Entanglement entropy in Loop Quantum Gravity

[marcus]

I downloaded William Donnelly's talk, given at Loops'07
Entanglement Entropy in Loop Quantum Gravity
http://www.matmor.unam.mx/eventos/loops07/talks/7B/Donnelly.pdf
maybe he will discuss it with us
I just had a look

 

[marcus]

Donnelly, if you feel like giving us a little intuitive idea of what the von Neumann entropy is or what the entanglement entropy is----and what the significance is that it turns out to be the same as usual black hole entropy---that would be great, if not that's OK too.

It's nice having people around here at PF who are going to conferences and giving papers.

 

[william donnelly]

Hi Marcus.

The von Neumann entropy is the natural generalization of entropy for quantum systems. It's defined in terms of the density matrix as

$$S(\rho) = -\text{Tr}(\rho \log \rho)$$​

This formula just means you take the eigenvalues of the density matrix $$p_1 \ldots p_n$$ and apply the usual Gibbs (or Shannon) entropy formula:

$$S(\rho) = - \sum p_i \log p_i$$​

The von Neumann entropy ranges from 0 for pure states, up to log(n) where n is the Hilbert space dimension.

The entanglement entropy is something that is unique to quantum mechanics. If you have a pair of quantum systems (A and B) in a pure state then the state of either subsystem is no longer described by a vector in a Hilbert space, but has to be described by a density matrix. This means that the entropy of the system (A and B) is zero, but the entropy of system A can be greater than zero. The von Neumann entropy of system A is called the entanglement entropy, because it quantifies the total amount of entanglement between systems A and B.

All this bears a resemblance to black hole entropy. A universe containing a black hole can be divided into two systems: everything inside the horizon, and everything outside the horizon. The combined system is in the vacuum state - which is a pure state with zero entropy - so the entire universe has zero entropy. But the observer outside the horizon sees a thermal state with positive entropy. So a natural explanation for this is phenomenon is that there is entanglement between the states inside the black hole and the states outside the black hole, which is causing the region outside the black hole to have entropy even when the total entropy of the universe is zero.

I think the idea that the entanglement entropy of the vacuum is the same thing as the mysterious Bekenstein-Hawking entropy was first proposed by Rafael Sorkin, but I haven't been able to find a copy of his original article. The first reference I have for this idea is described in this paper: http://prola.aps.org/abstract/PRD/v34/i2/p373_1

Hopefully this has clarified a few things. I'd like to talk about the connection to LQG, but it's probably best to wait until the paper is out.

William.

 

[olgranpappy]

The von Neumann entropy of system A is called the entanglement entropy, because it quantifies the total amount of entanglement between systems A and B.

But what do you mean by "entanglement?" That word seems too vivid for it's own good.

 

[George Jones]

But what do you mean by "entanglement?" That word seems too vivid for it's own good.

Actually, there is a standard definition of entanglement in quantum theory.

Suppose the state space for physical system 1 is $S_1$ and the state space for physical system 2 is $S_2.$ Then, the state space for the combined system is $S = S_1 \otimes S_2.$. If the state in the combined system cannot be written as the product of a state in $S_1$ with a state in $S_2$, then the system is said to be entangled.

For example consider the spins of two electrons. The state

$$\left| \psi \right> = \frac{1}{\sqrt{2}} \left( \left| \uparrow \right> \left| \downarrow \right> +\left| \downarrow \right> \left| \uparrow \right> \right)$$

is an entangled state.

 

[marcus]

But what do you mean by "entanglement?" That word seems too vivid for it's own good.

I agree with George. it is old well-established term. Here is an encyclopedia article
http://en.wikipedia.org/wiki/Quantum_entanglement

I would guess the recognition of entanglement came before 1935.
that was when Einstein and a couple of buddies wrote a paper about it.
Einstein didn't like that feature of Quant. Mech. and scornfully called it

Spuckhafte Fernwirkung

Spooky Longdistance Working (spooky action at a distance)

=============
that said, Grandfather, you have a valid point about VIVIDNESS.

there is a dilemma in scientific terminology about whether to say something clearly, in familiar words, or not

you are damned if you do and damned if you don't

if you say it clearly in words that make directly obvious what is happening (like entanglement of two widely separated objects with no obvious material connection)
then you might get accused of sensationalism
(but it is nature's fault for being sensational in her real everyday life)

or else you can use a LATIN OR GREEK word that doesn't directly convey the idea---that is not vivid or graphic or self-explanatory---and then you might get accused of being bookish, or of using jargon to alienate laymen and keep them in the dark.

