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arXiv:2610.31394

Krylov Complexity and Entropy of Composite Quantum Systems

Jeff Murugan, Victor Van Aswegen, Hendrik J R Van Zyl

hep-thquant-phSubmitted Oct 22, 2026• Announced Oct 23, 2026• 0 citations
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Abstract

We investigate the behaviour of Krylov complexity as the number of degrees of freedom of a quantum system increases. Specifically, we show that for tensor products of commuting subsystems, the leading excess complexity is controlled by a weighted variance of the subsystem Lanczos coefficients. This provides a quantitative measure of dynamical heterogeneity and a precise characterization of synchronous evolution. We also show that Krylov entropy is able to disambiguate the information associated with subsystems composition from that generated by dynamical mismatch, and that the same Lanczos variance controls the leading mismatch contribution to both complexity and the entropy. Our results establish excess complexity and Krylov entropy as complementary probes of the emergence of collective dynamics in large quantum systems.

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We investigate the behaviour of Krylov complexity as the number of degrees of freedom of a quantum system increases. Specifically, we show that for tensor products of commuting subsystems, the leading excess complexity is controlled by a weighted variance of the subsystem Lanczos coefficients. This provides a quantitative measure of dynamical heterogeneity and a precise characterization of synchronous evolution. We also show that Krylov entropy is able to disambiguate the information associated with subsystems composition from that generated by dynamical mismatch, and that the same Lanczos variance controls the leading mismatch contribution to both complexity and the entropy. Our results establish excess complexity and Krylov entropy as complementary probes of the emergence of collective dynamics in large quantum systems.
We study Krylov complexity for quantum systems whose Hamiltonians factorise as tensor products. We prove that complexity is superadditive under tensor products, C12≥C1+C2C_{12}\ge C_1+C_2C12​≥C1​+C2​, and identify a positive operator that quantifies the resulting excess complexity. The underlying mechanism is made transparent by introducing a Krylov graph representation in which tensor products generate a higher-dimensional lattice whose diagonal shells encode operator growth and binomial path multiplicities. In the continuum limit, Krylov dynamics reduces to diffusion on this graph, with superadditivity arising from geometric broadening across shells. Explicit examples illustrate how deviations from synchronous evolution generate bounded, oscillatory excess complexity.
This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around t=0t=0t=0. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a ``proof of principle'' for using Krylov complexity as a complete characterization of operator evolution in quantum systems.
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φφφ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φφφ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φφφ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.
In this work we study the relationship between quantum random walks on graphs and Krylov/spread complexity. We show that the latter's definition naturally emerges through a canonical method of reducing a graph to a chain, on which we can identify the usual Krylov structure. We use this identification to construct families of graphs corresponding to special classes of systems with known complexity features and conversely, to compute Krylov complexity for graphs of physical interest. The two main outcomes are the analytic computation of the Lanczos coefficients for the SYK model for an arbitrary number qqq of interacting fermions and the complete characterization of Krylov complexity for the hypercube graph in any number of dimensions. The latter serves as the starting point for an in-depth comparison between Krylov and circuit complexities as they purportedly arise in the context of black holes. We find that while under certain conditions Krylov complexity follows the growth and saturation pattern ascribed to such systems, the timescale at which saturation happens can generally be shorter than what is predicted by random unitary circuits, due to the effects of quantum speed-ups commonly occurring when comparing quantum and classical random walks.
Recently, the propagation of information through quantum many-body systems, developed to study quantum chaos, have found many applications from black holes to disordered spin systems. Among other quantitative tools, Krylov complexity has been explored as a diagnostic tool for information scrambling in quantum many-body systems. We introduce a universal limit to the growth of the Krylov complexity in dissipative open quantum systems by utilizing the uncertainty relation for non-Hermitian operators. We also present the analytical results of Krylov complexity for characteristic behavior of Lanczos coefficients in dissipative systems. The validity of these results is demonstrated by explicit study of transverse-field Ising model under dissipative effects.
This paper investigates the notion of Krylov complexity, a measure of operator growth, within the framework of 1-matrix quantum mechanics (1-MQM). Krylov complexity quantifies how an operator evolves over time by expanding it in a series of nested commutators with the Hamiltonian. We analyze the Lanczos coefficients derived from the correlation function, revealing their linear growth even in this integrable system. This growth suggests a link to chaotic behavior, typically unexpected in integrable systems. Our findings in both ground and thermal states of 1-MQM provide new insights into the nature of complexity in quantum mechanical models and lay the groundwork for further studies in more complex holographic theories.