Krylov-Ehrenfest Time Defines Limit of Classical-Quantum Complexity

Researchers at the University of Geneva and University of Southampton develop and study a classical Lanczos algorithm by replacing quantum commutators with Poisson brackets, allowing us to define Krylov complexity using the symplectic structure of phase space. This approach demonstrates that classical Krylov complexity accurately approximates quantum complexity during early-time chaotic dynamics, offering a characteristic scale for these systems. We define a Krylov-Ehrenfest time, which quantifies the eventual divergence of classical and quantum complexities. Applying this framework to the Lipkin-Meshkov-Glick and Feingold-Peres models, collective spin systems exhibiting distinct chaotic behaviors, the analysis shows that saddle points enhance the K-complexity saturation value locally at their energy shell in the LMG model, but leave the rest of the spectrum consistent with integrable behavior.

Classical Lanczos Algorithm Defines Krylov Complexity in Phase Space

A newly defined connection between classical and quantum mechanics reveals a Krylov-Ehrenfest time, which quantifies the eventual divergence of classical and quantum complexities. This advancement utilizes Poisson brackets in place of quantum commutators, enabling the definition of Krylov complexity through phase-space integrals that provide the inner product needed to define the Lanczos recursion. We show that, in theories with well-defined semiclassical limits, classical Krylov complexity accurately approximates quantum complexity at early times. This framework is formulated with microcanonical inner products, and the formulations agree in the classical limit.

Our microcanonical framework resolves this tension by showing that saddle points enhance the K-complexity saturation value locally at their energy shell in the LMG model, but leave the rest of the spectrum consistent with integrable behavior. This approach allows for a fine-grained study of complexity, energy shell by energy shell, revealing nuances previously obscured by global spectral averages.

Researchers are extending the reach of Krylov complexity, a tool for understanding chaotic systems, into the classical realm by reformulating the mathematics to operate on phase space rather than Hilbert space. Phase-space integrals now define the inner product necessary for the Lanczos recursion, a critical step in bridging the quantum and classical descriptions. This allows for the accurate approximation of quantum complexity by its classical counterpart during early-time chaotic dynamics. To test this framework, the researchers applied it to the Lipkin-Meshkov-Glick (LMG) and Feingold-Peres (FP) models, collective spin systems known to classicalize in the thermodynamic limit. This microcanonical approach allows for the isolation of phenomena tied to specific energies, such as unstable fixed points in classical dynamics.

N. De Ro at the University of Geneva and A. Sánchez-Garrido at the University of Southampton study a classical analogue of the Krylov framework, with Poisson brackets taking on the role of the quantum commutators and phase-space integrals furnishing the inner product needed to define the Lanczos recursion. This allows for an exploration of how accurately classical simulations can mirror quantum behavior, particularly in the initial stages of complex interactions.

Krylov-Ehrenfest Time Quantifies Classical-Quantum Complexity Divergence

Researchers are refining tools to measure the boundary between predictable classical systems and the inherently probabilistic world of quantum mechanics, with a newly defined Krylov-Ehrenfest time pinpointing when classical and quantum descriptions diverge. This time scale, derived from the mathematical framework of Krylov complexity, quantifies the depth within a Krylov chain, a sequence generated by repeated application of an operator, and translates to a well-known scale in generic chaotic systems. The work, detailed in recent publications, moves beyond simply acknowledging a classical limit to quantum systems; it defines a quantifiable point of separation. This convergence holds when formulated with microcanonical inner products, and the formulations agree in the classical limit.

Conventional Krylov complexity analyses often obscure crucial details by averaging across all energy levels; however, a new approach refines this by enabling a detailed, “energy shell by energy shell” investigation of complexity in both classical and quantum systems. This framework allows for a fine-grained study of complexity, energy shell by energy shell.

Researchers are extending the mathematical toolkit of quantum complexity to the classical realm, revealing surprising connections between seemingly disparate systems. Central to this advance is the adaptation of spherical harmonics to function as a classical analogue to the operator basis used in quantum theory; this allows for a phase-space formulation of Krylov complexity. This approach isn’t simply about finding a classical limit to quantum calculations, but rather constructing a fully-fledged classical framework for understanding complexity.

Application to Lipkin-Meshkov-Glick (LMG) and Feingold-Peres (FP) Models

These models serve as crucial test cases, allowing for direct comparison between quantum and classical Krylov dynamics. While the FP model exhibits spectral chaos across certain coupling values, the LMG model is notable for displaying a particular form of chaotic behavior.

The researchers demonstrated that dynamical instabilities are present, such as saddle points in the LMG phase space or the onset of chaos in the FP model, driving classical Krylov complexity to grow exponentially at early times. The microcanonical formulation of the classical LMG model revealed that while enhanced growth persists at the energy shell containing the saddle point, fluctuations suppress early-time growth away from it, offering a more nuanced understanding of complexity in integrable systems.

The team’s work studies the analysis of the Lipkin-Meshkov-Glick (LMG) model, a collective spin system previously known to display seemingly chaotic behavior despite its inherent integrability. Applying this framework to the LMG model, the researchers discovered that saddle points enhance the K-complexity saturation value locally at their energy shell in the LMG model, but leave the rest of the spectrum consistent with integrable behavior. This contrasts sharply with the Feingold-Peres (FP) model, which exhibits genuine chaos and uniform saturation across all energies. This nuanced understanding of the LMG model’s behavior offers a powerful tool for analyzing complex quantum systems and their classical limits.

The ability to accurately model complex systems is paramount across diverse fields, from materials science to financial forecasting, and recent advances in understanding chaotic dynamics are refining these capabilities. The Krylov-Ehrenfest time translates to a well-known scale in generic chaotic systems, suggesting a fundamental link between the two realms. Notably, the LMG model exhibits a unique behavior: saddle points enhance the K-complexity saturation value locally at its energy shell in the LMG model, but leave the rest of the spectrum consistent with integrable behavior.