An example is we say "hydrogen" and in German they say "Wasserstoff" (water-stuff, because if you burn it you get water).
We say "nitrogen" and they say "stink-stuff" because you can use it to make a lot of stinks. If I am mistaken will someone who knows better please correct me, or provide a better example.
this illustrates the two ways to go: you can be forthright direct or you can be Greeky-technical.

the main thing is once you have a term and it is accepted, never try to change it. trying to reform language only leads to more trouble
============

William Donnelly, thanks for your response. I assume you will be posting a preprint before long?

 

[Demystifier]

The entropy of the outside Hawking radiation can be identified with the entropy of entanglement between outgoing and ingoing particles of pair creation. But this is the entropy of outside Hawking radiation, not of the black hole. This is why BH entropy does not seem to be the entanglement entropy. If it still is, there must be something very subtle about it.

 

[william donnelly]

There really is a very good reason it is called entanglement entropy, which I completely glossed over. It's based on these theorems:

1. A pure state has zero entanglement entropy if and only if it is a product state. So the standard definition that George gave of "not entangled" is equivalent to "entanglement entropy zero". For the basic entangled state that George wrote down, the entanglement entropy is log(2), or 1 bit. This state is sometimes called an e-bit or "entangled bit".

2. Suppose Alice and Bob share n e-bits. Then they can (approximately) transform it into any state with entanglement entropy less than or equal to n. This is called entanglement of formation.

3. If Alice and Bob share a state with entanglement entropy n, then they can (approximately) transform it into n e-bits. This is called distillable entanglement.

These definitions are used most commonly in quantum computing. To someone building a quantum computer, entanglement is not a mysterious nonlocal force, but a useful resource for computation. And so it's important to know how much you have.

 

[Alamino]

I guess I miss the point in one thing. If you consider the black hole and the universe as two entangled systems, measuring the state of the universe would mean that you know the microstate of the black hole, isn't it? If you are external to the black hole, this cannot be, right? If someone could clarify it to me.

 

[olgranpappy]

Actually, there is a standard definition of entanglement in quantum theory.

Suppose the state space for physical system 1 is $S_1$ and the state space for physical system 2 is $S_2.$ Then, the state space for the combined system is $S = S_1 \otimes S_2.$. If the state in the combined system cannot be written as the product of a state in $S_1$ with a state in $S_2$, then the system is said to be entangled.

For example consider the spins of two electrons. The state

$$\left| \psi \right> = \frac{1}{\sqrt{2}} \left( \left| \uparrow \right> \left| \downarrow \right> +\left| \downarrow \right> \left| \uparrow \right> \right)$$

is an entangled state.

ah, okay--but it's still a pure state. Is an entangled state always a pure state?

...furthermore, that state is nothing more that the usual state of total spin 1 and z-projection 0. Similarly the other independent linear combination (the |0,0> state) is also what you would call an "entangled" state, but I would simply call them "states with S_z=0." Apparently there is no difference, except that you have invented some new language that seems to serve no purpose.

 

[william donnelly]

The entropy of the outside Hawking radiation can be identified with the entropy of entanglement between outgoing and ingoing particles of pair creation. But this is the entropy of outside Hawking radiation, not of the black hole. This is why BH entropy does not seem to be the entanglement entropy. If it still is, there must be something very subtle about it.

You're right that the entanglement entropy is the same as the entropy of the thermal atmosphere. But because the entanglement entropy is symmetric, it is also the entropy of the vacuum state restricted to the black hole interior. If we take this point of view, the entanglement entropy counts the states of the black hole interior that are correlated with the exterior.

 

[olgranpappy]

that said, Grandfather, you have a valid point about VIVIDNESS...
...
An example is we say "hydrogen" and in German they say "Wasserstoff" (water-stuff, because if you burn it you get water).

No need for formalities, call me "pappy."

As far as names go, "hydrogen" actually does translate to English as "water-stuff" and similarly "helium" translates as "sun-stuff."

 

[Demystifier]

You're right that the entanglement entropy is the same as the entropy of the thermal atmosphere. But because the entanglement entropy is symmetric, it is also the entropy of the vacuum state restricted to the black hole interior. If we take this point of view, the entanglement entropy counts the states of the black hole interior that are correlated with the exterior.

