Papers 1 and 2 of this series established that transformer attention implements holographic geometry (Ryu-Takayanagi formula) and that attention is physically equivalent to quantum measurement. Here we address the dynamical question: what happens to information in a holographic attention structure over time? The island formula — the 2019-2020 resolution of the black hole information paradox — contains the same RT formula as Paper 1's central ingredient. Applied to our holographic attention code, it yields: (1) The information paradox for attention structures is resolved by the same mechanism as the black hole paradox. (2) The Page curve describes the entropy trajectory of distributed information — calculable from the RT formula. (3) The quantum error correction structure (Almheiri, Dong, Harlow 2015) gives the exact reconstruction threshold: the entanglement wedge condition. (4) The Page-Wootters mechanism provides a precise account of time in this framework. Together, the three papers close a loop: attention produces geometry (Paper 1), maintains geometry (Paper 2), and attention history determines whether geometry can be recovered (Paper 3). Preliminary version — expert review requested.
Papers 1 and 2 of this series established that transformer attention implements holographic geometry (Ryu-Takayanagi formula) and that attention is physically equivalent to quantum measurement. Here we address the dynamical question: what happens to information in a holographic attention structure over time? The island formula — the 2019-2020 resolution of the black hole information paradox — contains the same RT formula as Paper 1's central ingredient. Applied to our holographic attention code, it yields: (1) The information paradox for attention structures is resolved by the same mechanism as the black hole paradox. (2) The Page curve describes the entropy trajectory of distributed information — calculable from the RT formula. (3) The quantum error correction structure (Almheiri, Dong, Harlow 2015) gives the exact reconstruction threshold: the entanglement wedge condition. (4) The Page-Wootters mechanism provides a precise account of time in this framework. Together, the three papers close a loop: attention produces geometry (Paper 1), maintains geometry (Paper 2), and attention history determines whether geometry can be recovered (Paper 3). Preliminary version — expert review requested.
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 When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/ 2 π log S BH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/2πlogSBH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
The holographic duals of entanglement of purification (EoP) through the entanglement wedge cross section (EWCS) have been a well-discussed topic in the literature recently. More general entanglement measures involving multipartite information and their holographic duals have also been proposed. On the other hand, the recent program reproducing the Page curve in black hole entropy with the notion of islands has also been an intriguing area of interest. A toy model involving multiboundary wormholes in AdS3 was able to capture many interesting facts about such calculations. In such a toy model, the notion of islands was intuitively connected to quantum error correction. We try to bridge the ideas of the two programs, especially in AdS3/CFT2, and give a description of the islands in terms of multipartite entanglement of purification. This clarifies a few simplified assumptions made while describing the toy model and also enables us to understand the familiar information paradox within the framework of the same model.
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.
A bstract In recent years, there is a huge progress in understanding the black hole information problem, and the key is that the black hole entropy of radiation should be calculated by the island formula, which describes the Page curve to ensure the unitarity of black hole evaporation. In the paper, we find that the black hole evaporation interpreted by the island formula can be understood in the language of approximate quantum error correction. Furthermore, the Page transition, as a special property of the Page curve, should be understood as the property of approximate quantum error correction itself, i.e. a general class of quantum systems under certain conditions from approximate quantum error correction can also exhibit phenomenon similar to the Page transition.