We consider the problem of how fast a quantum system can scramble (thermalize) information, given that the interactions are between bounded clusters of degrees of freedom; pairwise interactions would be an example Based on previous work, we conjecture: 1) The most rapid scramblers take a time logarithmic in the number of degrees of freedom 2) Matrix quantum mechanics (systems whose degrees of freedom are n by n matrices) saturate the bound 3) Black holes are the fastest scramblers in nature The conjectures are based on two sources, one from quantum information theory, and the other from the study of black holes in String Theory

Fast Scramblers

We compute the time-dependent entanglement entropy of a CFT which starts in relatively simple initial states. The initial states are the thermofield double for thermal states, dual to eternal black holes, and a particular pure state, dual to a black hole formed by gravitational collapse. The entanglement entropy grows linearly in time. This linear growth is directly related to the growth of the black hole interior measured along “nice” spatial slices. These nice slices probe the spacelike direction in the interior, at a fixed special value of the interior time. In the case of a two-dimensional CFT, we match the bulk and boundary computations of the entanglement entropy. We briefly discuss the long time behavior of various correlators, computed via classical geodesics or surfaces, and point out that their exponential decay comes about for similar reasons. We also present the time evolution of the wavefunction in the tensor network description.

Time Evolution of Entanglement Entropy from Black Hole Interiors

We describe the experience of an observer falling into a black hole using the AdS/CFT correspondence. In order to do this, we reconstruct the local bulk operators measured by the observer along his trajectory outside the black hole. We then extend our construction beyond the black hole horizon. We show that this is possible because of an effective doubling of the observables in the boundary theory, when it is in a purestate that is close to the thermal state. Our construction allows us to rephrase questions about information-loss and the structure of the metric at the horizon in terms of more familiar CFT correlators. It suggests that to precisely identify black-hole microstates, the observer would need to conduct measurements to an accuracy of $ {e^{{-{S_{\mathrm{BH}}}}}} $
 . This appears to be inconsistent with the “fuzzball” proposal, and other recent proposals in which pure states in the ensemble of the black hole are represented by macroscopically distinct geometries. Furthermore, our description of the black hole interior in terms of CFT operators provides a natural realization of black hole complementarity and a method of preserving unitarity without “firewalls.”

/pdf/an-infalling-observer-in-ads-cft-152bqarnq9.pdf

An infalling observer in AdS/CFT

We discuss universal properties of conformal field theories with holographic duals. A central feature of these theories is the existence of a low-lying sector of operators whose correlators factorize. We demonstrate that factorization can only hold in the large central charge limit. Using conformal invariance and factorization we argue that these operators are naturally represented as fields in AdS as this makes the underlying linearity of the system manifest. In this class of CFTs the solution of the conformal bootstrap conditions can be naturally organized in structures which coincide with Witten diagrams in the bulk. The large value of the central charge suggests that the theory must include a large number of new operators not captured by the factorized sector. Consequently we may think of the AdS hologram as an effective representation of a small sector of the CFT, which is embedded inside a much larger Hilbert space corresponding to the black hole microstates.

/pdf/emergent-spacetime-and-holographic-cfts-107xr4no9h.pdf

Emergent Spacetime and Holographic CFTs

We use the AdS/CFT correspondence to study the thermalization of a strongly coupled conformal field theory that is forced out of its vacuum by a source that couples to a marginal operator. The source is taken to be of small amplitude and finite duration, but is otherwise an arbitrary function of time. When the field theory lives on $R^{d-1,1}$, the source sets up a translationally invariant wave in the dual gravitational description. This wave propagates radially inwards in $AdS_{d+1}$ space and collapses to form a black brane. Outside its horizon the bulk spacetime for this collapse process may systematically be constructed in an expansion in the amplitude of the source function, and takes the Vaidya form at leading order in the source amplitude. This solution is dual to a remarkably rapid and intriguingly scale dependent thermalization process in the field theory. When the field theory lives on a sphere the resultant wave either slowly scatters into a thermal gas (dual to a glueball type phase in the boundary theory) or rapidly collapses into a black hole (dual to a plasma type phase in the field theory) depending on the time scale and amplitude of the source function. The transition between these two behaviors is sharp and can be tuned to the Choptuik scaling solution in $R^{d,1}$.

http://iopscience.iop.org/article/10.1088/1126-6708/2009/09/034/pdf

Weak Field Black Hole Formation in Asymptotically AdS Spacetimes

We derive a quantization formula of Bohr-Sommerfeld type for computing quasinormal frequencies for scalar perturbations in an AdS black hole in the limit of large scalar mass or spatial momentum. We then apply the formula to find poles in retarded Green functions of boundary CFTs on $R^{1,d-1}$ and $RxS^{d-1}$. We find that when the boundary theory is perturbed by an operator of dimension $\Delta>> 1$, the relaxation time back to equilibrium is given at zero momentum by ${1 \over \Delta \pi T} << {1 \over \pi T}$. Turning on a large spatial momentum can significantly increase it. For a generic scalar operator in a CFT on $R^{1,d-1}$, there exists a sequence of poles near the lightcone whose imaginary part scales with momentum as $p^{-{d-2 \over d+2}}$ in the large momentum limit. For a CFT on a sphere $S^{d-1}$ we show that the theory possesses a large number of long-lived quasiparticles whose imaginary part is exponentially small in momentum.

A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes

Quantum gravity in an AdS spacetime is described by an SU(N) Yang-Mills theory on a sphere, a bounded many-body system. We argue that in the high temperature phase the theory is intrinsically non-perturbative in the large N limit. At any nonzero value of the 't Hooft coupling λ, an exponentially large (in N 2 ) number of free theory states of wide energy range (or order N) mix under the interaction. As a result the planar perturbation theory breaks down. We argue that an arrow of time emerges and the dual string configuration should be interpreted as a stringy black hole.

https://iopscience.iop.org/article/10.1088/1126-6708/2007/12/027/pdf

The arrow of time, black holes, and quantum mixing of large N Yang-Mills theories

We derive a quantization formula of Bohr-Sommerfeld type for computing quasinormal frequencies for scalar perturbations in an AdS black hole in the limit of large scalar mass or spatial momentum. We then apply the formula to find poles in retarded Green functions of boundary CFTs on IR and IR× S. We find that when the boundary theory is perturbed by an operator of dimension ∆ ≫ 1, the relaxation time back to equilibrium is given at zero momentum by 1 ∆πT ≪ 1 πT . Turning on a large spatial momentum can significantly increase it. For a generic scalar operator in a CFT on IR, there exists a sequence of poles near the lightcone whose imaginary part scales with momentum as p d−2 d+2 in the large momentum limit. For a CFT on a sphere S we show that the theory possesses a large number of long-lived quasiparticles whose imaginary part is exponentially small in momentum.

Quantum gravity in an AdS spacetime is described by an SU(N) Yang-Mills theory on a sphere, a bounded many-body system. We argue that in the high temperature phase the theory is intrinsically non-perturbative in the large N limit. At any nonzero value of the 't Hooft coupling $\lambda$, an exponentially large (in N^2) number of free theory states of wide energy range (of order N) mix under the interaction. As a result the planar perturbation theory breaks down. We argue that an arrow of time emerges and the dual string configuration should be interpreted as a stringy black hole.

Inheritance principle and non-renormalization theorems at finite temperature

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A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes

The arrow of time, black holes, and quantum mixing of large N Yang-Mills theories

A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes

The arrow of time, black holes, and quantum mixing of large N Yang-Mills theories

Inheritance principle and non-renormalization theorems at finite temperature