arXiv subjects

Pawel Caputa

Publications and source records attributed to Pawel Caputa.

At least 19 recordsLinked to original sources

arXiv2026

Holographic Local Operator Quenches with Conserved Momentum and Spin

We investigate the holographic dictionary relating point particles carrying longitudinal momentum or angular momentum in asymptotically AdS$_3$ spacetimes to suitably regulated, time-evolved states created by local primary operators in the dual two-dimensional conformal field theory (2D CFT). We find that asymmetric left/right Euclidean smearing of local operators produces states carrying momentum, and the corresponding bulk excitation is a particle with conserved momentum. We compute the energy density and entanglement entropy in these states and in their dual back-reacted geometries, finding exact agreement between the CFT and gravity descriptions. We further extend this correspondence to particles with intrinsic spin, whose CFT duals are primary operators with unequal holomorphic and anti-holomorphic scaling dimensions. We again find a precise match between CFT and holographic calculations of energy densities and entanglement entropies. Finally, we explore applications of these setups beyond holography by deriving the evolution of R\'enyi entropies in 2D rational CFTs and introducing a new class of local quantum quench protocols with conserved longitudinal or angular momentum.

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arXiv2026

Wigner negativity in Krylov space and emergent semiclassicality

We propose that the Krylov basis gives a semiclassical representation of dynamics in general large-$N$, complex, many-body systems. As a probe of this semiclassicality, we study the growth of Wigner negativity -- a measure of the complexity of classical simulation -- under time evolution in Krylov space in several solvable models. We begin with 2d CFTs, initially in either the vacuum or the thermofield double state on a line excited by a primary operator. In both cases, Wigner negativity remains an $O(1)$ constant and does not grow at late times, indicating approximately classical dynamics in the Krylov basis. We then study random matrix theory with the maximally entangled state between two copies as the initial state. For general one-cut matrix models, we argue that Wigner negativity in the Krylov basis grows as $t^{1/2}$ at large $O(1)$ times but does not scale with the Hilbert space dimension, thus indicating semiclassical dynamics in Krylov space. Finally, in the double-scaled SYK model, we find an approximately classical phase (constant negativity) at early times and a semiclassical phase ($t^{1/2}$ growth) at late times. In all these examples, Wigner negativity either remains constant or grows slowly, demonstrating emergent semiclassicality of dynamics in Krylov space.

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arXiv2026

Complexity Inequalities for Quantum Subsystems

Motivated by the role of the holographic entropy cone in constraining the entanglement structure of states with classical gravitational duals, we investigate combinations of subsystem complexities associated with reduced density matrices in multipartite quantum systems. Focusing on subsystems composed of three disjoint regions, we introduce two quantities: a tripartite complexity, inspired by the tripartite information, and a complexity gap, designed to characterize emergent complexity in the full quantum state beyond that of its constituents. We study the sign structure of these quantities in three selected approaches to subsystem complexity. In holography, we employ the complexity=volume proposal in AdS spacetimes; for Gaussian many-body states, we use Fisher-Rao subsystem complexity; and we further develop a Krylov-space inspired, effective framework for reduced density matrices, which we test in few-qubit systems and coherent-state dynamics. Across all three approaches, we find that the tripartite complexity is not sign-definite in general. By contrast, the complexity gap exhibits a definite sign in every example we analyze, although the sign itself depends on the underlying notion of subsystem complexity. Our results suggest that the complexity gap could be a natural candidate building block for a prospective hierarchy of subsystem complexity inequalities.

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arXiv2026

Probing weak chaos in $\mathcal N=4$ super Yang-Mills and long-range spin chains

We study signatures of quantum chaos in finite-loop truncations of the planar dilatation operator in the $\mathfrak{su}(2)$ sector of $\mathcal N=4$ super Yang-Mills and its $\beta$-deformation. These truncations define holographically motivated long-range deformations of the nearest-neighbour XXX spin chain. At one-loop the model is integrable, while the all-loop planar theory is expected to again be integrable. Finite-loop truncations therefore provide a natural setting for investigating how chaotic behaviour emerges between these two integrable limits. We analyse this question using spectral statistics, eigenvector diagnostics and spread complexity. We find that the two- and four-loop truncations develop GOE-like level statistics at sufficiently large coupling but with features characteristic of weak integrability breaking. The integrability breaking at four-loops is weaker than at two-loops and the critical coupling at which chaos occurs is larger, at least for long spin chains. The three-loop truncation does not show the same onset of chaos in the range studied. Eigenvector diagnostics show that the corresponding eigenstates remain less random than GOE vectors, indicating weak ergodicity and multifractality. Finally, we can identify signatures of the eigenvalue and eigenvector chaos in the Krylov-space data. Namely, we demonstrate a correlation of the level spacing statistics with the peak of spread complexity and disorder on the Krylov chain. The delocalisation of the initial state in the Hamiltonian eigenbasis is shown to strongly affect the saturation of complexity. Our results suggest that finite-loop dilatation operators are not generic long-range spin chain Hamiltonians, but already display patterns consistent with the restoration of integrability in the all-loop planar theory.

