We investigate information flow and Page curves for tripartite systems. We prepare a tripartite system (say, A, B, and C) of a given number of states and calculate information and entropy contents by assuming random states. Initially, every particle was in A (this means a black hole), and as time goes on, particles move to either B (means Hawking radiation) or C (means a broadly defined remnant, including a non-local transport of information, the last burst, an interior large volume, or a bubble universe, etc.). If the final number of states of the remnant is smaller than that of Hawking radiation, then information will be stored by both of the radiation and the mutual information between the radiation and the remnant, while the remnant itself does not contain information. On the other hand, if the final number of states of the remnant is greater than that of Hawking radiation, then the radiation contains negligible information, while the remnant and the mutual information between the radiation and the remnant contain information. Unless the number of states of the remnant is large enough compared to the entropy of the black hole, Hawking radiation must contain information; and we meet the menace of black hole complementarity again. Therefore, this contrasts the tension between various assumptions and candidates of the resolution of the information loss problem.
We consider the entanglement island in a deformed Jackiw-Teitelboim black hole in the presence of the phase transition. This black hole has the van der Waals-Maxwell-like phase structure as it is coupled with a Maxwell field. We study the behavior of the Page curve of this black hole by using the island paradigm. In the fixed charge ensemble, we discuss different situations with different charges that influence the system's phase structure. There is only a Hawking-Page phase transition in the absence of charges, which leads to an unstable small black hole. Hence, the related Page curve does no…
The role played by the mutual information of subsystems on the Page curve is explored in this paper. With the total system consisting of the black hole and radiation, together with the inclusion of island, we observe that the vanishing of mutual information between $B_+$ and $B_-$ which in turn means the disconnected phase of the entanglement wedge corresponding to $B_+\cup B_-$, yields a time scale of the order of scrambling time. This results in a time independent expression for the fine grained entropy of Hawking radiation consistent with the correct Page curve. We also find corrections to …
We obtain the Page curves of an eternal Reissner-Nordström black hole in the presence of higher derivative terms in four dimensions. We consider two cases: gravitational action with general ${\cal O}(R^2)$ terms plus Maxwell term and Einstein-Gauss-Bonnet gravity plus Maxwell term. In both the cases entanglement entropy of the Hawking radiation in the absence of island surface is increasing linearly with time. After including contribution from the island surface, we find that after the Page time, entanglement entropy of the Hawking radiation in both the cases reaches a constant value which is …
The black hole singularity plays a crucial role in formulating Hawking's information paradox. The global spacetime analysis may be reconciled with unitarity by imposing a final state boundary condition on the spacelike singularity. Motivated by the final state proposal, we explore the effect of final state projection in two dimensional conformal field theories. We calculate the time evolution under postselection by employing the real part of pseudo-entropy to estimate the amount of quantum entanglement averaged over histories between the initial and final states. We find that this quantity pos…
The black hole firewall conjecture is based on Page curve hypothesis, which claims that entanglement between black hole and Hawking radiation is almost maximum. The hypothesis is inspired by Lubkin-Lloyd-Pagels-Page theorem for degenerate systems with zero Hamiltonian. Adopting canonical typicality for nondegenerate systems with nonvanishing Hamiltonians, the entanglement becomes nonmaximal,and energetic singularities (firewalls) do not emerge for general systems. For static thermal pure states of black hole and Hawking radiation, entanglement entropy equals thermal entropy of the smaller syst…
We consider a gravity theory coupled to matter, where the matter has a higher-dimensional holographic dual. In such a theory, finding quantum extremal surfaces becomes equivalent to finding the RT/HRT surfaces in the higher-dimensional theory. Using this we compute the entropy of Hawking radiation and argue that it follows the Page curve, as suggested by recent computations of the entropy and entanglement wedges for old black holes. The higher-dimensional geometry connects the radiation to the black hole interior in the spirit of ER=EPR. The black hole interior then becomes part of the entangl…
