State-dependent bulk-boundary maps and black hole complementarity

We provide a simple and explicit construction of local bulk operators that describe the interior of a black hole in the AdS/CFT correspondence. The existence of these operators is predicated on the assumption that the mapping of CFT operators to local bulk operators depends on the state of the CFT. We show that our construction leads to an exactly local effective field theory in the bulk. Barring the fact that their charge and energy can be measured at infinity, we show that the commutator of local operators inside and outside the black hole vanishes exactly, when evaluated within correlation functions of the CFT. Our construction leads to a natural resolution of the strong subadditivity paradox of Mathur and Almheiri et al. Furthermore, we show how, using these operators, it is possible to reconcile small corrections to effective field theory correlators with the unitarity of black hole evaporation. We address and resolve all other arguments, advanced in A. Almheiri et al. J. High Energy Phys. 09 (2013) 018 and D. Marolf and J. Polchinski, Phys. Rev. Lett. 111, 171301 (2013), in favor of structure at the black hole horizon. We extend our construction to states that are near equilibrium, and thereby also address the ``frozen vacuum'' objections of R. Bousso, Phys. Rev. Lett. 112, 041102 (2014). Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.