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arXiv:2511.02013 · Journal of High Energy Physics

Spread and circuit complexity as a measure of particle content and phase space fluctuations

Satyaki Chowdhury

quant-phSubmitted Nov 3, 2025• Announced Nov 5, 2025• 1 citation
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Abstract

In this work, we investigate the relation between different notions of quantum complexity, namely, circuit and spread complexity and physically meaningful quantities such as the particle content of the quantum state and the variances of position and momentum operators. Using a harmonic oscillator with time-dependent mass and frequency as a toy model, we show that both circuit and spread complexity at any instant is determined by the mean number of quanta and its rate of change. Furthermore, both complexity and its growth are directly linked to the variances of the position and momentum operators, providing a clear physical interpretation of complexity in terms of the state's excitation and phase-space fluctuation. Although the analysis is carried out for a single time-dependent oscillator, the results have direct relevance for quantum field theory in curved backgrounds, where individual field modes effectively behave as time-dependent oscillators. This offers new insights into how quantum complexity encodes particle production and phase space fluctuations in non-holographic systems. Finally, we establish a precise and potentially universal relation between spread and circuit complexity for the time evolved state suggesting deeper connections between different complexity measures in the context of field theories on curved backgrounds.

Comments
31 pages, 6 figures. Comments are welcome

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In this work, we investigate the relation between different notions of quantum complexity, namely, circuit and spread complexity and physically meaningful quantities such as the particle content of the quantum state and the variances of position and momentum operators. Using a harmonic oscillator with time-dependent mass and frequency as a toy model, we show that both circuit and spread complexity at any instant is determined by the mean number of quanta and its rate of change. Furthermore, both complexity and its growth are directly linked to the variances of the position and momentum operators, providing a clear physical interpretation of complexity in terms of the state's excitation and phase-space fluctuation. Although the analysis is carried out for a single time-dependent oscillator, the results have direct relevance for quantum field theory in curved backgrounds, where individual field modes effectively behave as time-dependent oscillators. This offers new insights into how quantum complexity encodes particle production and phase space fluctuations in non-holographic systems. Finally, we establish a precise and potentially universal relation between spread and circuit complexity for the time evolved state suggesting deeper connections between different complexity measures in the context of field theories on curved backgrounds.
This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.
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 three-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.
The determination of structures that can emerge out of the interactions among the constituents of a quantum many-body system is a foundational task of condensed matter physics.One of the most advanced proposals within this paradigm is the holographic generation of spacetime in some strongly coupled chaotic systems with particular patterns of entanglement and quantum complexity, which is historically motivated by the holographic principle and by the AdS/CFT correspondence.As a part of this scenario, this thesis is dedicated to the analysis of some concepts that are relevant to such proposal, where they are separately applied to some lattice models.Concretely, the matter is divided into three main studies: first, the calculation of the time-dependent circuit complexity in the Ising model with a periodically driven transverse field, where we establish the effectiveness of this quantity for the detection of nonequilibrium quantum phase transitions.Our results provide hints for understanding how universal features out of equilibrium are captured by the complexity of quantum states.Second, the derivation of a bound on the maximal rate of entanglement entropy production for a class of one-dimensional quantum circuits with periodic dynamics.An example of a circuit that saturates the bound is composed by parallel SWAP gates acting on entangled pairs.Out of inequalities obeyed by the entropy, in addition to considerations on multipartite entanglement, we indicate that the effect of a chaotic dynamics cannot result in the increase on the rate of entanglement production.Third, the construction of a class of supersymmetric many-body systems using symmetric inverse semigroups.For particular toy models built out of this structure, we study some questions regarding supersymmetric phases, integrability, disorder, spreading of quantum information and many-body localization.Finally, besides those three works, we include an essay addressing the problem of emergent holographic spaces out of two-dimensional conformal field theories, where the mediation between the two parts is performed by means of a Riemannian structure defined in the Hilbert space of the field theory.
Neutrino flavor oscillations, a fundamental process in particle physics, involve the transition between different neutrino flavors, governed by a complex relationship between mass and flavor eigenstates. Traditionally, probabilistic measures are used to study neutrino oscillations, but recent research suggests that quantum complexity, specifically quantum spread complexity, offers deeper insights, especially regarding charge-parity (CP) violation. Our work shows that quantum complexity can predict CP-violating phases that align with experimental observations from T2K and NOvA. This approach provides a new connection between quantum information theory and particle physics, revealing the potential of complexity in neutrino studies.
In the Wigner-Weyl phase space formulation of quantum mechanics, we analyse the problem of the spreading of an initial state or an initial operator under time evolution when described in terms of the Krylov basis. After constructing the phase space functions corresponding to the Krylov basis states generated by a Hamiltonian from a given initial state by using the Weyl transformation, we subsequently use them to cast the Krylov state complexity as an integral over the phase space in terms of the Wigner function of the time-evolved initial state, so that the contribution of the classical Liouville equation and higher-order quantum corrections to the Wigner function time evolution equation towards the Krylov state complexity can be identified. Next, we construct the double phase space functions associated with the Krylov basis for operators by using a suitable generalisation of the Weyl transformation applicable for superoperators, and use them to rewrite the Krylov operator complexity as an integral over the double phase space in terms of a generalisation of the usual Wigner function. These results, in particular, show that the complexity measures based on the expansion of a time-evolved state (or an operator) in the Krylov basis can be thought to belong to a general class of complexity measures constructed from the expansion coefficients of the time-dependent Wigner function in an orthonormal basis in the phase space, and help us to connect these complexity measures with measures of complexity of time evolved state based on harmonic expansion of the time-dependent Wigner function.
We relate the probability distribution of the work done on a statistical system under a sudden quench to the Lanczos coefficients corresponding to evolution under the postquench Hamiltonian. Using the general relation between the moments and the cumulants of the probability distribution, we show that the Lanczos coefficients can be identified with physical quantities associated with the distribution, e.g., the average work done on the system and its variance, as well as the higher order cumulants. In a sense this gives an interpretation of the Lanczos coefficients in terms of experimentally measurable quantities. Consequently, our approach provides a way towards understanding spread complexity, a quantity that measures the spread of an initial state with time in the Krylov basis generated by the postquench Hamiltonian, from a thermodynamical perspective. We illustrate these relations with two examples. The first one involves quench done on a harmonic chain with periodic boundary conditions and with nearest neighbor interactions. As a second example, we consider mass quench in a free bosonic field theory in ddd spatial dimensions in the limit of large system size. In both cases, we find out the time evolution of the spread complexity after the quench and relate the Lanczos coefficients with the cumulants of the distribution of the work done on the system.