In this work, we give two proposals regarding the status of connectivity of entanglement wedges and the associated saturation of mutual information. The first proposal has been given for the scenario before the Page time depicting the fact that at a particular value of the observer's time $t_b=t_R$ (where $t_R\llβ$), the mutual information $I(R_+:R_-)$ vanishes representing the disconnected phase of the radiation entanglement wedge. We argue that this time is the Hartman-Maldacena time at which the fine-grained entropy of radiation goes as $S(R)\sim \log(β)$, where $β$ is the inverse of Hawkin…
Black hole thermodynamics suggests that, in order to describe the physics of distant observers, one may model a black hole as a standard quantum system with density of states set by the Bekenstein-Hawking entropy $S_\mathrm{BH}$. This idea has long been considered to be in strong tension with Hawking's prediction that radiation from black holes is nearly thermal, and with low-energy gravity more generally. But the past two years have shown that low-energy gravity does offer a self-consistent description of black hole evaporation consistent with the above idea, and which in particular reproduce…
We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS$_2$), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS$_2$ with a vanishing dilaton, i.e. a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation, as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS$_2$ gravity. We…
A Page curve for an evaporating black hole in asymptotically flat spacetime is computed by adapting the Quantum Ryu-Takayanagi (QRT) proposal to an analytically solvable semi-classical two-dimensional dilaton gravity theory. The Page time is found to be one third of the black hole lifetime, at leading order in semi-classical corrections. A Page curve is also obtained for a semi-classical eternal black hole, where energy loss due to Hawking evaporation is balanced by an incoming energy flux.
Page curve for the fine grained entropy of the black hole radiation is obtained once we include the effect of a non vanishing quantum Fisher information in the evaporation data. This quantum Fisher information scales as the inverse of the original black hole entropy. The modified fine grained entropy can be naturally related to a relative entanglement entropy with the quantum Fisher information defining the curvature at Page point.
We calculate the time evolution of entanglement entropy in two dimensional conformal field theory with a moving mirror. For a setup modeling Hawking radiation, we obtain a linear growth of entanglement entropy and show that this can be interpreted as the production of entangled pairs. For a setup, which mimics black hole formation and evaporation, we find that the evolution follows the ideal Page curve. We perform these computations by constructing the gravity dual of the moving mirror model via holography. We also argue that our holographic setup provides a concrete model to derive the Page c…
Treating Maxwell soft hair as a transition function that relates U(1) gauge fields living in the asymptotic region and near-horizon region, the U(1) gauge parameter $ϕ$ naturally becomes a good label of those Maxwell soft hair degrees of freedom. This interpretation also builds the connection between Maxwell soft hair and U(1) edge modes living in the intermediate region, which admits a well-defined effective action description. We study the statistical properties by Euclidean path integral, which concludes that the soft hair density of state increases with black hole temperature. Hawking radi…
We study the Page curve and the information paradox for the BTZ black hole coupled to two thermal baths by applying the island paradigm. We prove that as the island locates outside the event horizon, the entanglement entropy of Hawking radiation for finite temperature black holes obeys the Page curve. However, for extremal rotating black holes, the Page time and the scrambling time become divergent. To avoid the ill-defined of the Page time, we consider the contribution of superradiance, which is a process that can extract rotational energy from black holes. The superradiance continues in a pe…
In this paper, we study entanglement islands and the Page curve in the eternal four-dimensional Schwarzschild black hole surrounded by finite temperature conformal matter. By finite temperature conformal matter we mean the matter described by the thermal density matrix, rather than the usually considered matter above the Fock vacuum. We take the matter and the black hole at different temperatures and calculate the entanglement entropy for such a setup using the s-wave approximation. As a result, we obtain that at late times the island prescription leads to the exponential growth of the entangl…
We study the black hole information problem within a semiclassically gravitating AdS$_d$ black hole coupled to and in equilibrium with a $d$-dimensional thermal conformal bath. We deform the bath state by a relevant scalar deformation, triggering a holographic RG flow whose "trans-IR" region deforms from a Schwarzschild geometry to a Kasner universe. The setup manifests two independent scales which control both the extent of coarse-graining and the entanglement dynamics when counting Hawking degrees of freedom in the bath. In tuning either, we find nontrivial changes to the Page time and Page …
