Entanglement negativity and replica symmetry breaking in general holographic states

The entanglement negativity is an element of(A:B) is a useful measure of quantum entanglement in bipartite mixed states. In random tensor networks (RTNs), which are related to fixed-area states, it was found in [arXiv:2101.11029] that the dominant saddles computing the even R & eacute;nyi negativity is an element of((2k)) generically break the Z(2k) replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed Z(2k) replica symmetry and proposed that the negativity was related to the entanglement wedge cross section. In this paper, we resolve this issue by showing that in general holographic states, the saddles computing is an element of((2k)) indeed break the Z(2k) replica symmetry. Our argument involves an identity relating is an element of((2k)) to the k-th R & eacute;nyi entropy on subregion AB & lowast; in the doubled state |rho AB > AA & lowast;BB & lowast;, from which we see that the Z(2k)replica symmetry is broken down to Z(k). For k<1, which includes the case of is an element of(A:B) at k=1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for is an element of((2k)) and show that it is given by a new saddle with multiple cosmic branes anchored to subregions A and B in the original state. Using our prescription, we reproduce known results for the PSSY model and show that our saddle dominates over previously proposed CFT calculations near k=1. Moreover, we argue that the Z(2k) symmetric configurations previously proposed are not gravitational saddles, unlike our proposal. Finally, we contrast holographic calculations with those arising from RTNs with non-maximally entangled links, demonstrating that the qualitative form of backreaction in such RTNs is different from that in gravity.