We consider a gravity theory coupled to matter, where the matter has a higher-dimensional holographic dual. In such a theory, finding quantum extremal surfaces becomes equivalent to finding the RT/HRT surfaces in the higher-dimensional theory. Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve, as suggested by recent computations of the entropy and entanglement wedges for old black holes. The higher-dimensional geometry connects the radiation to the black hole interior in the spirit of ER=EPR. The black hole interior then becomes part of the entanglement wedge of the radiation. Inspired by this, we propose a new rule for computing the entropy of quantum systems entangled with gravitational systems which involves searching for “islands” in determining the entanglement wedge.

Our earlier paper "Complexity Equals Action" conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the 'Wheeler-DeWitt' patch). We provide calculations for the results quoted in that paper; explain how it fits into a broader (tensor) network of ideas; and elaborate on the hypothesis that black holes are fastest computers in nature.

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti-de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.

As time passes, once simple quantum states tend to become more complex. For strongly coupled k-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we show that the same pattern is exhibited by a much simpler system-classical geodesics on a compact two-dimensional geometry of uniform negative curvature. This striking parallel persists whether the system is allowed to evolve naturally or is perturbed from the outside.

We consider a nearly-AdS 2 gravity theory on the two-sided wormhole geometry. We construct three gauge-invariant operators in NAdS 2 which move bulk matter relative to the dynamical boundaries. In a two-sided system, these operators satisfy an SL(2) algebra (up to non perturbative corrections). In a semiclassical limit, these generators act like SL(2) transformations of the boundary time, or conformal symmetries of the two sided boundary theory. These can be used to define an operator-state mapping. A particular large N and low temperature limit of the SYK model has precisely the same structure, and this construction of the exact generators also applies. We also discuss approximate, but simpler, constructions of the generators in the SYK model. These are closely related to the "size" operator and are connected to the maximal chaos behavior captured by out of time order correlators.