Quantum Relaxation Demonstrates Initial-State Typicality As System Size Increases, Enabling Re-examination of Liouvillian Gap and Maximal Relaxation Time

Relaxation, the process by which systems settle into equilibrium, underpins much of science and technology, yet the role of the starting conditions remains surprisingly unclear. Ruicheng Bao from The University of Tokyo, and colleagues, now demonstrate a remarkable phenomenon in complex systems, revealing that relaxation becomes largely independent of the initial state as the system grows larger. This research establishes a principle of ‘typicality’ in high-dimensional systems, proving that thermalisation occurs predictably above a certain temperature, regardless of how the process begins. The findings not only introduce new concepts, such as the ‘typical strong Mpemba effect’ and ‘typical relaxation time’, but also offer a pathway to accelerate relaxation processes and establish a reliable benchmark for assessing state preparation, moving beyond traditional worst-case scenarios.

Relaxation in open quantum systems represents a fundamental process within quantum science and technologies. Researchers systematically characterise the relaxation behaviour of generic initial states and uncover a typicality phenomenon in high-dimensional Hilbert spaces. This typicality manifests as a strong preference for initial states that rapidly drive the system towards its stationary state, irrespective of specific details. The study demonstrates that most initial states exhibit remarkably similar relaxation rates, converging towards a characteristic timescale determined by the system’s spectral properties. This finding challenges the conventional focus on specific initial states and suggests that relaxation behaviour is largely independent of initial state preparation, offering a new perspective on the dynamics of open quantum systems.

Dissipative Ising Model, Derivations and Analysis

This document details the theoretical and numerical foundations of a study investigating accelerated relaxation in dissipative quantum systems. The research explores a phenomenon where a system can reach equilibrium faster under certain conditions, specifically within the Transverse Field Ising Model with dissipation. The document provides detailed mathematical derivations, explanations of analytical choices, and supporting numerical results. It offers a deep dive into the technical aspects of the research and its underlying principles. The document establishes the importance of understanding the conditions under which accelerated relaxation occurs and presents a detailed theoretical framework, including the use of the Lindblad equation to describe the time evolution of the quantum system under dissipation.

The authors perform an eigenvalue analysis to understand the relaxation dynamics and employ perturbation theory to analyze how small changes in the system affect the eigenvalues and relaxation rates. A significant portion of the work focuses on the normalization of eigenvectors and its impact on the scaling of the results, comparing different normalization schemes and discussing their implications. The team also derives conditions for the concentration of specific modes, crucial for understanding the relaxation dynamics, and establishes scaling conditions for the variance of certain observables. The document presents numerical simulations of the dissipative Transverse Field Ising Model, demonstrating how the variance of a specific observable changes with system size and temperature. Supporting figures illustrate the numerical results and validate the theoretical analysis. Additional details and explanations are provided in appendices to offer further clarity.

High-dimensional open quantum systems exhibit relaxation that becomes nearly independent of the initial state as system size increases, under verifiable conditions. The research proves this typicality applies to many thermalisation processes above a size-independent temperature. These findings extend typicality to transient open quantum dynamics, identifying a class of systems where the established understanding of relaxation timescales requires re-examination. The work formalises these observations with two new concepts, the “typical strong Mpemba effect” and the “typical relaxation time”, providing a new framework for understanding relaxation behaviour. Beyond these conceptual advances, the results provide a scalable route to accelerating relaxation.

Typicality Dictates Open Quantum System Relaxation

This research establishes the phenomenon of initial-state typicality in the relaxation of open quantum systems, demonstrating that, as system size increases, relaxation behaviour becomes largely independent of the initial state under specific conditions. This finding challenges the conventional reliance on established measures of relaxation processes, suggesting these metrics require re-evaluation. To address this, the team introduced the ‘typical strong Mpemba effect’ as a diagnostic tool and proposed the ‘typical relaxation time’ as a more relevant timescale when considering typicality. Beyond these conceptual advances, the work provides a practical basis for accelerating relaxation and offers a new benchmark for assessing the efficiency of algorithms, such as quantum Gibbs samplers.

The researchers acknowledge that typicality does not hold universally, and further investigation is needed to understand relaxation behaviour in fluctuation-dominated regimes. Future research directions include constructing physical models that exhibit a clear transition into the typicality regime, exploring potential universal statistical laws governing relaxation in atypical scenarios, and investigating the implications of typicality for metastable states in quantum computation. The newly defined typical relaxation time offers a powerful tool that may enable proofs of rapid mixing under broader conditions and redefine performance benchmarks across quantum science, including state preparation and quantum simulations.