We argue that the `island conjecture' and the replica wormhole derivation of the Page curve break monogamy of entanglement through allowing black hole interior states to be non-classically correlated while also pairwise entangled with radiation states. The reason is that quantum degrees of freedom (present in any half of a Hawking pair) cannot all be identified with the environment at semi-classical pair production, and can only be fixed relative to a subsystem, as required for the Page curve, by correlations equivalent to entanglement - regardless of what those correlations are attributed to. This implies that the recent gravity (replica wormhole) and holographic (island conjecture) derivations of the Page curve entail new physics not yet properly taken into account.
We argue that the `island conjecture' and the replica wormhole derivation of the Page curve break monogamy of entanglement through allowing black hole interior states to be non-classically correlated while also pairwise entangled with radiation states. The reason is that quantum degrees of freedom (present in any half of a Hawking pair) cannot all be identified with the environment at semi-classical pair production, and can only be fixed relative to a subsystem, as required for the Page curve, by correlations equivalent to entanglement - regardless of what those correlations are attributed to. This implies that the recent gravity (replica wormhole) and holographic (island conjecture) derivations of the Page curve entail new physics not yet properly taken into account.
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.
Abstract Recent work has shown how to understand the Page curve of an evaporating black hole from replica wormholes. However, more detailed information about the structure of its quantum state is needed to fully understand the dynamics of black hole evaporation. Here we study entanglement negativity, an important measure of quantum entanglement in mixed states, in a couple of toy models of evaporating black holes. We find four phases dominated by different types of geometries: the disconnected, cyclically connected, anti-cyclically connected, and pairwise connected geometries. The last of these geometries are new replica wormholes that break the replica symmetry spontaneously. We also analyze the transitions between these four phases by summing more generic replica geometries using a Schwinger-Dyson equation. In particular, we find enhanced corrections to various negativity measures near the transition between the cyclic and pairwise phase.
In a close look at the replica wormhole derivation of the island conjecture, we note a discrepancy in the derived matrix eigenvalues in the limit where the Hawking radiation entropy goes to zero. The small corrections to the density matrix for the radiation infer that tr(ρR2), equally a sum over the eigenvalues squared, must be small (≪1) at all times. This is in contradiction with an ensemble dominated by a density matrix with an eigenvalue close to unity. Also, the radiation entropy has been approximated by the ensemble averaged entropy, as in a final state analysis. For ρR=ρR(σ) with σ the set of variables used in the average, the ensemble average is only an accurate estimate of the entropy after σ has been measured. At the formation of each Hawking pair, there is a distribution over σ that affects the entropy. Without a real time determination of σ, the radiation entropy follows Hawking's result during the evaporation. A question remains of how quickly such determinations can be assumed to occur. A complete argument for that the radiation entropy follows the Page curve should address this.
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.
Abstract Of course not, but if one believes that information cannot be destroyed in a theory of quantum gravity, then we run into apparent contradictions with quantum theory when we consider evaporating black holes. Namely that the no-cloning theorem or the principle of entanglement monogamy is violated. Here, we show that neither violation need hold, since, in arguing that black holes lead to cloning or non-monogamy, one needs to assume a tensor product structure between two points in space-time that could instead be viewed as causally connected. In the latter case, one is violating the semi-classical causal structure of space, which is a strictly weaker implication than cloning or non-monogamy. This is because both cloning and non-monogamy also lead to a break-down of the semi-classical causal structure. We show that the lack of monogamy that can emerge in evaporating space times is one that is allowed in quantum mechanics, and is very naturally related to a lack of monogamy of correlations of outputs of measurements performed at subsequent instances of time of a single system. This is due to an interesting duality between temporal correlations and entanglement. A particular example of this is the Horowitz-Maldacena proposal, and we argue that it needn’t lead to cloning or violations of entanglement monogamy. For measurements on systems which appear to be leaving a black hole, we introduce the notion of the temporal product, and argue that it is just as natural a choice for measurements as the tensor product. For black holes, the tensor and temporal products have the same measurement statistics, but result in different type of non-monogamy of correlations, with the former being forbidden in quantum theory while the latter is allowed. In the case of the AMPS firewall experiment we find that the entanglement structure is modified, and one must have entanglement between the infalling Hawking partners and early time outgoing Hawking radiation which surprisingly tames the violation of entanglement monogamy.
Black holes are windows into the deepest structures of reality, where quantum mechanics, thermodynamics, and geometry converge. In this pedagogical preprint we explore how quantum entanglement entropy provides a microscopic foundation for black hole thermodynamics. Starting from rst principles of quantum information theory, we introduce entanglement entropy, the area law, and Hawking radiation, then proceed to modern results: the Ryu-Takayanagi formula, quantum extremal surfaces, the Page curve, replica wormholes, and the island formula. Throughout, the emphasis is on physical intuition. No prior knowledge of string theory or loop quantum gravity is assumed.