While working with AdS/CFT, I am trying to look at the nature of the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula in AdS/CFT, which is the statement that $S(\rho _{A}|\sigma _{A})=S_{\text{bulk}}(\rho _{a}|\sigma _{a})$, where $A$ is a boundary subregion (and $\bar{A}$ is its boundary complement), and $a$ is the bulk subregion dual to $A$ bound by the entanglement wedge $\mathcal{E}_{W}(A)$ found by the separating Ryu-Takayanagi surface $\gamma _{A}$ (resp. $\bar{a}$ is the bulk complement). I understand that this can be derived from the FLM (Faulkner-Lewkowycz-Maldacena) prescription by doing perturbations. That is, considering a perturbation $\delta \rho _{A}$, FLM becomes $$S(\rho _{A}+\delta \rho _{A})-S(\delta \rho _{A})\sim \mathrm{Tr}(\delta \rho _{A}K_{A})-\frac{1}{2}\mathrm{Tr}(\rho _{A}^{-1}\delta \rho _{A}^{2}),$$ but the second term cannot be dropped when $\rho _{A}$ is small. Now, it is said that violations can be found in PSSY and tensor networks. My question is, when it is said that tensor networks pose an issue with JLMS, how is this so? (Note: I do not have much experience with tensor networks.)
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$\begingroup$ I'm offering a bounty for 50 points from my own reputation. Now if the answers are not good enough, what do I do with this 50 points that I just lost? Hopefully someone can answer it instead. $\endgroup$– VaibhavKCommented May 7 at 18:56
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$\begingroup$ Where can this statement be found? Tensor networks is simply a parametrization of a vector in the Hilbert space, so without further specifications that statement is pretty void. It amounts to say that there exist states that violate that conjecture. $\endgroup$– lcvCommented May 7 at 19:22
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$\begingroup$ One possible bit if information is that general tensor network states (as opposed to MPS) can be constructed that have nearly maximal entanglement. $\endgroup$– lcvCommented May 7 at 19:24
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$\begingroup$ @lcv Not sure where this statement is precisely said, but in general people talk about corrections to the JLMS formula arising from JT gravity coupled to EOTW branes. What conjecture are you saying will be violated? $\endgroup$– VaibhavKCommented May 8 at 11:48
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$\begingroup$ When you say violations you mean violations of what? $\endgroup$– lcvCommented May 8 at 15:06
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