Black holes and the butterfly effect
TL;DR: In this article, the authors used holography to study sensitive dependence on initial conditions in strongly coupled field theories and showed that the effect of the early infalling quanta relative to the t = 0 slice creates a shock wave that destroys the local two-sided correlations present in the unperturbed state.
Abstract: We use holography to study sensitive dependence on initial conditions in strongly coupled field theories. Specifically, we mildly perturb a thermofield double state by adding a small number of quanta on one side. If these quanta are released a scrambling time in the past, they destroy the local two-sided correlations present in the unperturbed state. The corresponding bulk geometry is a two-sided AdS black hole, and the key effect is the blueshift of the early infalling quanta relative to the t = 0 slice, creating a shock wave. We comment on string- and Planck-scale corrections to this setup, and discuss points that may be relevant to the firewall controversy.
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TL;DR: In this paper, a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom is given, based on plausible physical assumptions, establishing this conjecture.
Abstract: We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent λ
L
≤ 2πk
B
T/ℏ. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.
2,216 citations
Cites background from "Black holes and the butterfly effec..."
...J H E P 0 8 ( 2 0 1 6 ) 1 0 6 becomes more and more complicated, and the delicate local correlations present in the thermofield double state will be destroyed [16], causing F to become small....
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TL;DR: In this paper, the authors studied the quantum mechanical model of $N$ Majorana fermions with random interactions of a few Fermions at a time (Sachdev-Ye-Kitaev model) in the large N$ limit.
Abstract: The authors study in detail the quantum mechanical model of $N$ Majorana fermions with random interactions of a few fermions at a time (Sachdev-Ye-Kitaev model) in the large $N$ limit. At low energies, the system is strongly interacting and an emergent conformal symmetry develops. Performing technical calculations, the authors elucidate a number of properties of the model near the conformal point.
1,953 citations
TL;DR: In this paper, the authors show that the EPR pair can be interpreted as maximally entangled states of two black holes, and they suggest that similar bridges might be present for more general entangled states.
Abstract: General relativity contains solutions in which two distant black holes are connected through the interior via a wormhole, or Einstein-Rosen bridge. These solutions can be interpreted as maximally entangled states of two black holes that form a complex EPR pair. We suggest that similar bridges might be present for more general entangled states. In the case of entangled black holes one can formulate versions of the AMPS(S) paradoxes and resolve them. This suggests possible resolutions of the firewall paradoxes for more general situations.
1,446 citations
Cites background or methods from "Black holes and the butterfly effec..."
...Using arguments similar to those of [32] one could argue that the state that results from adding particles to the thermofield double state will be singular if we evolve it to the future by a sufficiently long time amount....
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...Instead, if we perform E at very early time, in the bottom left part of the Penrose diagram, we can create a big disturbance in A due to the blueshift factor [32]....
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TL;DR: The hypothesis that black holes are the fastest computers in nature is discussed and the conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that is called a Wheeler-DeWitt patch is illustrated.
Abstract: We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti-de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.
933 citations
Cites background from "Black holes and the butterfly effec..."
...In [30] (see also [7, 31–34]), the field theory was perturbed with a small thermal-scale operator....
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TL;DR: In this paper, a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom is given. But this bound depends on the assumption that the influence of chaos on the commutator can develop no faster than exponentially with Lyapunov exponent.
Abstract: We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent $\lambda_L \le 2 \pi k_B T/\hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.
860 citations
References
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TL;DR: The algorithm can be used as a building block for solving other distributed graph problems, and can be slightly modified to run on a strongly-connected diagraph for generating the existent Euler trail or to report that no Euler trails exist.
13,828 citations
TL;DR: It is argued that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy.
Abstract: A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.
4,395 citations
TL;DR: In this paper, a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence, and the relation between the entropy and central charges in 4D conformal fields is examined.
Abstract: This is an extended version of our short report [1], where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.
2,310 citations
"Black holes and the butterfly effec..." refers methods in this paper
...It was proposed by Ryu and Takayanagi [7, 8] (RT) that the entanglement entropy of a region A in a CFT in a state |ψ〉 is given by the area (in Planck units) of the minimal area codimension two spacelike surface whose asymptotic boundary is the boundary of A in the geometry dual to |ψ〉....
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...It was proposed by Ryu and Takayanagi [5] (RT) that the entanglement entropy of a region A in a CFT in a state |ψ〉 is given by the area (in Planck units) of the minimal area codimension two spacelike surface whose asymptotic boundary is the boundary of A in the geometry dual to |ψ〉....
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...Employing the RT proposal [7, 8] and its time-dependent extension [10], we can compute the entropy SΩ of...
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TL;DR: In this paper, a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001 is proposed to understand the time-dependence of entropy in generic quantum field theories.
Abstract: With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.
2,047 citations
"Black holes and the butterfly effec..." refers methods in this paper
...The RT proposal has been extended to nonstatic geometries in [10]....
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...Employing the RT proposal [7, 8] and its time-dependent extension [10], we can compute the entropy SΩ of...
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TL;DR: The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho) = alphaL - gamma + ..., where the ellipsis represents terms that vanish in the limit L --> infinity.
Abstract: We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator rho for the degrees of freedom in the interior. The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho)=alphaL-gamma+[centered ellipsis], where the ellipsis represents terms that vanish in the limit L-->[infinity]. We show that -gamma is a universal constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for gamma in terms of properties of the superselection sectors of the medium.
2,028 citations
"Black holes and the butterfly effec..." refers methods in this paper
...= tr[ρAB(U) (2)]− 2 |A|2 tr[ρAB] + 1 |A|4 tr[1] (A....
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...Entropy of entanglement of the ground state can be used as a diagnostic of topological order in gapped quantum systems [1, 2]....
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