Core Concepts

The evolution speed of a subsystem, measured by the rate of change in its reduced density matrix, serves as a reliable indicator of relaxation in quantum many-body systems, as demonstrated across various models.

Abstract

Zhang, J., Rajabpour, M. A., Heyl, M., & Khasseh, R. (2024). Subsystem Evolution Speed as Indicator of Relaxation. *arXiv preprint arXiv:2410.17798*.

This paper introduces a novel method for assessing relaxation in isolated quantum many-body systems by analyzing the evolution speed of subsystems, aiming to provide a tool that bypasses the limitations of traditional approaches relying on local operators or prior knowledge of the steady state.

The researchers define the evolution speed of a subsystem as the rate of change of its reduced density matrix over time, utilizing the trace distance as a measure of distance between quantum states. They then investigate the behavior of this metric in various quantum models, including the chaotic Ising chain, XXZ chains with and without many-body localization (MBL), and the transverse field Ising chain, comparing their findings with established relaxation indicators.

The study reveals that the subsystem evolution speed decreases as the overall system size increases in systems approaching relaxation, particularly for subsystems smaller than half the total system size. This trend is consistent with the behavior of the subsystem trace distance to the steady state and aligns with the predictions of the eigenstate thermalization hypothesis (ETH). The researchers demonstrate the robustness of their method across different initial states and models, including those exhibiting thermalization, integrability, and MBL.

The subsystem evolution speed offers a reliable and accurate indicator of relaxation in quantum many-body systems, independent of specific local operators or prior knowledge of the steady state. This approach provides a valuable tool for investigating relaxation dynamics in complex quantum systems, potentially offering insights into the transition from integrable to ergodic dynamics.

This research introduces a novel perspective on analyzing relaxation in quantum systems, potentially impacting the study of quantum dynamics, thermalization, and the characteristics of many-body localized phases. The proposed method's simplicity and reliance solely on the system's state make it a powerful tool for exploring a wide range of quantum phenomena.

While the study validates the method across several models, further investigation into more complex systems, such as those with higher dimensions, long-range interactions, and diverse symmetries, is necessary to confirm its general applicability. Additionally, a deeper theoretical understanding of the relationship between subsystem evolution speed and fundamental dynamical processes in quantum systems is crucial for fully leveraging its potential.

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Stats

For subsystems smaller than half the size of the total system, the subsystem evolution speed follows similar dynamics to that of the subsystem trace distance.
As the total system size increases, the evolution speed of sufficiently small subsystems decreases, indicating the system's progression towards a steady state.

Quotes

"In this letter, we propose an alternative relaxation indicator that relies solely on the quantum system’s state, eliminating the necessity for local operators or a predefined steady state."
"This indicator quantifies the rate of change of the time-dependent subsystem RDM, thereby representing the subsystem’s evolution speed in a manner analogous to the geometric approach to the quantum speed limit."
"As a subsystem approaches a steady state, its evolution speed vA(t) progressively decreases in the thermodynamic limit and ultimately approaches zero."

Key Insights Distilled From

by Jiaju Zhang,... at **arxiv.org** 10-24-2024

Deeper Inquiries

Applying the subsystem evolution speed method to open quantum systems presents a fascinating challenge. Here's a breakdown of the complexities and potential adaptations:
Challenges:
Non-Unitary Dynamics: Open quantum systems evolve non-unitarily due to entanglement with the environment. The simple relationship between evolution speed and energy fluctuations, derived for closed systems, no longer holds.
Steady State Ambiguity: The nature of the steady state in open systems is highly dependent on the system-environment interaction. It might not be a thermal state or a well-defined GGE.
Information Leakage: The environment acts as a sink for information. Assessing relaxation solely based on the system's state might not capture the complete picture.
Adaptations:
Modified Evolution Speed: Instead of the trace distance, one could employ measures tailored for open systems, such as the Bures metric or quantum relative entropy, to define a modified evolution speed.
Effective Hamiltonians: In certain regimes, open system dynamics can be approximated by effective non-Hermitian Hamiltonians. The evolution speed concept might be adaptable using these effective descriptions.
Environment Monitoring: To fully understand relaxation, it might be necessary to track information flow to the environment. This could involve studying the evolution speed of the combined system and environment or focusing on specific environmental degrees of freedom.
Research Directions:
Markovian vs. Non-Markovian: Investigating how the evolution speed behaves in both Markovian (memoryless) and non-Markovian open systems would be insightful.
Dissipative Phase Transitions: Exploring the evolution speed near dissipative phase transitions, where the steady state changes qualitatively, could reveal new universal behavior.
Experimental Feasibility: Assessing the feasibility of measuring the adapted evolution speed in experimental platforms for open systems, such as trapped ions or superconducting qubits, is crucial.

