Entanglement Negativity and Defect Extremal Surface

We would like to thank the referee for helpful and constructive suggestions. We have implemented all five points the referee recommends for improvement. More specifically,

(1) We agree to clarify the relation between our set up and the inherent CFT perspective. To our best knowledge, the inherent CFT comes from the old idea that the AdS bulk bounded by Karch-Randall brane (as well as the asymptotic boundary) is expected to be dual to two CFTs and the one on the brane may be called as an inherent CFT. This is not our perspective in this paper. By partial reduction we perform explicit dimension reduction (which may not be applicable in higher dimensions) for the AdS$_3$ gravity action between Karch-Randall brane and the tensionless brane. The resulting 2d gravity is therefore equivalent to 3d gravity in that region, which means that we do not need additional duality to translate this part AdS gravity to some inherent CFT (also to avoid double-counting). In our set up we treat the brane CFT as a bulk defect representing some bulk degrees of freedom from the beginning. Our perspective has received a bunch of tests, such as refs.[46, 47, 48]. We took the referee's suggestion and added some detail discussion on the distinction between our set up and the inherent CFT perspective at the end of Section 4.

(2) We agree to add the justification why the factorizations work. Specifically, for eq.(50) in v2, we first use doubling trick to convert 2-point BCFT correlator to 4-point chiral CFT correlator. Then we calculate the 4-point function assuming large $c$ limit and vacuum block dominance, following the approach in [arXiv:1303.6955] by Hartman. We numerically checked that the dominate channels are indeed the ones corresponding to the holographic configurations. The details of the calculation are included in the new Appendix B. The result of this 4-point function is given in eq.(51) of the current version and one can see that it coincides with 3-point function. The factorizations can also be justified in the same way in other places, such as eq.(135) and eq.(142) in v2.

(3) We agree to include a twist operator insertion at the endpoint of the left subsystem and the referee is right. We corrected our calculation in section 7.3 and updated our results in the current version.

(4) We agree to remove the inverse doubling trick around eq.(134, 135) of v2 and the referee is correct. For the justification of the factorization, precisely the same check can be done by assuming large $c$ and vacuum block dominance as we did in response (2). We include the details of the calculation in the new Appendix B.

(5) We agree to make it clear how the OPE coefficient is obtained. Our OPE coefficient was obtained by matching the half of $n=1/2$ R\'enyi reflected entropy and the entanglement negativity. However we stress that the equality between half of $n=1/2$ R\'enyi reflected entropy and the entanglement negativity is an assumption. The details are included in Appendix A of the current version. The final OPE coefficient is given by taking $m = 1,\ n = 1/2$ for eq.(4.37) of Dutta and Faulkner [arXiv:1905.00577]. We also add footnote 2 to emphasize that the relation between half of $n=1/2$ Renyi reflected entropy and the entanglement negativity is an assumption.

We also corrected many English as well as a few typos of the previous version. We hope the improved version met the clarification the referee suggested and became clearer and more precise.

(1) Discussion about our set up and inherent CFT perspective was added in the end of Section 4 (Page 11).

(2) Justification of why the factorizations work was added after Figure 4 (Page 13). The new Appendix B was added to include the numerical check for such factorization.

(3) The referee's suggestion was taken and accordingly the corrections were made for Section 7.3 (Page 25). Other relevant places were also corrected accordingly.

(4) The previous word "inverse doubling trick" was corrected (eq.(136), Page 30). The justification of factorization follows the same way in response (2).

(5) A derivation of OPE coefficient was added in Appendix A and footnote 2 (Page 6) was added to emphasize that the relation between half of $n=1/2$ Renyi reflected entropy and the entanglement negativity is still an assumption.