Entanglement Wedge Reconstruction and the

Information ParadoxGeoff Penington, Stanford University

Based on:Entanglement Wedge Reconstruction and the Information Paradox. GP. arXiv:1905.08255.

Replica Wormholes and the Black Hole Interior. GP, S. Shenker, D. Stanford, Z. Yang. arXiv:1911.11977.

See also:

The Entropy of Bulk Quantum Fields and the Entanglement Wedge of an Evaporating Black Hole. A. Almheiri, N. Engelhardt, D. Marolf, H. Maxfield. arXiv:1905.08762.

The Page Curve of Hawking Radiation From Semiclassical Geometry. A. Almheiri, R. Mahajan, J. Maldacena, Y. Zhao. arXiv:1908.10996.

Replica wormholes and the entropy of hawking radiation. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, A. Tajdini. arXiv:1911.12333.

Other important work by: Akers, Harlow, Bousso, Tomasevic, Chen, Fisher, Hernandez, Myers, Ruan, Rozali, Van Raamsdonk, Sully, Waddell, Wakeham

Talk will be very achronological

The claim:❑ We can derive a unitary Page curve, and the escape of information in the

Hawking radiation (specifically the Hayden-Preskill decoding criterion) directly from a gravitational description of the evaporation.

❑ The original focus was on black holes in AdS/CFT, but the arguments don’t require a CFT, or string theory, or even anti-de Sitter space. In their most pure form, all we need is a gravitational path integral.

❑ Major “but”: the calculations only work when we sum over arbitrary topologiesin the gravitational path integral, including spacetime wormholes that connect distant regions.

❑ This is the “right” thing to do, but is known to cause problems of its own, particularly in AdS/CFT when the boundary theory is supposed to factorise. This is intimately related to the discovery that JT gravity has a disorder-averaged boundary dual.

The plan:

❑ Part I: start with a very simple but unrealistic toy model of black hole evaporation, where we can calculate everything completely explicitly.

❑ Part II: move to more realistic models, including evaporating four-dimensional black holes, at the cost of being somewhat less explicit.

arXiv:1911.11977

arXiv:1905.08255

Entropies and Observables❑ The von Neumann entropy is not an observable.

❑ However, the integer n Renyi entropies

are proportional to the logarithm of an observable on n copies of the system.

❑ The key idea: the gravitational path integral should include topologies that connect the different copies via spacetime wormholes.

❑ Eventually, we can calculate the von Neumann entropy by analytically continuing the Renyi entropies to n=1.

A Very Simple Model: Pure JT gravity plus EOW Branes

Euclidean

Analogue for Hawking radiation: add degrees of freedom to the EOW brane (interior modes) that are maximally entangled with a reference system

Lorentzian

A Very Simple Information Paradox

Euclidean Lorentzian

=

i

i

=

(Simple) information paradox when 𝑘 ≫ 𝑒𝑆𝐵𝐻 . Entanglement entropy seemingly becomes larger than the Bekenstein-Hawking entropy

Orthogonal basis for internal stateof EOW brane

i

i

Seeminglymaximally mixed

Calculating the PurityCalculate the purity 𝑇𝑟(𝜌𝑅

2) using a Euclidean path integral, where we sum over all topologies with the correct boundary conditions:

Two copies of the original black hole

Two black holes connected by a wormhole

=1

k

~ 𝑒−𝑆0

Very small but doesn’t decay at large k

𝑖,𝑗

Calculating the von Neumann EntropyIn general, there are a lot of topologies that can contribute to Tr 𝜌𝑅

𝑛 . However, in the limit where k is very large/small one of two families of topologies dominates

Small k

1

𝑘𝑛−1=

Disconnected topology

Gives 𝑆 = log 𝑘

Large k

Connected topologyGives 𝑆 = 𝑆𝐵𝐻

~ 𝑒− 𝑛−1 𝑆0

Calculating the von Neumann Entropy

❑ Connected topology has a 𝑍𝑛 replica symmetry.

❑ After quotienting by this symmetry, we get roughly the original black hole geometry, except that there is a conical singularity at the fixed point of the replica symmetry

❑ In the limit 𝑛 → 1, the singularity vanishes

❑ Von Neumann entropy given by the “area” of replica fixed point, which is the bifurcation surface

Much more to say about this model!

❑ Simple enough that we can do the full path integral, rather than just looking at classical saddle points

❑ Can use tools from free probability theory to find the corrections to the von Neumann entropy, and even the full entanglement spectrum, near the Page transition, when the Renyi entropies are not dominated by a single topology.

❑ Transition is complicated, with seven distinct phases. However the main qualitative features agree with expectations. In particular, there are

𝑂(1/ 𝐺𝑁) corrections near the transition from energy fluctuations.

❑ We can also explicitly see how operators acting on the radiation (Petz map) can secretly measure the interior.

❑ This gives a new, direct derivation of entanglement wedge reconstruction (valid far beyond this simple model)

Time to talk about actual evaporating black holes❑ Bad news: no one has found analytical replica wormhole geometry for integer 𝑛 ≥ 2 for any more realistic model.