This point of view seems fine.
But the entropy of the thermal atmosphere does not scale as the BH surface. Or doesn't it?

 

[George Jones]

Is an entangled state always a pure state?

I am not familiar with quantum entanglement of mixed states (William probably is), but there is a definition in this case as well, and it seems to be what I would expect. Scroll down to the Ensembles section from the wikipedia link given by Marcus in post #6.

...furthermore, that state is nothing more that the usual state of total spin 1 and z-projection 0. Similarly the other independent linear combination (the |0,0> state) is also what you would call an "entangled" state, but I would simply call them "states with S_z=0."

In the example I chose, the state spaces are irreducible representation spaces of the (cover of) the rotation group. The product representation space is not irreducible, but can written as a direct sum of irreducible representation, i.e.,

$$D_{1/2} \otimes D_{1/2} = D_0 \oplus D_1$$.

I just happened to chose a state in $D_1$, but I didn't have to do this. I could have chosen

$$\left| \Psi \right> = \frac{1}{\sqrt{2}} \left| \uparrow \right> \left| \uparrow \right> + \frac{1}{2} \left| \uparrow \right> \left| \downarrow \right> - \frac{1}{2} \left| \downarrow \right> \left| \uparrow \right>,$$

which, if I have done things correctly, is an entangled state that is a linear comnination of states from $D_1$ and $D_0.$

The state $| \Psi >$ is entangled ecause it cannot be written in the form $| \psi > | \phi >.$

Apparently there is no difference, except that you have invented some new language that seems to serve no purpose.

Entanglement has nothing to with angular momentum per se, and so can't, in general, be described by the language of angular momentum. Although my example involved angular momentum, the original definition of entanglement that I gave did not involve angular momentum. The original EPR example did not involve angular momentum. The example using spins was popularized, I think, by David Bohm.

I didn't invent the language, so I would prefer that you say "except that new language has been invented that ..." I don't know when the term "entangled" came into use either in research or in texts. I have a text published in 1986 that I thought used the term, but upon checking I find that although it discusses entangled states, it doesn't call them entangled states. I do have a 1995 "text" that uses the term extensively.

In my opinion, the language does serve a useful purpose. Quantum entanglement is used in stuff like quantum teleportation and quantum cryptography, and is involved in a substantial amount of quantum "weirdness".

 

[william donnelly]

I guess I miss the point in one thing. If you consider the black hole and the universe as two entangled systems, measuring the state of the universe would mean that you know the microstate of the black hole, isn't it? If you are external to the black hole, this cannot be, right? If someone could clarify it to me.

It's true that by measuring one half of an entangled state you can learn about the other half. But entanglement can't be used to send signals, so there is no violation of causality. In particular, there's no way that someone on the other side of the horizon could send a message back to the exterior using entanglement.

 

[william donnelly]

ah, okay--but it's still a pure state. Is an entangled state always a pure state?

No. One can also define what it means for a mixed state to be entangled, but the theory is more complicated because there are several different useful measures of entanglement. For pure states the situation is much simpler, because these measures are all equal to the entanglement entropy.

 

[william donnelly]

This point of view seems fine.
But the entropy of the thermal atmosphere does not scale as the BH surface. Or doesn't it?

It does! If it were otherwise I'm not sure anyone would take entanglement entropy seriously as a candidate for the black hole entropy.

The problem is that you get a divergent answer unless there is some kind of minimal length scale for the entropy to depend on.
So you have to put the theory on some kind of lattice. This was done in http://arxiv.org/abs/hep-th/9303048" .
You get the Bekenstein-Hawking entropy back when the lattice spacing is about the Planck length.

There have also been attempts to compute entanglement entropy in a black hole background instead of flat space. The problem is that the lattice procedure is not Lorentz-invariant. For a black hole the situation is even more complicated, and it isn't clear what coordinates one should use to introduce a cutoff.

 

[Demystifier]

I see, thanks.

 

[Demystifier]

I see also the following problem with the idea that BH entropy is the entanglement entropy. Assume that a BH is formed at some time t. At that time, BH starts to emit thermal Hawking radiation. But before this radiation become radiated, there can be no entanglement entropy. On the other hand, BH should have an entropy even then. How do you comment on this?

Besides, I would appreciate any comments on my (few days old) resolution of the BH information paradox:
http://xxx.lanl.gov/abs/0708.0729
(It does not attempt to explain why entropy is proportional to the surface, but rather how complete BH evaporation can be consistent with unitarity.)