Yes, a decrease in subsystem evolution speed might not always guarantee relaxation, especially in systems far from equilibrium. Here are some alternative interpretations:
Prethermalization Plateaus: In certain systems, the evolution speed might decrease and seemingly plateau at a non-equilibrium value for long times before eventually relaxing to the true steady state. This phenomenon, known as prethermalization, can occur due to the presence of approximate conservation laws or emergent symmetries.
Slow Dynamics and Metastability: Systems with complex energy landscapes, such as glassy systems, can exhibit extremely slow dynamics and get trapped in metastable states. The evolution speed might decrease significantly in these metastable states, mimicking relaxation, even though the system hasn't reached true equilibrium.
Finite-Size Effects: For small system sizes, a decrease in evolution speed might be a finite-size artifact. As the system size increases, the evolution speed might saturate at a non-zero value, indicating the absence of true relaxation in the thermodynamic limit.
Non-Ergodic Behavior: In systems with many degrees of freedom and weak interactions, ergodicity might break down. The system's dynamics could be confined to a limited portion of phase space, leading to a decrease in evolution speed without exploring the full range of accessible states.
Distinguishing Relaxation:
To differentiate true relaxation from these alternative scenarios, it's crucial to:
Examine Long-Time Behavior: Simulations or experiments should be performed for sufficiently long times to rule out prethermalization plateaus or slow dynamics.
System Size Scaling: Analyzing the evolution speed as a function of system size is essential to identify finite-size effects and extrapolate to the thermodynamic limit.
Other Relaxation Indicators: Combining the evolution speed analysis with other relaxation indicators, such as the behavior of local observables or entanglement measures, can provide a more comprehensive picture.

Applying the concept of subsystem evolution speed to the entire universe is highly speculative but thought-provoking. Here are some potential implications and limitations:
Implications:
Cosmic Expansion and Evolution Speed: The universe's expansion could be seen as a form of "evolution" in a cosmological sense. Investigating whether the evolution speed of cosmic subsystems (e.g., galaxy clusters) decreases or increases over cosmological timescales might provide insights into the dynamics of dark energy and the universe's fate (e.g., Big Freeze, Big Rip).
Information Content and Entropy: The evolution speed of cosmic subsystems could be related to the rate of information processing or entropy generation within them. A decreasing evolution speed might suggest a universe tending towards a state of maximum entropy (heat death), while an increasing speed could indicate ongoing structure formation or other non-equilibrium processes.
Quantum Nature of Cosmology: Applying quantum information-theoretic concepts like evolution speed to cosmology could shed light on the quantum nature of spacetime and gravity, potentially offering clues about the very early universe and quantum gravity theories.
Limitations:
Observational Constraints: Measuring the evolution speed of cosmic subsystems over cosmological timescales is practically impossible with current technology. We can only observe snapshots of the universe at different epochs.
Fundamental Assumptions: The concept of a closed quantum universe relies on assumptions about the nature of quantum mechanics, gravity, and the universe's boundary conditions, which are still under active debate.
Emergent Description: The evolution speed, as defined for quantum systems, might not be directly applicable to the universe as a whole, which could be governed by emergent laws and principles beyond our current understanding.
Research Avenues:
Theoretical Frameworks: Developing theoretical frameworks that connect quantum information theory, cosmology, and quantum gravity is crucial to explore these ideas rigorously.
Cosmological Simulations: Numerical simulations of cosmic structure formation, incorporating quantum effects, could provide insights into the evolution speed of cosmic subsystems.
Analog Gravity Systems: Studying analog gravity systems, such as condensed matter systems that mimic aspects of gravity, might offer experimental insights into the interplay between quantum mechanics and spacetime.

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