❑ Numerical results for the SYK model suggest that the physics is inherently messy, with complicated backreaction related to the fast scrambling behaviour.

❑ We saw in the simple model that the von Neumann entropy was given by the area of an extremal surface in the unbackreacted geometry.

❑ This is true very generally: the von Neumann entropy is given by the generalised entropy of the replica fixed point (FLM 2013).

❑ To be a saddle point of the gravitational action, the replica fixed point needs to be a classical extremal surface.

❑ When we include the matter action, we find that it instead needs to be a quantum extremal surface.

Summary of QES prescription/ “Island rule”

Replica trick calculations imply that the entropy of the black hole is given by the generalised entropy of the minimal quantum extremal surface

As in Rob’s talk, we allow a black hole to evaporate using absorbing boundary conditions, and then calculate entropies for the black hole and radiation systems.

The entropy of the Hawking radiation is given by the same rule, except that we need to include the Hawking radiation in the bulk entropy. The purity of the global bulk state implies that the two entropies are equal.

Each system encodes its entanglement wedge (the bulk domain of dependence bounded by minimal QES).

The Empty Surface

Obvious QES: the empty set(corresponds to the disconnected topologies in the replica trick calculations)

Generalised entropy = semiclassical entropy of the Hawking radiation (area is zero)

There also exists a non-empty quantum extremal surface that lies just inside the event horizon, at an infalling time exactly the scrambling time in the past of the ‘current’ boundary time, with generalised entropy given by the Bekenstein-Hawkingentropy

A claim

At the Page time, this becomes the minimal QES (corresponds to the transition to a fully connected replica wormhole topology)

As the black hole continues to evaporate, the RT surface tracks along the horizon, travelling on a spacelike trajectory

This first-order phase transition in the minimal QES gives the Page curve for the entanglement entropy

If no greybody factors are present (e.g. in two dimensions), the quantum extremal surface can be found analytically

When there are greybody factors, we can still show that a quantum extremal exists at an infalling time one scrambling time (plus unknown subleading corrections) in the past

Both calculations are annoyingly technical. If you care, ask me afterwards!

Done in my paper and (for JT gravity) by AEMM

Finding the nonempty Quantum Extremal Surface

Hayden-Preskill

Initially, the worldline of the diary is in the entanglement wedge of the CFT: no information about the diary has escaped in the Hawking radiation

Suppose we throw a diary into the black hole (after the Page time)

Hayden-Preskill

However as the black hole continues to evaporate, the RT surface continues to track along the horizon

After waiting for more than the scrambling time, the worldline of the diary will be in the entanglement wedge of the radiation

The state of the diary has escaped in the Hawking radiation

The firewall paradox

The interior partners of late time Hawking radiation are also encoded in the early Hawking radiation. We have therefore derived the ER=EPR resolution to the firewall paradox.

(Full story is more complicated with several important subtleties, but quantum extremality magically ensures that everything works out and you exactly avoid any firewall paradox.)

How does the information get out?

The entanglement between the Hawking radiation and the interior does not depend on the initial state of the black hole or any diary that was thrown in

We’ve explained how the final state of the radiation can be pure, but not how it can encode the information that was thrown into the black hole (as implied by entanglement wedge reconstruction).

Entanglement wedge reconstruction is potentially state dependent (arXiv:1807.06041, arXiv:1911.11977) .

Entanglement between Hawking radiation interior is independent of the initial black hole microstate.

But the encoding of the interior in early radiation depends on the initial state.

Hence, the Hawking radiation provides new information about the initial state to an observer with access only to the radiation

Answer: State Dependence

Thank you!

Derivation of location of quantum extremal surface (with greybody factors)

Step 1: The backreacted metric

Ingoing Vaidya metric:

Step 1: The backreacted metric

Near horizon region:

Over timescales of interest:

Step 1: The backreacted metric

Near horizon region:

Outgoing lightrays labelled by Kruskal-like coordinate U:

Interior:

Step 2: The bulk entropyOnly modes with low angular momentum are able to escape the black hole. (Problem is effectively two-dimensional). Also near the horizon outgoing/ingoing modes are decoupled.

Outgoing modes near the horizon are locally in the vacuum state ( = Rindler modes in thermofield double state)

The modes outside the horizon are partially reflected back into the black hole (greybodyfactors)

Important: entanglement in the thermofield double state is local. No entanglement when separation:

Step 3: The quantum extremal surfaceBecomes small when

Independent of v for

No greybody factors:

With greybody factors for :

Step 3: The quantum extremal surface

( for )

Implicitly we assumed that the cut-offs on the outgoing/ingoing modes were constant in units of U, v. But the proper cut-off:

Each angular momentum mode has a universal divergence , so shifting to a constant proper cut-off, increases by What about the entropy

production of the black hole?

Step 3: The quantum extremal surface

Positive constant

Scrambling time!

Must exist solution

Lots of algebra: