Law of entanglement
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Understanding the Law of Entanglement in Quantum Theory
Introduction to Entanglement Theory and Thermodynamics
Entanglement is a fundamental concept in quantum theory and a crucial resource in quantum information science. The manipulation of entanglement is governed by specific laws, particularly the non-increase of entanglement under local operations on constituent particles. These laws draw formal analogies to the second law of thermodynamics, where entropy determines the accessibility of states. However, unlike thermodynamics, entanglement manipulation under local operations exhibits fundamental irreversibility, preventing a straightforward order similar to the second law.
Reversible Theory of Entanglement
A reversible theory of entanglement can be established when considering all non-entangling transformations. In this context, the asymptotic relative entropy of entanglement plays a role analogous to entropy in thermodynamics. This approach is useful for general resource theories and quantum information theory, providing a rigorous relationship with thermodynamics .
Entanglement and Thermodynamical Laws
The laws of entanglement theory can be seen as equivalent to thermodynamical laws if the theory is made reversible by adding certain bound entangled states as a free resource during entanglement manipulation. However, this reversibility is not generally achievable, which has significant implications for the thermodynamics of entanglement.
Area Laws for Entanglement Entropy
In quantum many-body systems, physical interactions are typically local, leading to a decay of correlation functions and scaling laws for entanglement entropy. The entanglement entropy of ground states often scales with the boundary area of a subregion rather than its volume, known as "area laws." These laws have significant implications in fields such as black hole physics, quantum information science, and quantum many-body physics .
Implications of Area Laws
Area laws for entanglement entropy are crucial for the numerical simulation of lattice models. They help quantify the effective degrees of freedom needed in simulations, making it possible to efficiently simulate quantum states using methods like matrix-product states and entanglement renormalization .
First Law of Entanglement
The first law of entanglement has been used to derive the linearized Einstein equations of holographic dual spacetimes. In Quasi-Topological gravity, the first law of entanglement can be explicitly derived using the Iyer-Wald formalism. This law also extends to boundary conformal field theories (BCFT), providing constraints for a broad range of covariant bulk Lagrangians .
Entanglement as Geometry and Flow
The connection between the area law for entanglement and geometry can be represented using a generalized adjacency matrix. This matrix coincides with the mutual information between pairs of sites, providing a natural entanglement contour. For one-dimensional conformal invariant systems, the generalized adjacency matrix is given by the two-point correlator of an entanglement current operator, suggesting that entanglement current may give rise to a metric entirely built from entanglement.
Conclusion
The law of entanglement in quantum theory reveals deep connections with thermodynamics, geometry, and information theory. While the manipulation of entanglement under local operations exhibits fundamental irreversibility, a reversible theory can be established under certain conditions. Area laws for entanglement entropy play a crucial role in understanding the scaling behavior of quantum systems and have significant implications for numerical simulations. The first law of entanglement provides insights into the dynamics of holographic dual spacetimes and boundary conformal field theories, further enriching our understanding of quantum entanglement.
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