String Theory Seminars

The Crossed Product Construction from Holographic Quantum Error-Correcting Codes

by Dr. Ronak Soni (Cambridge University, UK)

Thursday, March 16, 2023 from to (Asia/Kolkata)
at A304 and on zoom
Recently, the old observation that the generalised entropy of a horizon --- the sum of the area and the entropy of exterior fields --- is better defined than either of the objects that make it up has been put on firmer footing by Witten et al. The main observation is that including fluctuations in the ADM charges in the algebra of matter fields changes the type of algebra from type III_1 --- which doesn't have an entropy function --- to type II --- which does --- using the crossed product construction. We show that a holographic quantum error-correcting (QEC) code naturally has a structure analogous to the crossed product. When the encoded algebra is type I, we show that ‘boundary modular flow’ in the code makes it into a direct sum of type I factors. In the limit where we take the encoded algebra towards a type III_1 algebra, a certain recasting of the direct sum becomes a type II algebra. We are able to recover both type II_\infty --- relevant for black holes --- as well as type II_1 algebras -- relevant for cosmological horizons in de Sitter space --- by imposing different constraints on the area operator in the code. These results indicate that the holographic QEC code might be useful in contexts where the 'boundary' is actually an observer, like the static patch of de Sitter.