Black hole microstate in AdS/CFT

[atyy]

I've asked this before, but am still confused.

In AdS/CFT, a typical statement like that on p6 of Hubeny and Rangamani's http://arxiv.org/abs/1006.3675 is that a large black hole in the bulk corresponds to a thermal state in the CFT.

However, if the CFT is the complete physics, then I'd expect a bulk black hole to correspond to a boundary pure state. Takayangai and Ugajin http://arxiv.org/abs/1008.3439 have an example of this. Are there others?

Also, what is the relationship between the black hole/thermal state and black hole/pure state pictures - is every pure state "in" the boundary canonical ensemble a bulk black hole?

 

[Finbar]

I've asked this before, but am still confused.

In AdS/CFT, a typical statement like that on p6 of Hubeny and Rangamani's http://arxiv.org/abs/1006.3675 is that a large black hole in the bulk corresponds to a thermal state in the CFT.

However, if the CFT is the complete physics, then I'd expect a bulk black hole to correspond to a boundary pure state. Takayangai and Ugajin http://arxiv.org/abs/1008.3439 have an example of this. Are there others?

Also, what is the relationship between the black hole/thermal state and black hole/pure state pictures - is every pure state "in" the boundary canonical ensemble a bulk black hole?

A large black hole in thermal equilibrium, i.e. far from extremal, should correspond to a state with a large entropy in both the bulk and the boundary. When we reach a near extremal state and thermal equilibrium breaks down one would hope that the CFT would allow the underlying microdegrees of freedom to be counted. In a sense all stable classical black holes should correspond to mixed states.

 

[atyy]

A large black hole in thermal equilibrium, i.e. far from extremal, should correspond to a state with a large entropy in both the bulk and the boundary. When we reach a near extremal state and thermal equilibrium breaks down one would hope that the CFT would allow the underlying microdegrees of freedom to be counted. In a sense all stable classical black holes should correspond to mixed states.

But what about say, Samir Mathur's work? Surely, if we are still sticking with quantum mechanics and unitary evolution for the time being, the whole universe must be in pure state, and then by coarse graining for some observers in a subsystem, a black hole appears.

 

[Finbar]

One can only write down a pure state if we know the underlying degrees degrees of freedom. For non-perturbative gravity we don't know these yet we only know semi-classical solutions to the effective theory. These solutions are therefore coarse grained and can only be understood as a mixed state.


Sure the universe is in a pure state with respect to some degrees of freedom just like a gas at finite temperature is in a pure state with respect to its underlying atoms and molecules. Remember in QM the word "state" tells us only our knowledge of the system with respect to the observables we can measure. Saying a black hole is in a pure or a mixed state is only a statement of our knowledge of the black hole.

 

[atyy]

One can only write down a pure state if we know the underlying degrees degrees of freedom. For non-perturbative gravity we don't know these yet we only know semi-classical solutions to the effective theory. These solutions are therefore coarse grained and can only be understood as a mixed state.


Sure the universe is in a pure state with respect to some degrees of freedom just like a gas at finite temperature is in a pure state with respect to its underlying atoms and molecules. Remember in QM the word "state" tells us only our knowledge of the system with respect to the observables we can measure. Saying a black hole is in a pure or a mixed state is only a statement of our knowledge of the black hole.

OK, but in AdS/CFT we are supposed to know the underlying degrees of freedom, so what is a pure state that corresponds to a large black hole in that picture?

 

[atyy]

Some earlier papers that looked at black holes in AdS/CFT are:
-Balasubramanian et al, Holographic Probes of Anti-de Sitter Spacetimes, http://arxiv.org/abs/hep-th/9808017
-Maldacena, Eternal black holes in Anti-de-Sitter, http://arxiv.org/abs/hep-th/0106112v6

These new papers seem to have a different set up, making use of a quantum quench (http://arxiv.org/abs/cond-mat/0503393) in which the initial pure state is not an eigenstate of the Hamiltonian:
-Takayanagi and Ugajin, Measuring Black Hole Formations by Entanglement Entropy via Coarse-Graining, http://arxiv.org/abs/1008.3439
-Abajo-Arrastia et al, Holographic Evolution of Entanglement Entropy, http://arxiv.org/abs/1006.4090
-Albash and Johnson, Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches, http://arxiv.org/abs/1008.3027

For example, T & U say "In summary, in the time-dependent background which describes a thermalization of a pure state, the von-Neumann entropy of the total system is indeed vanishing also in the gravity side. This contrasts strikingly with the setup of an eternal AdS black hole [29] as it is dual to a mixed state in a thermal CFT and has a non-vanishing thermal entropy ..."

 

[atyy]

T & U say that their proposal is a "a toy holographic dual of black hole creations and annihilations" and that it is "manifestly free from the black hole information problem". So maybe it would be interesting to compare their work with Samir Mathur's fuzzball proposal for black hole microstates. Here are some reviews of the current state of that approach:

The information paradox: A pedagogical introduction http://arxiv.org/abs/0909.1038
Fuzzballs and the information paradox: a summary and conjectures http://arxiv.org/abs/0810.4525

 

[genneth]

To have a black hole terminating the AdS space, you need to be considering finite temperatures in the CFT. So then you're mapping a mixed state to a mixed state, no problems. At zero temperature, the AdS space does not get terminated by a black hole.

 

[Physics Monkey]

This can be a very subtle issue.

An eternal black hole actually has two asymptotic regions which correspond to two copies of the conformal field theory in an entangled state. The state of a single copy of the CFT is truly mixed and thermal because it is entangled with its partner. This is called the thermofield double.

On the other hand, a black hole that formed via collapse may look at late times almost exactly like an eternal black hole, yet we expect that the state of the whole CFT is pure. The Takayanagi paper you mentioned is one proposal to see this fact based on a speculative extension of the formalism for entanglement entropy to time dependent backgrounds.

It seems that the presence of a black hole at late times can always be interpreted in some coarse grained sense as a thermal state. However, the ultimate fate of the state, whether it is mixed or pure depends on the detailed history. In fact, it seems as if the same geometry can actually encode multiple kinds of entropy: microscopic entanglement entropy, coarse grained entanglement entropy, thermal entropy, etc. In this way of thinking, different geometrical features of the same time dependent geometry give answers to different entropy related questions. However, this is still an open question.

 

[atyy]

To have a black hole terminating the AdS space, you need to be considering finite temperatures in the CFT. So then you're mapping a mixed state to a mixed state, no problems. At zero temperature, the AdS space does not get terminated by a black hole.

An eternal black hole actually has two asymptotic regions which correspond to two copies of the conformal field theory in an entangled state. The state of a single copy of the CFT is truly mixed and thermal because it is entangled with its partner. This is called the thermofield double.

So it looks like in the eternal black hole, the full system of two CFTs is still in a pure state, and the mixed thermal state of one CFT is obtained because its density matrix is a reduced density matrix obtained by tracing over the other CFT.

Is it the case that a mixed state can always be obtained as a reduced density matrix, so that the mixed states in AdS/CFT always have a pure state of a larger system behind them?

I guess what's unintuitive is that for an ordinary system in a mixed state, it seems reasonable to imagine that I can always treat it as a subsystem, with the larger system presumably pure. But I kinda thought the AdS bulk in AdS/CFT is "the whole universe". I guess the eternal black hole with its double CFT shows there's more to the "universe" than the bulk of one CFT.

In fact, it seems as if the same geometry can actually encode multiple kinds of entropy: microscopic entanglement entropy, coarse grained entanglement entropy, thermal entropy, etc. In this way of thinking, different geometrical features of the same time dependent geometry give answers to different entropy related questions. However, this is still an open question.

In the eternal black hole, it looks like the black hole entropy is calculated via the entanglement entropy of the 2 CFTs - but is that the microscopic or coarse grainted entanglement entropy?

 

[genneth]

On the other hand, a black hole that formed via collapse may look at late times almost exactly like an eternal black hole, yet we expect that the state of the whole CFT is pure. The Takayanagi paper you mentioned is one proposal to see this fact based on a speculative extension of the formalism for entanglement entropy to time dependent backgrounds.

A non-eternal black hole has a CFT on the boundary? Really? That sounds wrong to me, based on dimensional grounds, i.e. when things are collapsing there should be some time/length scales in the problem.

 

[Physics Monkey]

A CFT refers to a whole set of states, not just to the vacuum. You can always put a CFT at finite temperature. Equivalently for local observables, you can always put a CFT into a superposition of highly excited states that look locally thermal (almost all states are of this form). The effective temperature is determined by the average energy density in this pure state.

In the context of holographic duality, there is always a CFT "on the boundary" no matter what the bulk is doing. Different bulk geometries correspond to different states or dynamical processes in that CFT.

 

[atyy]

The effective temperature is determined by the average energy density in this pure state.

I see, that makes sense. Any recommendations for reading? Presumably one can have different "local" temperatures if the energy density is inhomogeneous?