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arXiv2025

Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition

We introduce generalisations of von Neumann entanglement entropy that are invariant with respect to certain scale transformations. These constructions are based on the Unit-Invariant Singular Value Decomposition (UISVD) in its left-, right-, and bi-invariant incarnations, which are variations of the standard Singular Value Decomposition (SVD) that remain invariant under the corresponding class of diagonal transformations. These measures are naturally defined for non-Hermitian or rectangular operators and remain useful when the input and output spaces possess different dimensions or metric weights. We apply the UISVD entropy and discuss its advantages in the physically interesting framework of Biorthogonal Quantum Mechanics, whose important aspect is indeed the behaviour under scale transformations. Further, we illustrate features of UISVD-based entropies in other well-known setups, from simple quantum mechanical bipartite states to random matrices relevant to quantum chaos and holography, and in the context of Chern-Simons theory. In all cases, the UISVD yields stable, physically meaningful entropic spectra that are invariant under rescalings and normalisations.

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arXiv2025

Variations on a Theme of Krylov

Spread complexity uses the distribution of support of a time-evolving state in the Krylov basis to quantify dispersal across accessible dimensions of a Hilbert space. Here, we describe how variations in initial conditions, the Hamiltonian, and the dimension of the Hilbert space affect spread complexity and Krylov basis structure. We introduce Koherence, the entropy of coherence between perturbed and unperturbed Krylov bases, which can, e.g., quantify dynamical amplification of differences in initial conditions in chaos. To illustrate, we show that dynamics on SL(2,R), SU(2), and Heisenberg-Weyl group manifolds, often used as paradigmatic settings for contrasting chaotic and integrable (semi-)classical behavior, display distinctively different responses to variations of the initial state or Hamiltonian. We then describe a lattice model that displays linear growth of spread complexity, saturating for bounded lattices and continuing forever in a thermodynamic limit. The latter example illustrates a breakdown of continuum/classical effective descriptions of complexity growth in bounded quantum systems.

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arXiv2025

Symmetry-Resolved Spread Complexity

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.

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arXiv2025

Gauge Invariants at Arbitrary $N$ and Trace Relations

We investigate conformal field theories with gauge group $U(N)$ at arbitrary rank $N$, focusing on the role of trace relations in determining the structure of the Hilbert space. Working in the free trace algebra without imposing relations, we identify a class of evanescent states that vanish at finite $N$. Using the Koszul complex of [1], we implement trace relations systematically via ghosts and a fermionic charge $Q_b$. This framework allows us to define and compute transition amplitudes between evanescent and physical states, which we show correspond precisely to ordinary CFT amplitudes analytically continued in $N$. Our results provide a direct algebraic realization of the proposals which realize trace relations in the bulk as over-maximal giant gravitons [1-3] and establish analytic continuation in $N$ as a powerful tool for understanding finite-$N$ effects.

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arXiv2025

Growth of block diagonal operators and symmetry-resolved Krylov complexity

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.

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arXiv2025

Complexity of PXP scars revisited

We revisit a quantum quench scenario in which either a scarring or thermalizing initial state evolves under the PXP Hamiltonian. Within this framework, we study the time evolution of spread complexity and related quantities in the Krylov basis. We find that the Lanczos coefficients $b_n$, as functions of the iteration number $n$, exhibit a characteristic arched growth and decay, followed by erratic oscillations which we refer to as buttress. The arched profile predominantly arises from contributions within the quantum many-body scar subspace, while the buttress is linked to thermalization dynamics. To explain this behavior, we utilize the representation theory of $\mathfrak{s}l_3(\mathbb{C})$, allowing us to decompose the PXP Hamiltonian into a linear component and a residual part. The linear term governs the formation and width of the arch, and we observe that that there exists a threshold of arch width which determines whether a given initial state exhibits scarring. Meanwhile, the residual term accounts qualitatively for the emergence of the buttress. We estimate an upper bound for the extent of the buttress using Lucas numbers. Finally, we demonstrate that spread complexity oscillates periodically over time for scarred initial states, whereas such oscillations are suppressed in thermalizing cases.

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arXiv2025

Quantum complexity in gravity, quantum field theory, and quantum information science

Quantum complexity quantifies the difficulty of preparing a state or implementing a unitary transformation with limited resources. Applications range from quantum computation to condensed matter physics and quantum gravity. We seek to bridge the approaches of these fields, which define and study complexity using different frameworks and tools. We describe several definitions of complexity, along with their key properties. In quantum information theory, we focus on complexity growth in random quantum circuits. In quantum many-body systems and quantum field theory (QFT), we discuss a geometric definition of complexity in terms of geodesics on the unitary group. In dynamical systems, we explore a definition of complexity in terms of state or operator spreading, as well as concepts from tensor-networks. We also outline applications to simple quantum systems, quantum many-body models, and QFTs including conformal field theories (CFTs). Finally, we explain the proposed relationship between complexity and gravitational observables within the holographic anti-de Sitter (AdS)/CFT correspondence.

