors becomes more intricate. In general, the symmetry-resolved Krylov complexity depends on the charge sector, although in some cases this dependence disappears, leading to a form of Krylov complexity equipartition. Our analysis lays the groundwork for a broader application of symmetry structures in the study of Krylov space complexities with implications for thermalization and universality in many-body quantum systems.
We study symmetry-resolved Krylov complexity in finite-dimensional chaotic quantum many-body systems. When both the Hamiltonian and the initial operator commute with a conserved charge, the operator dynamics decomposes into independent symmetry sectors, each with its own Krylov chain. We show that, after saturation, the unresolved Krylov complexity is additive over symmetry sectors. In the absence of additional Liouvillian degeneracies, the late-time contribution of a sector with Hilbert-space dimension dq is controlled by dq(dq−1), leading to a dimension-weighted equipartition that approaches the simple large-sector scaling dq2/∑q′dq′2. This late-time rule differs from the early-time weighted-average discussed in the literature and is governed instead by the dimensions of the accessible operator spaces. We support the analytic prediction with numerical studies of the real and complex SYK models, a chaotic bosonic spin model, and the mixed-field Ising chain. Our results show that resolving exact symmetries is essential for interpreting the saturation value of Krylov complexity as a diagnostic of chaotic operator growth.
The symmetry-resolved Krylov complexity is a useful tool for studying the chaotic properties of systems endowed with symmetries. We investigate the conditions under which an invariant Hermitian operator would have the symmetry-resolved Krylov complexity in a charge subspace identical to the Krylov complexity of the full operator, and find a necessary and sufficient condition that ensures this matching at all temperatures. Further, we study the Krylov complexity of the Uncoloured Tensor Model, a disorder-free kin of the SYK Model, which has a plethora of symmetries. We identify charge subspaces of the same operator in which equipartition holds, as well as those where it doesn't, along with observations on the underlying mechanism behind these. We also find that, within the computational limits, the Krylov complexity, averaged over the symmetry subspace, is bounded above by that of the operator in the full space.
A bstract In this work, we investigate the impact of conserved charges on the dynamics of spread complexity of quantum states. Building on the notion of symmetry-resolved Krylov complexity [1], we extend the framework to general quantum states and analyze the relation between the total spread complexity and its decomposition into fixed-charge sectors. After exploring a range of analytical examples and using orthogonal polynomial approach, we identify conditions under which spread complexity exhibits equipartition across sectors. Finally, we discuss quantum speed limits that constrain the growth of complexity in the presence of conserved charges.
This research release presents an analytic proof of the Caputa-Di Giulio-Loc conjecture on symmetry-resolved Krylov complexity and quantum-state spread complexity in finite dimensions. For every real time t, the total complexity satisfies C(t) ≥ Σ_q p_q C_q(t), where C_q(t) is the normalized complexity within sector q and p_q is its initial probability weight. The theorem holds for finite-dimensional Hermitian generators with orthogonal reducing sectors, including degenerate spectra and terminating Lanczos chains. It requires no assumptions of quantum chaos, spectral randomness, or late-time equilibration. The conjecture appears in “Growth of block-diagonal operators and symmetry-resolved Krylov complexity,” Physical Review Research 7, 043055 (2025), DOI: 10.1103/9v9v-54zv, and “Symmetry-resolved spread complexity,” JHEP 02 (2026), 189, DOI: 10.1007/JHEP02(2026)189. The result establishes a general relationship between full-system quantum dynamics and calculations performed separately within symmetry sectors. Its practical importance is that sector computations provide rigorous lower bounds on total complexity at every time. It also identifies precisely when resolving the symmetry leaves complexity unchanged. These conclusions are relevant to mathematical physics, operator growth, quantum many-body dynamics, and the interpretation of Krylov complexity as a diagnostic of quantum chaos. The release proves a stronger statement than the original mean-complexity inequality: the full Krylov-index distribution dominates the probability-weighted sector distribution in first-order stochastic order. Consequently, the inequality extends to every nondecreasing function of the Krylov index. Additional results include: • Two complementary proofs using nested Krylov-space projections and best polynomial approximation in spectral measures.• Concavity under mixing of normalized spectral measures.• Exact nonnegative gap identities, including a moment-Gram formulation valid for singular matrices.• Fixed-time equality criteria and a finite-dimensional characterization of equality at all times.• An explicit three-node example and reproducible numerical diagnostics.• An extension to bounded self-adjoint generators, with explicit domain and completeness assumptions for unbounded generators. The standalone release includes the PDF manuscript, editable LaTeX, complete searchable Markdown proofs, Python verification code, citation metadata, an expert review guide, and machine-readable theorem, assumption, dependency, and proof records for research discovery and AI-assisted retrieval. Research status: a complete analytic proof is supplied for the stated finite-dimensional theorem. Independent peer review and proof-assistant verification have not been completed. Worldwide novelty and priority are not certified; the unrestricted infinite-dimensional conjecture is not claimed to be resolved.
We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics, and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues toward computation of Lanczos coefficients, and it sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples, among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy, and capacity of entanglement.
We investigate operator growth in quantum systems with two-dimensional Schrödinger group symmetry by studying the Krylov complexity. While feasible for semisimple Lie algebras, cases such as the Schrödinger algebra which is characterized by a semi-direct sum structure are complicated. We propose to compute Krylov complexity for this algebra in a natural orthonormal basis, which produces a pentadiagonal structure of the time evolution operator, contrasting the usual tridiagonal Lanczos algorithm outcome. The resulting complexity behaves as expected. We advocate that this approach can provide insights to other non-semisimple algebras.