We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the 2-dimensional asymptotically-flat space. The islands and Page curves of eternal, evaporating and bath-removed black holes are investigated. The different theories in this parameter class are identified through field redefinitions which leave the island invariant. The Page transition is found to occur at the first a third of the black hole life time in the evaporating case for this family of solutions. In addition, we consider gluing the equilibrium black hole and the evaporating one along a null tr…
We reformulate recent insights into black hole information in a manner emphasizing operationally-defined notions of entropy, Lorentz-signature descriptions, and asymptotically flat spacetimes. With the help of replica wormholes, we find that experiments of asymptotic observers are consistent with black holes as unitary quantum systems, with density of states given by the Bekenstein-Hawking formula. However, this comes at the cost of superselection sectors associated with the state of baby universes. Spacetimes studied by Polchinski and Strominger in 1994 provide a simple illustration of the as…
One-shot decoupling is a powerful primitive in quantum information theory and was hypothesized to play a role in the black hole information paradox. We study black hole dynamics modeled by a trilinear Hamiltonian whose semiclassical limit gives rise to Hawking radiation. An explicit numerical calculation of the discretized path integral of the S-matrix shows that decoupling is exact in the continuous limit, implying that quantum information is perfectly transferred from the black hole to radiation. A striking consequence of decoupling is the emergence of an output radiation entropy profile tha…
By applying the island rule proposed recently, we compute the entanglement entropy of Hawking radiation and study the Page curve for the eternal black holes in massive gravity. We investigate for both the neutral and charged black holes which the corresponding results of Schwarzschild and Reissner-Nordström black holes are restored in the limit of massless graviton. We show for the neutral and non-extremal charged black holes that the island is not formed at the early times of the evaporation and hence the entanglement entropy increases linearly in time. However, for the extremal charged black…
In the spirit of studying the information paradox as a scattering problem, we pose and answer the following questions: i) What is the scattering amplitude for $N$ particles to emerge from a large black hole when two energetic particles are thrown into it? ii) How long would we have to wait to recover the information sent in? The answer to the second question has long been expected to be Page time, a quantity associated with the lifetime of the black hole. We answer the first by evaluating an infinite number of `ladder of ladders' Feynman diagrams to all orders in $M_{Pl}/M_{BH}$. Such processe…
We compute the entanglement negativity for various pure and mixed state configurations in a bath coupled to an evaporating two dimensional non-extremal Jackiw-Teitelboim (JT) black hole obtained through the partial dimensional reduction of a three dimensional BTZ black hole. Our results exactly reproduce the analogues of the Page curve for the entanglement negativity which were recently determined through diagrammatic technique developed in the context of random matrix theory.
We propose a model with multiple qubits that reproduces the thermal properties of 4-dimensional (4-dim) Schwarzschild black holes (BHs) by simultaneously taking account of the emission of Hawking particles and the zero-energy soft hair evaporation at horizon. The results verify that the entanglement entropy between a qubit and other subsystems, including emitted radiation, is much larger than the BH entropy analogue of the qubit, as opposed to the Page curve prediction. Our result suggests that early Hawking radiation is entangled with soft hair, and that late Hawking radiation can be highly e…
Quantum corrections to the entanglement entropy of matter fields interacting with dynamical gravity have proven to be very important in the study of the black hole information problem. We consider a one-particle excited state of a massive scalar field infalling in a pure AdS$_3$ geometry and compute these corrections for bulk subregions anchored on the AdS boundary. In the dual CFT$_2$, the state is given by the insertion of a local primary operator and its evolution thereafter. We calculate the area and bulk entanglement entropy corrections at order $\mathcal{O}(N^0)$, both in AdS and its CFT…
Understanding the entanglement of radiation in QFT has been a long standing challenge, with implications ranging from black hole thermodynamics to quantum information. We demonstrate how the case of the free fermion in $1 + 1$ dimensions reveals the details of the density matrix of the radiation produced by a moving mirror. Using the resolvent method rather than standard CFT techniques we derive the Rényi entropies, modular Hamiltonian and flow of the radiation, and determine when mirrors generate unitary transformations.