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arXiv2025

Local Quenches from a Krylov Perspective

In this work, we investigate local quench dynamics in two-dimensional conformal field theories using Krylov space methods. We derive Lanczos coefficients, spread complexity, and Krylov entropies for local joining and splitting quenches in theories on an infinite line, a circle, a finite interval, and at finite temperature. We examine how these quantities depend on the central charge of the underlying conformal field theory and find that both spread complexity and Krylov entropy are proportional to it. Interestingly, Krylov entropies evolve logarithmically with time, mirroring standard entanglement entropies, making them useful for extracting the central charge. In the large central charge limit, using holography, we establish a connection between the rate of spread complexity and the proper momentum of the tip of the end-of-the world brane, which probes the bulk analogously to a point particle. Our results further demonstrate that spread complexity and Krylov entropies are powerful tools for probing non-equilibrium dynamics of interacting quantum systems.

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arXiv2024

Thermal Pseudo-Entropy

In this work, we develop a generalisation of the thermal entropy to complex inverse temperatures, which we call the thermal pseudo-entropy. We show that this quantity represents the pseudo-entropy of the transition matrix between Thermofield Double states at different times. We have studied its properties in various quantum mechanical setups, Schwarzian theory, Random Matrix Theories, and 2D CFTs, including symmetric orbifolds. Our findings indicate a close relationship between the averaged thermal pseudo-entropy and the spectral form factor, which is instrumental in distinguishing chaotic and integrable models. Moreover, we have observed a logarithmic scaling of this quantity in models with a continuous spectrum, with a universal coefficient that is sensitive to the scaling of the density of states near the edge of the spectrum. Lastly, we found the connection between the real and imaginary parts of the thermal pseudo-entropy through the Kramers-Kronig relations.

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arXiv2024

Spread Complexity Rate as Proper Momentum

We demonstrate a precise relation between the rate of complexity of quantum states excited by local operators in two-dimensional conformal field theories and the radial momentum of particles in 3-dimensional Anti-de Sitter spacetimes. Similar relations have been anticipated based on qualitative models for operator growth. Here, we make this correspondence sharp with two key ingredients: the precise definition of quantum complexity given by the spread complexity of states, and the match of its growth rate to the bulk momentum measured in the proper radial distance coordinate.

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arXiv2024

Musings on SVD and pseudo entanglement entropies

Pseudo-entropy and SVD entropy are generalizations of the entanglement entropy that involve post-selection. In this work we analyze their properties as measures on the spaces of quantum states and argue that their excess provides useful characterization of a difference between two (i.e. pre-selected and post-selected) states, which shares certain features and in certain cases can be identified as a metric. In particular, when applied to link complement states that are associated to topological links via Chern-Simons theory, these generalized entropies and their excess provide a novel quantification of a difference between corresponding links. We discuss the dependence of such entropy measures on the level of Chern-Simons theory and determine their asymptotic values for certain link states. We find that imaginary part of the pseudo-entropy is sensitive to, and can diagnose chirality of knots. We also consider properties of these entropy measures for simpler quantum mechanical systems, such as generalized SU(2) and SU(1,1) coherent states, and tripartite GHZ and W states.

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arXiv2024

Krylov complexity and gluon cascades in the high energy limit

We point out an interesting connection between the mathematical framework of the Krylov basis, which is used to quantify quantum complexity, and the entanglement entropy in high-energy QCD. In particular, we observe that the cascade equation of the dipole model is equivalent to the $SL(2,R)$ Schrodinger equation in the Krylov basis. Consequently, the Krylov complexity corresponds to the average distribution of partons and the Krylov entropy is the counterpart the entanglement entropy computations of \cite{Kharzeev:2017qzs}. Our work not only brings new tools for exploring quantum information and complexity in QCD, but also gives hope for experimental tests of some of the recent, physical probes of quantum complexity.

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arXiv2024

Krylov complexity of density matrix operators

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ($C_K$) and Spread complexity ($C_S$) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between $C_K$ and $2C_S$. Furthermore, for maximally entangled pure states, we find that the moment-generating function of $C_K$ becomes the Spectral Form Factor and, at late-times, $C_K$ is simply related to $NC_S$ for $N\geq2$ within the $N$-dimensional Hilbert space. Notably, we confirm that $C_K = 2C_S$ holds across all times when $N=2$. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

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arXiv2023

Krylov complexity of modular Hamiltonian evolution

We investigate the complexity of states and operators evolved with the modular Hamiltonian by using the Krylov basis. In the first part, we formulate the problem for states and analyse different examples, including quantum mechanics, two-dimensional conformal field theories and random modular Hamiltonians, focusing on relations with the entanglement spectrum. We find that the modular Lanczos spectrum provides a different approach to quantum entanglement, opening new avenues in many-body systems and holography. In the second part, we focus on the modular evolution of operators and states excited by local operators in two-dimensional conformal field theories. We find that, at late modular time, the spread complexity is universally governed by the modular Lyapunov exponent $λ^{mod}_L=2π$ and is proportional to the local temperature of the modular Hamiltonian. Our analysis provides explicit examples where entanglement entropy is indeed not enough, however the entanglement spectrum is, and encodes the same information as complexity.

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