Abstract This article reviews the progress in our understanding of the reconstruction of the bulk spacetime in the holographic correspondence from the dual field theory including an account of how these developments have led to the reproduction of the Page curve of the Hawking radiation from black holes. We review quantum error correction and relevant recovery maps with toy examples based on tensor networks, and discuss how it provides the desired framework for bulk reconstruction in which apparent inconsistencies with properties of the operator algebra in the dual field theory are naturally resolved. The importance of understanding the modular flow in the dual field theory has been emphasized. We discuss how the state-dependence of reconstruction of black hole microstates can be formulated in the framework of quantum error correction with inputs from extremal surfaces along with a quantification of the complexity of encoding of bulk operators. Finally, we motivate and discuss a class of tractable microstate models of black holes which can illuminate how the black hole complementarity principle can emerge operationally without encountering information paradoxes, and provide new insights into generation of desirable features of encoding into the Hawking radiation.
Abstract This article reviews the progress in our understanding of the reconstruction of the bulk spacetime in the holographic correspondence from the dual field theory including an account of how these developments have led to the reproduction of the Page curve of the Hawking radiation from black holes. We review quantum error correction and relevant recovery maps with toy examples based on tensor networks, and discuss how it provides the desired framework for bulk reconstruction in which apparent inconsistencies with properties of the operator algebra in the dual field theory are naturally resolved. The importance of understanding the modular flow in the dual field theory has been emphasized. We discuss how the state-dependence of reconstruction of black hole microstates can be formulated in the framework of quantum error correction with inputs from extremal surfaces along with a quantification of the complexity of encoding of bulk operators. Finally, we motivate and discuss a class of tractable microstate models of black holes which can illuminate how the black hole complementarity principle can emerge operationally without encountering information paradoxes, and provide new insights into generation of desirable features of encoding into the Hawking radiation.
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.
Tensor networks have previously been used to construct holographic quantum error correcting codes (HQECC), which are toy models of the AdS/CFT correspondence. HQECC have been able to exhibit many of the interesting features of the duality, such as complementary recovery and redundant encoding. However, in previous HQECC the boundary Hamiltonian which results from mapping a local bulk Hamiltonian to the boundary is a non-local Hamiltonian, with global terms that act over the entire boundary. In this work, we combine HQECC with recently developed Hamiltonian simulation theory to construct a bulk-boundary mapping in which local bulk Hamiltonians map to local boundary Hamiltonians. This allows us to construct a toy model of a complete holographic duality between models, not just states and observables. This mapping of local Hamiltonians extends the toy models to encompass the relationship between bulk and boundary energy scales, and in particular the mapping between bulk and boundary time dynamics. We discuss what insight this gives into toy models of black hole formation.
Abstract When the bulk geometry in AdS/CFT contains a black hole, boundary subregions may be sufficient to reconstruct certain bulk operators if and only if the black hole microstate is known, an example of state dependence. Reconstructions exist for any microstate, but no reconstruction works for all microstates. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction α of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the α-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being ‘pushed’ through the black hole itself. Finally, we show that black holes provide the first ‘explicit’ examples of capacity-achieving α-bit codes. Unintuitively, Hawking radiation always reveals the α-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in 1/N.
Central phenomena in quantum information processing– from the coding gainof quantum error-correcting codes, to the information mapping between boundaryand bulk in the AdS/CFT holographic duality, to the Page curve of informationrecovery during black hole evaporation– can be unified under the concept of “information compression cost”. This paper proposes that any quantum informationoperation must pay an irreversible entropy production jointly determined by Landauer’s principle and the Bekenstein bound, and that systems under these costconstraints seek to minimize a cost functional representing the difference betweendescription complexity and predictive (or protective) capability. The encodingprocess of stabilizer codes is modeled as the information compression required toprotect logical qubits against noise. The Ryu-Takayanagi formula in holographicduality indicates that the boundary theory must pay an information maintenancecost proportional to the area of the extremal surface in order to maintain its effective description of the bulk geometry. The Page curve of the black hole informationparadox is interpreted as a first-order kinetic phase transition on the cost landscapebetween two minima– one corresponding to an information-lost state and the otherto an information-recovered state. This framework integrates quantum error correction, the holographic principle, and the black hole information problem into aunified picture of information economics, and presents quantitative relations thatare amenable to further testing.
A bstract Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.
Abstract: It was shown by [2] how bulk operators in the AdS/CFT correspondence can be represented on the boundary analogously to the way logical qubits are represented in an encoded subspace in quantum error correction. Then in [1] holographic tensor networks that serve as toy models of the bulk boundary. This paper reviews some of the developments of [1] and [2]. Then it is demonstrated explicitly how to construct perfect tensors, which are essential to the tensor networks mentioned in [2]. Lastly a new example of a holographic quantum error-correcting code based on an eight index perfect tensor is presented.