![]() 14 to find quantum corrections to the prescription.ĭiscussions of area laws such as equations (1) and (2) have so far been constrained to the von Neumann entropy. Similar techniques were used in refs 11, 12, 13 to generalize the Ryu–Takayanagi prescription to cases where the bulk theory involves higher derivative gravity, and used in ref. Nonetheless, Lewkowycz and Maldacena 9 overcame this difficulty and showed that the Ryu–Takayanagi prescription (2) follows from gauge/gravity duality, by applying the replica trick and generalizing the Euclidean method developed in refs 2, 10 of calculating gravitational entropies to cases without a U(1) symmetry. ![]() In more general cases, there is no U(1) symmetry to facilitate such a derivation. Not surprisingly, this horizon is mapped back to the Ryu–Takayanagi minimal surface in the original problem. The latter problem was then solved by gauge/gravity duality, which tells us that the thermal state of the CFT is dual to a hyperbolic black hole in the bulk, and the thermal entropy is given by the area of the black hole horizon according to equation (1). This elegant prescription for holographic entanglement entropy was initially proven in the special case of spherical entangling regions in the vacuum state of a conformal field theory (CFT), by employing a U(1) symmetry to map the problem to one of finding the thermal entropy of the CFT on a hyperboloid 8. Here we follow the standard terminology of referring to the dual spacetime in which the gravitational theory lives as the bulk, and identify the spacetime in which the QFT lives with the asymptotic boundary of the bulk spacetime. In particular, this means that the minimal surface is anchored at the entangling surface ∂ A. The minimal surface is constrained to be at a moment of time-reflection symmetry in the bulk and homologous to the entangling region A. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity. Applying this we provide the first holographic calculation of mutual Rényi 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 Rényi 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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