Many proposed quantum mechanical models of black holes include highly nonlocal interactions. The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity. Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information. It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster. In this article, we address the conjecture from two directions. First, we exhibit two examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph. Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss. Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian. The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.
More indirectly, while most work prior to @cite_40 argued that black holes held information for an amount of time comparable to the black hole lifetime, if not forever, occasional hints were found that information might leak out faster @cite_35 . Reversing the reasoning, one could interpret such arguments as evidence in favour of the fast scrambling conjecture.
@cite_35 @cite_40
1979169890 2129872856
We investigate a recently proposed model for a full quantum description of two-dimensional black hole evaporation, in which a reflecting boundary condition is imposed in the strong-coupling region. It is shown that in this model each initial state is mapped to a well-defined asymptotic out state, provided one performs a certain projection in the gravitational zero mode sector. We find that for an incoming localized energy pulse, the corresponding outgoing state contains approximately thermal radiation, in accordance with semiclassical predictions. In addition, our model allows for certain acausal strong-coupling effects near the singularity that give rise to corrections to the Hawking spectrum and restore the coherence of the out state. To an asymptotic observer these corrections appear to originate from behind the receding apparent horizon and start to influence the outgoing state long before the black hole has emitted most of its mass. Finally, by putting the system in a finite box, we are able to derive some algebraic properties of the scattering matrix and prove that the final state contains all initial information. We study information retrieval from evaporating black holes, assuming that the internal dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has already proceeded past the half-way'' point, where half of the initial entropy has been radiated away, then additional quantum information deposited in the black hole is revealed in the Hawking radiation very rapidly. Information deposited prior to the half-way point remains concealed until the half-way point, and then emerges quickly. These conclusions hold because typical local quantum circuits are efficient encoders for quantum error-correcting codes that nearly achieve the capacity of the quantum erasure channel. Our estimate of a black hole's information retention time, based on speculative dynamical assumptions, is just barely compatible with the black hole complementarity hypothesis.