Both reflected entropy and entanglement negativity provide valid measures of entanglement between subsystems of a mixed state. For general 2D eternal black holes coupled with CFT matters in large
c limit, we perform the replica-trick computation and find that both effective Renyi reflected entropy and effective entanglement negativity can be expressed in terms of the combination of modified backreacting cosmic branes in
AdS3 bulk. We then develop a holographic scheme to calculate effective reflected entropy and entanglement negativity for general 2D eternal black holes coupled with CFT matters in large
c limit. Using the scheme, we check the consistency condition of the island formulae for entanglement negativity and reflected entropy. We find that the combinations of modified backreacting cosmic branes in
AdS3 bulk from the two island proposals of entanglement negativity exactly match with each other. Finally, we study the saturation of the reflected entropy inequality.
We obtain the holographic entanglement negativity for bipartite mixed states at a finite temperature in baths described by conformal field theories dual to configurations involving two communicating black holes in a braneworld geometry. In this context, we analyze the mixed state entanglement structure characterized by the information transfer between the black holes. The model corresponds to a configuration of two dimensional eternal Jackiw-Teitelboim (JT) black holes in a braneworld geometry involving two Planck branes coupled through shared bath systems described by
CFT2s. Our results reproduce analogue of the Page curves for the entanglement negativity obtained earlier in the context of random matrix theory and from geometric evaporation in JT black hole configurations.
A bstract The entanglement negativity
E E ( 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 ref. [1] that the dominant saddles computing the even Rényi negativity
E(2k) E 2 k generically break the ℤ 2 k replica symmetry. This calls into question previous calculations of holographic negativity using 2D CFT techniques that assumed ℤ 2 k 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
E(2k) E 2 k indeed break the ℤ 2 k replica symmetry. Our argument involves an identity relating
E(2k) E 2 k to the k -th Rényi entropy on subregion AB ∗ in the doubled state
∣ρAB⟩AA∗BB∗ ρ AB A A ∗ BB ∗ , from which we see that the ℤ 2 k replica symmetry is broken down to ℤ k . For k < 1, which includes the case of
E E ( A : B ) at k = 1/2, we use a modified cosmic brane proposal to derive a new holographic prescription for
E(2k) E 2 k 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 2 k 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.
We propose a doubly holographic version of the semiclassical island formula for the entanglement negativity in the framework of the defect AdS/BCFT correspondence where the anti--de Sitter (AdS) bulk contains a defect conformal matter theory. In this context, we propose a defect extremal surface (DES) formula for computing the entanglement negativity modified by the contribution from the defect matter theory on the end-of-the-world brane. The equivalence of the DES proposal and the semiclassical island formula for the entanglement negativity is demonstrated in
AdS3/BCFT2 framework. Furthermore, in the time-dependent
AdS3/BCFT2 scenarios involving eternal black holes in the lower dimensional effective description, we investigate the time evolution of the entanglement negativity through the DES and the island formulas and obtain the analogs of the Page curves.
Recent work has demonstrated the need to include contributions from entanglement islands when computing the entanglement entropy in quantum field theory states coupled to regions of semiclassical gravity. We propose a new formula for the reflected entropy that includes additional contributions from such islands. We derive this formula from the gravitational path integral by finding additional saddles that include generalized replica wormholes. We also demonstrate that our covariant formula satisfies all the inequalities required of the reflected entropy. We use this formula in various examples that demonstrate its relevance in illustrating the structure of multipartite entanglement that are invisible to the entropies.
We study how mixed-state correlations reorganize across island transitions in Jackiw-Teitelboim gravity coupled to a non-gravitating bath. Using the partial-dimensional-reduction realization of double holography, we compute reflected entropy for adjacent and disjoint radiation regions in the vacuum and in the thermofield-double state of an eternal black hole. Reflected entropy probes the internal partition of the entanglement wedge and therefore resolves structure that is invisible to the ordinary Page curve: its dominant saddle can change even when the entropy saddle of the union remains fixed, and it vanishes at leading order when the entanglement wedge disconnects. We show that the lower-dimensional reflected-island description agrees, to leading order in the dynamical brane fluctuation, with the higher-dimensional defect-extremal-surface picture. The finite-angle comparison further suggests a projection-covariant identification of island and brane endpoints at linear order. The resulting Page-like curves and Markov gaps provide a geometric probe of the reorganization of the purification and of non-Markovian, multipartite correlation structure during island transitions.