Algebraic perturbation theory: traversable wormholes and generalized entropy beyond subleading order

The crossed product has recently emerged as an important tool in high-energy theory. We combine this with another powerful tool, namely pertubation theory, and study the crossed product algebra of a system under a deformation, relating the structure of deformed observables to that of the undeformed theory. In particular, we derive the change in the von Neumann entropy of the type II algebras, and demonstrate that our approach allows one to formally compute this to arbitrarily high orders in perturbation theory. As a concrete example, we apply this machinery to the case of a double-trace deformation of the thermofield double state in AdS/CFT, which is dual to a traversable wormhole in the bulk, obtaining several new contributions to the generalized entropy relative to the original work by Gao, Jafferis, and Wall. We comment on the relevance of this framework for black hole evaporation and interiors, as well as on the applicability of the algebraic approach to quantum gravity more generally.