Emergence of a second law of thermodynamics in isolated quantum systems

Abstract: The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schr\"odinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value -- their equilibrium value. This ultimately raises the question: in what sense does the entropy of an isolated quantum system increase over time? For classical systems, one introduces the assumption of a low entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the equilibration on average of observables. While the statistical behavior of observable expectation values is well-established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends towards its equilibrium value in the course of the system's unitary evolution. These results also support recent findings which question the necessity of non-integrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations which expose the underlying reversible evolution in accordance with the derived bounds.

• The paper shows that observable-based entropy increases in isolated quantum systems, formulating a quantum version of the second law.
• The paper employs quantitative analysis and numerical simulations via a quantum Ising model to establish equilibration bounds.
• The paper highlights that a low-entropy initial state is key to reconciling quantum reversibility with classical thermodynamic irreversibility.

Emergence of a Second Law of Thermodynamics in Isolated Quantum Systems

The paper "Emergence of a Second Law of Thermodynamics in Isolated Quantum Systems" by Florian Meier et al. addresses a fundamental conundrum in the intersection of quantum mechanics and thermodynamics: the reconciliation of thermodynamic irreversibility with quantum mechanical reversibility. The research provides a robust theoretical framework demonstrating that isolated quantum systems, despite the reversibility of their unitary evolution, adhere to a variant of the second law of thermodynamics. This is achieved by evaluating entropy through the lens of observables rather than the entire microscopic state.

Key Contributions

1. Observable-Based Entropy Increase: The paper introduces the concept of entropy increase in terms of observables. The authors derive novel bounds for the equilibration of quantum systems relative to specific observables, demonstrating that observable-based entropy tends towards equilibrium values over time. This notion of observability is pivotal, as it circumvents the constancy of von Neumann entropy in unitary evolution by focusing on a coarse-grained view of the system.
2. Quantitative Analysis: The authors quantitatively connect the statistical behavior of expectation values to entropy increase. They present bounds that show entropy relative to an observable equilibrates within the unitary evolution framework. Specifically, they prove that the Shannon observable entropy is on average close to its equilibrium value, thus formulating a second law within quantum mechanics.
3. Role of Initial State: A critical component of this research is the assumption of a low-entropy initial state, known as the past hypothesis. By incorporating both initial conditions and observable-based measurements, the paper aligns with classical thermodynamic views that entropy increases from a low-entropy past.
4. Numerical Simulations: Using a quantum Ising model, the authors illustrate their theoretical findings. The numerical results depict how observable entropy increases to an equilibrium value for a quantum Ising chain. These results simulate the theoretical framework and show transient entropy fluctuations, supporting the paper's bounds.

Implications and Future Directions

The implications of this research are profound, particularly in understanding quantum thermodynamics and developing quantum technologies. The reconciliation of irreversibility with quantum mechanics inspires several future research directions:

• Quantum Information Theory: This framework could be expanded to explore connections with quantum information theory, particularly concerning entropic uncertainty relations and the thermodynamics of quantum information processing.
• Practical Applications: As quantum systems are increasingly pivotal in technological advances, understanding their thermodynamic properties could lead to innovations in quantum computing and energy-efficient quantum devices.
• Further Exploration of Non-Integrability: The findings question the need for non-integrability as a prerequisite for quantum thermalization. Investigating the role of integrability and effective dimension in other quantum systems could yield new insights into the dynamics of quantum equilibrium.

In summary, this paper challenges and extends our comprehension of thermodynamic laws in quantum physics by proposing an observable-based method to account for entropy increase. It navigates the intricate landscape of quantum mechanics to uphold a nuanced version of the second law, potentially bridging gaps between quantum physics and thermodynamics while setting a foundation for future explorations in quantum technology and information science.