The interior of a pure-state black hole is known to be reconstructed from the Petz map by collecting a sufficiently large amount of the emitted Hawking radiation. This was established based on the Euclidean replica wormhole, which comes from an ensemble averaging over gravitational theories. On the other hand, this means that the Page curve and the interior reconstruction are both ensemble averages; thus, there is a possibility of large errors. In the previous study [R. Bousso and M. Miyaji, Fluctuations in the entropy of Hawking radiation], it was shown that the entropy of the Hawking radiation has fluctuation of order e−SBH , thus is typical in the ensemble. In the present article, we show that the fluctuations in the information recovery are similarly suppressed by the factor e−SBH by studying the fluctuations of the relative entropy difference in the encoding map and the entanglement fidelity of the Petz map. This result establishes the typicality of the information recovery in the ensemble. In addition, we also compute the entanglement loss of the encoding map. Published by the American Physical Society 2024
The interior of a pure-state black hole is known to be reconstructed from the Petz map by collecting a sufficiently large amount of the emitted Hawking radiation. This was established based on the Euclidean replica wormhole, which comes from an ensemble averaging over gravitational theories. On the other hand, this means that the Page curve and the interior reconstruction are both ensemble averages; thus, there is a possibility of large errors. In the previous study [R. Bousso and M. Miyaji, Fluctuations in the entropy of Hawking radiation], it was shown that the entropy of the Hawking radiation has fluctuation of order e−SBH , thus is typical in the ensemble. In the present article, we show that the fluctuations in the information recovery are similarly suppressed by the factor e−SBH by studying the fluctuations of the relative entropy difference in the encoding map and the entanglement fidelity of the Petz map. This result establishes the typicality of the information recovery in the ensemble. In addition, we also compute the entanglement loss of the encoding map. Published by the American Physical Society 2024
We use the gravitational path integral (GPI) to compute the fluctuations of the Hawking radiation entropy around the Page curve in a two-dimensional model introduced by Penington Before the Page time, we find that δS=e−S/2 , where S is the black hole entropy. This result agrees with the Haar-averaged entropy fluctuations in a bipartite system. After the Page time, we find that δS∼e−S , up to a prefactor that depends logarithmically on the width of the microcanonical energy window. This is not symmetric under exchange of subsystem sizes and so does not agree with the Haar average for a subsystem of fixed Hilbert space dimension. The discrepancy can be attributed to the fact that the black hole Hilbert space dimension is not fixed by the state preparation: even in a microcanonical ensemble with a top-hat smearing function, the GPI yields an additive fluctuation in the number of black hole states. This result, and the fact that the Page curve computed by the GPI is smooth, all point towards an ensemble interpretation of the GPI. Published by the American Physical Society 2024
A bstract Recent work has shown how to obtain the Page curve of an evaporating black hole from holographic computations of entanglement entropy. We show how these computations can be justified using the replica trick, from geometries with a spacetime wormhole connecting the different replicas. In a simple model, we study the Page transition in detail by summing replica geometries with different topologies. We compute related quantities in less detail in more complicated models, including JT gravity coupled to conformal matter and the SYK model. Separately, we give a direct gravitational argument for entanglement wedge reconstruction using an explicit formula known as the Petz map; again, a spacetime wormhole plays an important role. We discuss an interpretation of the wormhole geometries as part of some ensemble average implicit in the gravity description.
The black hole information puzzle arises from a discrepancy between conclusions drawn from general relativity and quantum theory about the nature of the radiation emitted by a black hole. According to Hawking's original argument, the radiation is thermal and its entropy thus increases monotonically as the black hole evaporates. Conversely, due to the reversibility of time evolution according to quantum theory, the radiation entropy should start to decrease after a certain time, as predicted by the Page curve. This decrease has been confirmed by new calculations based on the replica trick, which also exhibit its geometrical origin: spacetime wormholes that form between the replicas. Here we analyse the discrepancy between these and Hawking's original conclusions from a quantum information theory viewpoint, using in particular the quantum de Finetti theorem. The theorem implies the existence of extra information, W, which is neither part of the black hole nor the radiation, but plays the role of a reference. The entropy obtained via the replica trick can then be identified to be the entropy S(R∣W) of the radiation conditioned on the reference W, whereas Hawking's original result corresponds to the non-conditional entropy S(R). The entropy S(R∣W), which mathematically is an ensemble average, gains an operational meaning in an experiment with N independently prepared black holes: For large N, it equals the normalised entropy of their joint radiation, S(R1⋯RN)/N. The discrepancy between this entropy and S(R) implies that the black holes are correlated. The replica wormholes may thus be interpreted as the geometrical representation of this correlation. Our results also suggest a many-black-hole extension of the widely used random unitary model, which we support with non-trivial checks.
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.
The black hole information paradox is one of the important questions of physics in the last few decades. We study the reconstruction of the bulk operators in AdS/CFT when the geometry contains a black hole. The black hole exterior can be mapped to the CFT via a very simple Petz map which coincides with the HKLL map reconstruction of the black hole exterior. For the interior modes of the bulk theory, using the definition of the Petz recovery channel in modular theory, we can find the mapping from the black hole interior to the dual boundary theory. In the case of the evaporating black hole, it is expected that the interior modes map to some operators that have support only on the bath system, the cavity that absorbs the Hawking radiation. The most important observation that we have here is that in the case that we have a typical black hole microstate in the bulk, the CFT dual of the interior modes that we can find using the Petz recovery channel are exactly the operators that so-called "mirror operator " in the Papadodimas-Raju proposal. The main idea of the PR proposal is to focus on a code subspace of the CFT theory, which is created by acting with a small algebra on the corresponding pure state and then fnd the CFT description of the interior operator in a state-dependent manner just in the chosen code subs. Therefore, we can interpret Papadodimas- Raju proposal as an example of the Petz map reconstruction. It may help us answer some open questions about their procedure.
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.