Holographic entanglement obeys strong subadditivity

Aron Wall

Aron Wall is a member of the School of Natural Sciences at the Institute for Advanced Study. In his spare time he blogs at Undivided Looking. He was the 2013 recipient of the Bergmann-Wheeler thesis prize, which is sponsored by Classical and Quantum Gravity.

Gauge-gravity duality allows us to calculate properties of certain quantum field theories (QFT) from classical general relativity. One famous piece of this conjecture, due to Ryu and Takayanagi, relates the entanglement entropy in a QFT region to the area of a surface in the gravitational theory. In addition to being a clue about quantum gravity, this proposal is one of the few tools which allow us to calculate entanglement entropy analytically. Since the entanglement entropy is of increasing interest for field theory and condensed matter applications, it is important to check if the conjecture is true.

One important property of the entropy is strong subadditivity (SSA). This quantum inequality says that the sum of the entropies in two regions is always greater than the sum of the entropies of their union and intersection. My article uses proof techniques in classical general relativity to show that holographic entanglement entropy obeys SSA.

In the case where everything is static, the entropy is given by the area of a minimal surface on a static time slice. In that case, Matt Headrick provided a simple picture proof that SSA is true (Figure).

Strong Subadditivity

The figure shows space at one time. The horizontal line at the base represents the QFT, whose space is subdivided into regions A, B, C. The solid lines are the surfaces ed and cf used to calculate the entropies of the regions AB and BC respectively. By reconnecting the lines, one obtains surfaces cd and ef, which must have more area than the minimal surfaces anchored to B and ABC

But if we want to consider dynamical spacetimes, or QFT regions which are not on a static (t = constant) time slice, then Hubeny, Ragamani, and Takayanagi have conjectured that we need to use extremal surfaces, whose area is constant for first order variations. But it is harder to prove interesting results about extremal surfaces, since they are neither global minima nor global maxima.

My contribution was to reformulate the definition of the surface. I considered the maximin surface, which is defined by minimizing the area on a given time slice, but then choosing the time slice to maximize that minimum value. It turns out that this definition is equivalent to the extremal surface definition on physically reasonable spacetimes, but it makes it much easier to prove general results like SSA.

I also proposed that if someone tells you all about the QFT region, you can reconstruct the gravitational spacetime all the way up to the holographic entropy surface, but no further. My results are consistent with this hypothesis, but more work is needed to see if it is really true.

Read the full article in Classical and Quantum Gravity:
Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy
Aron C Wall
2014 Class. Quantum Grav. 31 225007

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