Sub-Ads scale resolution of pluperfect tensor network I am currently reading article "Bidirectional holographic codes and sub-AdS locality". [I]
This article presents tensor networks which are built from so called pluperfect tensors. 
Authors claim that this kind of tensor network(tn) may "describe geometry at sub-Ads scale"
I am trying to show that we can define sub-Ads scale for this type of tn in case of const. time slice of $Ads_{3}$. (In my understanding it means that we can show that distance between two nodes in this tn can be smaller than $l_{Ads}$)
 I am using approach from article "Consistency Conditions for an AdS/MERA Correspondence"[II].
In this article authors compare lengths of two curves ($\gamma_{1,2}$) in Poincare Ads and in MERA correspondently. To calculate length of curves in MERA they introduce two length scales $L_{1}$ and $L_{2}$.  These distances will eventually be defined in terms of $l_{Ads}$.In this article it is also shown that no change of coordinates can make $L_{1}$ and $L_{2}$ be smaller than  $l_{Ads}$. 
 
 In case of tensor network formed by pluperfect tensors for Poincare disc looks like(left image from [I], right one is toy model,I am currently working with) 
To build analogy with MERA I am trying to introduce two length scales(I introduce only two because tn should describe 2-dim plane).
Let's define black lines to represent (vertical) distance between layers ($L_{2}$) and coloured ones - (horizontal) distance between nodes within layer ($L_{1}$). The problem is that this definition doesn't give us consistent way to define number of nodes as a function of layer. (For 1st we 5 nodes, for 2nd - 25, for 3d - 95). This problem becomes even worse if I try to use this approach for left picture.
  Is there any strict way to define such tensor network layer by layer?
May be there exists simpler way to prove that this pluperfect tensor network admits sub-Ads scale resolution. If there is one I would like to know about it as well.
 
 A: In order to answer this question one needs to follow different strategy. 
 First let us consider tensor network formed by so called perfect tensors(PTN). We will investigate problem of sub-AdS scale resolution for it. This kind of TN is introduced in the article Toy models for the bulk/boundary correspondence. Once this problem is resolved the same problem for tensor network formed by pluperfect tensors may  be seen just as a generalization of above example. 
In case of PTN one starts from constant time slice of $AdS_3$ given by Poincare disc. Now one needs to consider uniform tessellation of this disc. General idea is that Poincare disc in this case will be filled by polygons of equal area. There are many different ways to do so depending of particular form of polygon. Here is an example:

If we associate one perfect tensor to each polygon and connect them we will obtain following tensor network: 

So now we can say that there is certain area associated to each tensor in tensor network. It may be shown that this area has to be of order of $ l^2_{AdS}$.This is why we can conclude that we are not allowed to have sub-AdS scale in our tensor network in aforementioned setup. 
Obvious resolution of this "problem" will be to introduce non uniform tessellation of Poincare disc. However if we try to do so for PTN we will almost instantly arrive to following problem.  By increasing number of tensors in the bulk we increase number of uncontracted(red) bulk legs. Hence at some point dimension of bulk Hilbert space will become larger than dimension of boundary Hilbert space. Yet in this case our tensor network can't be used to mimic AdS/CFT anymore. 
If instead of perfect tensors we consider pluperfect ones we will be able to avoid this problem. Because of certain properties of new setup one will be able to define so called "physical" bulk Hilbert space which will always have smaller dimension than boundary one. That is why nonuniform tessellation is allowed for pluperfect tensor networks. Hence one can have sub-AdS scale resolution in this new setup. Example of nonuniform tessellation may be seen in following image:

