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 (by 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.)

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.)

While working with AdS/CFT, I am trying to look at the nature of the JLMS (by Jafferis, Lewkowycz, Maldacena & Suh) 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 (by 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.)

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 (by 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.)