9/3/2023 0 Comments Renyi entropyOur result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the more » boundary of region we are studying. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. « lessĮntanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. Specifically, we argue that quantum entanglement on the open string side is mapped onto quantum entanglement on the closed string side and briefly comment on the implications of our result for physical holographic theories where entanglement has been argued to be crucial ingredient for the emergence of classical geometry. Our results provide a framework for efficiently studying Re´nyi entropies and understanding entanglement structures in strongly coupled systems and quantum = ^3$$, the emergent picture is rather general. Applying this we provide the first holographic calculation of mutual Re´nyi information between two disks of arbitrary dimension. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Re´nyi entropies. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area.
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