Area laws in quantum systems: mutual information and correlations (0704.3906v2)

Abstract: The holographic principle states that on a fundamental level the information content of a region should depend on its surface area rather than on its volume. This counterintuitive idea which has its roots in the nonextensive nature of black-hole entropy serves as a guiding principle in the search for the fundamental laws of Planck-scale physics. In this paper we show that a similar phenomenon emerges from the established laws of classical and quantum physics: the information contained in part of a system in thermal equilibrium obeys an area law. While the maximal information per unit area depends classically only on the number of microscopic degrees of freedom, it may diverge as the inverse temperature in quantum systems. A rigorous relation between area laws and correlations is established and their explicit behavior is revealed for a large class of quantum many-body states beyond equilibrium systems.

• The paper establishes that quantum systems in equilibrium obey area laws, with mutual information scaling with the boundary rather than the volume.
• The paper reveals that mutual information decays with distance, governed by a characteristic correlation length that quantifies entanglement.
• The paper proves that mixed PEPS satisfy area laws, enabling the modeling of complex quantum states as finitely correlated systems.

Analysis of "Area laws in quantum systems: mutual information and correlations"

This paper addresses the concept of area laws as they pertain to quantum systems, specifically focusing on mutual information and correlations. The authors examine how quantum systems in equilibrium adhere to an area law, a principle often associated with the holographic theory that suggests the information content of a region is related to its boundary area rather than its volume. The research provides a comprehensive paper of this phenomenon in non-critical and critical quantum systems, discussing the conditions under which area laws hold and the implications of a finite correlation length.

The authors delve into the distinctions between classical and quantum systems in relation to correlation scaling, demonstrating that while classical systems typically follow a volume law, non-critical quantum lattice systems adhere to an area law where information content scales with the boundary. For quantum systems, they conclude that the divergence from classical systems is related to quantum entanglement and the nature of correlations, as quantified by the mutual information.

The paper explores how mutual information serves as a pivotal measure of entanglement and correlation in quantum systems, equating under certain conditions to the entanglement entropy. The mutual information framework provides insights into how correlations decay with distance and how an area law can be rigorously proven at any finite temperature. A central finding is that the mutual information between two regions decreases as the distance between them increases and this decay is governed by a characteristic correlation length.

Another significant component of the paper is the examination of projected entangled pair states (PEPS) as they pertain to area laws. The paper proves that an area law is satisfied by all mixed PEPS, indicating that these states have a representation as finitely correlated states (FCS) in one dimension. This offers a powerful tool for understanding the structure and behavior of quantum states beyond Gibbs states.

Among the speculative conclusions, the authors assert that in one-dimensional systems, saturation of mutual information implies the system's structure can be captured by a quantum Markov chain, highlighting the critical role of mutual information in characterizing complex quantum states. The paper further explores the implications of mutual information on correlation lengths, with mutual information providing a bound on traditional correlation functions.

The findings from this research have substantial implications for theoretical and practical applications in quantum information theory and condensed matter physics. Understanding area laws in quantum systems aids in developing efficient computational methods for simulating quantum systems, particularly those based on tensor networks such as PEPS. This work sets a foundation for future exploration into critical vs. non-critical behaviors, entropy scaling, and possible extensions of the area law to more generalized quantum systems and interactions.

In conclusion, the paper embodies a significant stride in elucidating the properties of quantum correlations and entanglement, highlighting the profound distinctions between classical and quantum regimes. Future research could extend these findings to higher dimensions or non-equilibrium states, explore the role of mutual information in other forms of quantum entanglement, or explore practical implementations in quantum computing and information technologies.