Time evolution of entanglement entropy from black hole interiors

By Thomas Hartman, J. Maldacena. 2013

746 citations recorded by the source corpus.

A bstractWe 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f3bf896648e074234ffcc7521760b51e68e765f1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. R. Myers
2. D. Stanford
3. D. Harlow
4. S. Raju
5. J. Maldacena
6. T. Takayanagi
7. L. Susskind
8. Shira Chapman

In Nuclear and High Energy Physics

1. R. Myers
2. D. Harlow
3. T. Takayanagi
4. D. Stanford
5. S. Raju
6. J. Maldacena
7. L. Susskind
8. Thomas Hartman

In Statistical and Nonlinear Physics

1. R. Myers
2. D. Harlow
3. S. Raju
4. D. Stanford
5. T. Takayanagi
6. J. Maldacena
7. Shira Chapman
8. H. Marrochio

Citations in the corpus

• Chaos and complexity by design (2016, background): “These proposals are motivated by the fact that the black hole interior continues to grow as the state evolves long past the time entropic quantities equilibrate [77].”
• Complexity and Shock Wave Geometries (2014, background): “Hartman and Maldacena [1] have proposed a tensor network (TN) picture to illustrate the evolution of ERBs.5 Figure 5 shows the evolution of the Hartman-Maldacena TN for the ERB in the case of a 1+1 dimensional boundary theory (D = 3).”
• Information Transfer with a Gravitating Bath (2020, methodology): “Naively, this would introduce an information paradox [43].”
• Holographic Quantum Matter (2016, background): “We will now describe this connection, following the exposition of [332].”
• Entanglement Entropy at Large Central Charge (2013, background): “The result also applies to problems related by a conformal mapping, including the (possibly time dependent) entanglement entropy of a CFT at finite temperature where it is very natural to consider multiple intervals [9, 10, 11].”
• Black hole collapse in the 1/c expansion (2016, background): “Such probes have been discussed in CFT before [51, 52], though not in detail for black holes formed by gravitational collapse.”
• The holographic entropy cone (2015, methodology): “Notably, when the formula is applied to the case of the two-sided black hole geometry, one finds that the Einstein-Rosen bridge encodes entanglement of thermofield double degrees of freedom in the boundary CFT [6, 7].”
• Surface/State Correspondence as a Generalized Holography (2015, methodology): “d in [4] (see also [5] for a modified version of this conjecture) that the AdS/CFT can be interpreted as a real space renormalization flow called entanglement renormalization or MERA [6]. Refer to e.g. [7, 8, 9, 10, 11, 12, 13] for more 1 progresses in this direction. Our proposal in this paper is hig”
• On volumes of subregions in holography and complexity (2016, background): “On the other hand, thermal equilibrium is achieved much quicker, and the entanglement entropies saturate [34–37].”
• Integral geometry and holography (2015, background): “3 The The analogy between holography and tensor networks was further advanced in [22–26]; see also [27] for a critique of associating MERA directly with the spatial slice of AdS3.”
• Jerusalem Lectures on Black Holes and Quantum Information (2014, methodology): “Two especially interesting examples of this are its use to study the two-sided AdS Schwarzschild geometry, by Hartman and Maldacena [139], and perturbations thereof, by Shenker and Stanford [60, 61].”
• Inconsistency of islands in theories with long-range gravity (2021, background): “In particular, if one takes R to be the union of a region on one asymptotic boundary with a similar region on the other asymptotic boundary, then at late enough times, one finds a phase transition for the surface X between what is called a Hartman-Maldacena surface [90], which runs from ∂R on one bo”
• Addendum to computational complexity and black hole horizons (2014, methodology): “The vertical entanglement can be evaluated using the Ryu-Takayanagi [3] minimal surface hypotheses as explained by Hartman and Maldacena [4].”