I will first motivate the benefits on taking the algebraic approach to understand AdS/CFT and how an algebra of local operators of QFT has knowledge of a spacetime region. Utilizing both operator algebras and quantum information theory, I will explain a new framework for understanding nonperturbative gravitational aspects of bulk reconstruction with a finite or infinite-dimensional boundary Hilbert space. This will be understood as approximate recovery containing gravitational errors, and the bulk reconstruction in this context will be understood using the privacy/correctability correspondence.