# What does this question about entanglement and classical geometry mean?

Below is the question from Andy Strominger's presentation at the String 2014 conference. The question was asked by credible physicist Ashoke Sen as an important question.

"What is the precise relation between quantum entanglement and classical geometry?"

Could somebody describe the question and explain why it is an important question and its implications?

• I wasn't there but one can guess this is a reference to the "ER=EPR" conjecture. – SM Kravec Jul 9 '14 at 23:22
• @SM Kravec. I do not think so. It is from a completely different context than EPR and the like.. – Sbaniala Jul 9 '14 at 23:33
• These kind of ideas (ER=EPR, etc...), to my knowledge, has begun with this paper of Mark Van Raamsdonk – Trimok Jul 10 '14 at 9:45

There exists a precise way of calculating the entanglement entropy in a conformal field theory via the Ryu-Takayanagi (RT) prescription in the context of the AdS/CFT correspondence.

The RT prescription says that the entanglement entropy of a sub-system $A$ in the CFT$_{d+1}$ that lives on the boundary of AdS$_{d+2}$ is given by the minimal area surface ($\gamma_A$) that hangs from the boundary/perimeter of the sub-system $A$ in the bulk AdS$_{d+2}$. Note that this is co-dimension 2 surface in AdS$_{d+2}$.

$$S_A = \frac{\text{Area}(\gamma_A)}{4G_N^{(d+2)}}$$

Entanglement entropy (EE) is typically a difficult quantity to calculate in quantum field theory (even in free field theories). However, the RT proposal provides a very simple and elegant formulation to calculate EE in field theories with a holographic dual. The proposal is quite successful in the sense that it has reproduced the area-law for EE and obey strong-subadditivity as well.

EE is a quantum mechanical property. However, it is quite amazing that an object in classical geometry (a minimal surface) captures it.

The RT proposal is not proven. It has survived numerous tests in terms of bulk-boundary matchings. I think, what Sen is trying to ask is : can we understand why an object in classical differential geometry will capture a quantum mechanical quantity?

References :

• I heard that in the paper written by Lewkowycz and Maldacena, Ryu-Takayanagi conjecture was proved (in some sense). But I'm not sure whether it is really proof or not, because I didn't see it yet. – Frame Feb 5 '16 at 14:42