Holography, localisation of information and subregions

Locality is undoubtedly an integral part of both quantum theory and general relativity. On the other hand, a holographic theory like the AdS/CFT implies that the bulk quantum gravitational degrees of freedom are encoded at spatial infinity in the boundary theory. Even though this statement is a claim at the non perturbative level, there are still remnants of this property in the perturbative limits of quantum gravity. This is primarily due to the gravitational Gauss law, which prevents us from defining strictly local operators. Since including gravity in the description is requiring the theory to be invariant under coordinate transformations, the physical operators need to be diffeomorphism invariant. This condition, implemented by Gauss law, demands the operators be dressed to the boundary and include a gravitational version of the Wilson line extending all the way to infinity, therefore demands them to be non-local. Towards resolving this tension, we propose candidate operators that bypass this requirement and are local and diffeomorphism invariant at the same time, in the AdS/CFT context. These operators still satisfy a version of gravitational Gauss law, as they are interpreted to be dressed with respect to the features of the states. Therefore, the states these operators are defined on are states that break the symmetries of the theory and have ’features’. These states are in general high energy states with large variance and correspond to non trivial semiclassical geometries in the bulk. This proposal will also help resolve paradoxes raised concerning the island proposal. In addition, this enables one to discuss subregions, their associated subsystems and localization of information more concretely in perturbative quantum gravity. In the second part, we will be mostly concerned with a bulk sub region called an AdS-Rindler wedge. We will use what is called the Petz map, borrowed from the quantum information and quantum computing community, to explicitly reconstruct this bulk subregion from its boundary dual subregion. This agrees with the earlier conjecture on the bulk subregion reconstruction and the proposal that, due to the quantum error correcting nature of gravity, the Petz map can be used to reconstruct the entanglement wedge. In addition, we study the algebra of operators in the AdS Rindler wedge precisely, both in the bulk and the boundary dual. Using the crossed product construction and a novel method of renormalizing the Ryu Takayanagi surface, we show how including gravitational corrections modifies the algebra to a more manageable one where we can define density matrices and von Neumann entropy. Finally, we study a particular representation of algebra of operators in general backgrounds, called the covariant representation, in situations where gravitational interactions are present. This representation will illuminate what the crossed product construction is in physical terms.