Quantum Error Correction and Holography, Krylov Complexity, and continuous Tensor Networks

In this thesis, we study a couple of different but related questions: holography and quantum error correction, the gravitationally dressed operators and locality in holographic theory, introducing generalized versions of the class of continuous matrix product state as the most famous class of tensor network, and the notion of Krylov complexity of matrix quantum mechanics. In the first part, we use the notion of the Petz map and apply it to reconstruct operators in the entanglement wedge. Moreover, for geometries that contain black holes, we generalize the notion of the Petz map by itself and reconstruct black hole interior modes. In particular, we show that the Petz map reconstruction of the black hole interior is equivalent to the Papadodimas-Raju Proposal. In the second part, within the AdS/CFT correspondence, we identify a class of CFT operators which represent diff-invariant and approximately local observables in the gravitational dual. The interpretation of these observables is that they are not gravitationally dressed with respect to the boundary, but instead to features of the state. We also provide evidence that there are bulk observables whose commutator vanishes to all orders in 1/N with the entire algebra of single-trace operators defined in a space-like separated time-band. In the third part, we defined two new classes of continuous tensor networks by generalizing the class of continuous matrix product states. One is appropriate to describe the ground state of the relativistic field theory at strong coupling and the other is proper for the theories on compact spacetime. We conjecture that the possible bulk dual of it in the AdS/CFT can be an evaporating black hole microstates with the end of the word branes. In the last project, we study the Krylov complexity for 1-matrix quantum mechanics. In the ground state, the Lancsoz coefficients have linear behavior. The bn coefficients have a positive slope while an coefficients have a negative slope. In the thermal states, the an coefficients are zero while bn have two branches of linear growth. Although matrix quantum mechanics is a solvable theory rather than chaotic, we find the linear behavior of the Lancsoz coefficients of this theory.