Andreas Karch

This paper summarizes recent discussions of the Page curve and the information paradox, and responds to the reasoning and examples from arXiv:2506.04311. We review arguments demonstrating that in quantum gravity the algebra of observables at infinity is complete, both in AdS and in asymptotically flat space. This completeness implies that the bulk Hilbert space in quantum gravity does not factorize along the radial direction, undermining a key common assumption in Hawking's argument for information loss and in initial derivations of the Page curve. As a consequence, in a standard theory of gravity, information does not ``emerge'' from a black hole in the manner suggested by the Page curve; rather, it is already encoded in asymptotic observables. Relatedly, the full black hole interior, and not just an ``island'', can be reconstructed from exterior data. Page curves and islands can be obtained by removing the Hamiltonian from the exterior algebra. This may be implemented operationally by restricting access to part of the asymptotic region (a detector with a ``blind spot'') or, in the special case of null infinity in asymptotically flat spacetimes, by formally discarding the Hamiltonian from the set of observables despite its physical accessibility. Such Page curves describe only the redistribution of information between measured and unmeasured degrees of freedom, rather than fundamental information recovery. Finally, Page curves and islands also arise when a black hole is coupled to a nongravitational bath, a setup that yields a nonstandard theory of gravity. We show how, even in this setting, the unusual localization of information in gravity provides a concrete physical mechanism for information transfer from the gravitational system into the bath.

A bstract: The entanglement entropy of intervals in 1 + 1 interface CFTs is modified in two ways compared to a CFT without interface: there is a finite boundary entropy contribution, and, for an interval with an endpoint at the interface, the coefficient of the logarithmically divergent contribution -which is usually proportional to the central charge of the CFT- is modified to an effective central charge. We show that the latter modification can be understood as a limit of the former using holographic duals of interface CFTs. Furthermore, we show that a finite “boundary entropy” contribution also appears in intervals that do not cross the interface and it is needed to ensure strong subbaditivity of the entanglement entropy.

A bstract: We exploit the holographic realization of a conformal theory coupled to an external bath realized via a double trace deformation and its gravity dual in terms of transparent boundary conditions in order to map out some basic dissipative properties of this simple open holographic system. In particular, we determine the energy transmission coefficient across the boundary, discover a novel duality relating weak and strong coupling to the external bath, and quantify the dissipation in the system by working out the quasi normal modes.

The circuit complexity of time-evolved pure quantum states grows linearly in time for an exponentially long time. This behavior has been proven in certain models, is conjectured to hold for generic quantum many-body systems, and is believed to be dual to the long-time growth of black hole interiors in AdS/CFT. Achieving a similar understanding for mixed states remains an important problem. In this work, we study the circuit complexity of time-evolved subsystems of pure quantum states. We find that for greater-than-half subsystem sizes, the complexity grows linearly in time for an exponentially long time, similarly to that of the full state. However, for less-than-half subsystem sizes, the complexity rises and then falls, returning to low complexity as the subsystem equilibrates. Notably, the transition between these two regimes occurs sharply at half system size. We use holographic duality to map out this picture of subsystem complexity dynamics and rigorously prove the existence of the sharp transition in random quantum circuits. Furthermore, we use holography to predict features of complexity growth at finite temperature that lie beyond the reach of techniques based on random quantum circuits. In particular, at finite temperature, we argue for an additional sharp transition at a critical less-than-half subsystem size. Below this critical value, the subsystem complexity saturates nearly instantaneously rather than exhibiting a rise and fall. This novel phenomenon, as well as an analogous transition above half system size, provides a target for future studies based on rigorous methods.

A bstract: In 1+1 dimensional conformal field theory with a boundary the boundary contribution to the entanglement entropy is determined by a single number g effectively counting the boundary degrees of freedom. In contrast, in 1+1 dimensional interface CFTs the corresponding quantity is a non-trivial function depending on the position of the interval relative to the interface, giving access to much more detailed information about the defect. In this work we determined this g -function in several examples using holography and derive some of its basic properties from holography and strong subadditivity.

We use the ideas behind the duality web to construct numerous conformal field theories mediating the phase transitions between various symmetry broken and topological phases. In particular we obtain the full field theory version of the Kennedy Tasaki transformation, mapping a gapless theory mediating a topological phase transition of symmetry protected topological orders to a standard symmetry breaking one in a 1+1 dimensional ℤ ₂ × ℤ ₂ gauge theory. When we consider all possible discrete gauging operations, we obtain bosonic and fermionic webs with 9 critical theories per web, each connecting 4 separate gapped phases, some of them topological. Bosonization maps the two webs into each other. In addition to discussing the multi-critical theory connecting the four gapped phases in each phase diagram, we discuss the partially gapped theories connecting two of those four. Some of these are gapless symmetry protected topological phases.

We calculate the transmission coefficient of the super-Janus interface conformal field theory, both at weak and at strong coupling, where latter is described holographically as a domain-wall solution on AdS₂ × S ² × M ₄ × Σ. Surprisingly we find perfect agreement between the free and strong coupling answer, mirroring a similar unexpected equivalence previously found for the entanglement entropy.