Bit Threads, AdS/CFT and de Sitter

 Recently, I came across a very striking formulation of Ryu-Takayanagi's result in AdS/CFT due to Freedman and Headrick [1604.00354]. While I came across this from reading Frederick and Hubeny's paper on Covariant Bit Threads [2208.10507] (and also from Susskind and Shaghoulian's paper on de Sitter entanglement [2201.03603]), I felt that this proposal has some things to be understood even in the framework of AdS/CFT. The idea is summarised as follows: due to a max-flow-min-cut theorem for Riemannian manifolds, one can state the Ryu-Takayanagi area-minimizing term as a max-flow for bit threads. 

Start from Ryu-Takayanagi (where $\sim $ denotes that the surface is homologous to the boundary subregion):

\[S(\partial R)=\max _{\Sigma }\min _{\mathcal{X}\sim \partial R} \text{Area of } \mathcal{X}\;.\]

Usually, the term $\max _{\Sigma }$ is redundant and is instead used in the HRT prescription for covariant holographic entanglement entropy. From the max-flow-min-cut principle, the term $\min _{\mathcal{X}\sim \partial R}$ can be replaced by maximizers of the flow $v$, which has field lines defined as bit threads. These have two properties: (1) they have a fixed ``width" of $1/4G_{N}$, and (2) they cannot cross a horizon. Ryu-Takayanagi then becomes

\[S(\partial R)=\max _{\# \text{ threads}} \int v\;,\]

where by maximizing flows $v$ we mean the number of threads that leave the boundary subregion. These bit threads have an additional property: they always start and end on the boundary. Ryu-Takayanagi then takes the above form, and is fully expressed in terms of the maximum $\#$ of bit threads leaving the subregion into the bulk, forming a ``bottleneck". This is preserved from the max-flow-min-cut theorem, and so the definition of Ryu-Takayanagi is still the same. 

Susskind and Shaghoulian introduced this aspect of bit threads and entanglement entropy in de Sitter space, which is an interesting read. In de Sitter, one does not have this conformal boundary nature to describe a nice picture for entanglement entropy. Instead, as Susskind and Shaghoulian pointed out, one can define a Ryu-Takayanagi formula in static patch by identifying a surface between the stretched horizons to find the entanglement of the pode-antipode system. This reduces to the familiar Gibbons-Hawking entropy,

\[S=\frac{\text{Area of } H_{\Lambda }}{4G_{N}}\;.\]

In the bit threads formulation, one now has the ability to define two proposals: first, a monolayer proposal, where each horizon emits bit threads towards the other component in a single-layered fashion, and second, a bilayer proposal, where the largest component (usually the cosmic horizon) emits bit threads in a double-layered fashion. Due to this, one has bit threads going (1) to the other component, and (2) ``backwards" towards the pode or antipode. Of course, these bit threads find a bottleneck of zero area (a discrepancy taken into account while making the condition of the surface homologous to the horizons), and therefore the monolayer and the bilayer proposals are equivalent. For, say, the Schwarzschild-de Sitter case, the second layer encounters an ER bridge (since the black hole in say the pode would imply a black hole in the antipode in an entangled state), where the second bottleneck is located. However, we are far from understanding whether there are other information theoretic aspects in de Sitter. In AdS/CFT, presumably things like strong subadditivity can be explained in bit threads, but maybe with semiclassical corrections there are other things to take into account. In de Sitter, this becomes even more complicated. However, this is a nice proposal and I believe some more attention should be given to these things.