Recent developments on black holes have shown that a unitarity-compatible Page curve can be obtained from an ensemble-averaged semi-classical approximation. In this paper, we emphasize (1) that this peculiar manifestation of unitarity is not specific to black holes, and (2) that it can emerge from a single realization of an underlying unitary theory. To make things explicit, we consider a hard sphere gas leaking slowly from a small box into a bigger box. This is a quantum chaotic system in which we expect to see the Page curve in the full unitary description, while semi-classically, eigenstate…
In this work new interaction terms between two SYK models are proposed, which allow to define an interaction picture such that it is possible to calculate exactly the vacuum state's time evolution. It is shown that the vacuum evolves as a time dependent $SU(2)$ squezed state. The time dependent entanglement entropy is calculated and it has the same form of the Page curve of Black Hole formation and evaporation.
Asymptotic Causal Diamonds (ACDs) are a natural flat space analogue of AdS causal wedges, and it has been argued previously that they may be useful for understanding bulk locality in flat space holography. In this paper, we use ACD-inspired ideas to argue that there exist natural candidates for Quantum Extremal Surfaces (QES) and entanglement wedges in flat space, anchored to the conformal boundary. When there is a holographic screen at finite radius, we can also associate entanglement wedges and entropies to screen sub-regions, with the system naturally coupled to a sink. The screen and the b…
Seeking for the proper situation of Emergent Gravity, we recent reveal that the entropic mechanism only happens when extremal surfaces are varied, which is similar to the non-gravitational-bath-coupled setup of the island development. In this paper, we consider perturbing thin shell state outside horizon during equilibrating, to find the evolution of these Euclidean states requires an external force to be $ F_{ex}\propto T_{H}δA(μ_a)$, proportional to area variation of the apparent horizon which could transform into the actual event horizon as extremal surface. We analogize the falling tendenc…
We study the Page curve in the four dimensional asymptotically flat eternal Garfinkle-Horowitz-Strominger dilaton black holes by applying the "quantum extremal surface" prescription. The results demonstrate that without island, the entanglement entropy of Hawking radiation is proportional to time and divergent at late times. While taking account of the emergence of the island extending slightly outside the event horizon, the entanglement entropy of Hawking radiation stops growing and eventually reaches a saturation value, which is twice of the Bekenstein-Hawking entropy and consistent with the…
The problem of complete evaporation of Schwarzschild black holes raised by Hawking is that one has an explosion of the temperature $T=1/8πM$ for vanishing black hole mass M. We consider the Reissner-Nordstrom black hole and note that if mass $M$ and charge Q<M satisfy the bound Q>M-C M^3, C>0 for small M then the complete evaporation of black holes without blow-up of temperature is possible. We describe curves on the surface of state equations such that the motion along them provides complete evaporation. The radiation entropy follows the Page curve and vanishes at the end of evaporation. Simi…
We study the generalized entanglement entropy in the higher dimensional two-sided eternal black hole by double holography. By introducing an end-of-the-world ETW brane, which defines the time-dependent effective Hawking radiation region, we find a new type of Ryu-Takayanagi surface besides the Hartman-Maldacena surface and the island Ryu-Takayanagi surface known previously. We study the phase transition among the three Ryu-Takayanagi surfaces at different temperatures and obtain the phase diagram as well as the Page curve.
If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, w…
Motivated by the quantum process of black hole evaporation and its implications for symmetries, we consider a qubit system with a random dynamics as a toy model of black hole. We compute its symmetry-resolved entropies and discuss its implications. We first consider the case where charges are conserved and compute the symmetry-resolved entropies. We derive a symmetry-resolved analogue of the Page curve. We then consider the case where symmetry is explicitly broken and charges are no longer conserved. It serves as a toy model for global symmetry breaking in black hole evaporation. Despite the s…
