# Can PEPS explain the holographic principle in quantum gravity?

Condensed matter physicists have shown using quantum information that in many condensed matter systems, entanglement entropy only scales as the area of the boundary, and not the volume. This is the basis for the density matrix renormalization group and Projected Entangled Pair States (PEPS). Does this also explain the holographic principle in quantum gravity?

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What do you mean by explain? Most meanings of that word don't make any sense here. –  Marek Mar 13 '11 at 9:48
I don't understand what you are asking at all. The area law scaling of entanglement entropy in the quantum many body systems that you are mentioning is the basis for the DMRG, MPS, PEPS etc, but I don't see how these variational constructions for the quantum state of these systems could "explain" the holographic principle in QG. I think that the question could be better posed as: Can the holographic principle in QG give some theoretical support to variational methods in quantum many body systems such as PEPS? –  xavimol Mar 13 '11 at 12:31
PEPS == Pair Entangled Product States, correct? I edited the question to make this clear. –  user346 Mar 13 '11 at 14:32
@Deepak: isn't it projected entangled pair states? Today is the first time I heard the term PEPS but google (and many articles it provides) suggests this is the most likely meaning :) –  Marek Mar 13 '11 at 15:48
@Deepak and @Marek: Yes it is true, the correct term is Projected Entangled Pairs (PEPS) states that is a generalization of Matrix Product states (MPS,that is directly related with density matrix renormalization group). –  xavimol Mar 13 '11 at 16:26
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## 1 Answer

Nope, the very fact that the entanglement entropy is - naturally - proportional to the surface area does not explain the holographic principle because the holographic principle, reduced to the corresponding entropy bound, implies that the total entropy of one of the systems can't exceed $A/4G$ where $A$ is the surface. The entanglement entropy is just one tiny term of the entropy, a degree of correlation between two subsystems, so its being proportional to the area is a much weaker and less surprising statement than the holographic principle.

Quite generally, non-gravitational physical systems are simply not holographic in the space (bulk) that they occupy. Only (quantum) gravitational systems obey the holographic principle. In the context of AdS/CFT or AdS/anything, the holographic principle is encoded in the description of the CFT or anything by its having a secret extra (fifth) dimension of spacetime.

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Nice answer @Lubos. Some people are trying to reinterpret quantum information based renormalization group procedures in quantum many body physics (an example is PEPS) in terms of holography. These procedures support the area law scaling of entanglement entropy in low dimensional systems. However, despite the interesting or even the usefulness (in terms of being an effective tool to compute things) that this interpretation can be, it is questionable how these facts could "explain" something that, at this time, has the status of principle. –  xavimol Mar 14 '11 at 13:03
This answer sounds good, but there's a much better reason for why these have to be different results. PEPS shows that the ground state and low-energy excited states of a condensed matter system have entanglement entropy proportional to the surface area, but it's still true that an arbitrary state could have entanglement entropy proportional to the volume. For the black hole, even an arbitrary state can only have entanglement entropy (or indeed total entropy) proportional to the surface area. –  Peter Shor Dec 4 at 12:29
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