Islands for Entanglement Negativity

TL;DR

This work develops two complementary island constructions for entanglement negativity in quantum field theories coupled to semiclassical gravity and validates them in JT gravity with baths. Proposal-I expresses the island contribution as a specific algebraic sum of generalized Renyi entropies of order half, while Proposal-II uses backreacted EWCS area plus bulk entanglement negativity and is shown to be equivalent to half the generalized Renyi reflected entropy. Applying both approaches to extremal and eternal JT black holes with large central charge CFT$_2$ matter, the authors find exact agreement across disjoint, adjacent, and single-interval configurations, providing strong consistency checks. The results support a doubly holographic picture and are corroborated by a replica wormhole derivation, indicating robust structural connections between mixed-state entanglement measures and island contributions in gravity-coupled QFTs.

Abstract

We advance two alternative proposals for the island contributions to the entanglement negativity of various pure and mixed state configurations in quantum field theories coupled to semiclassical gravity. The first construction involves the extremization of an algebraic sum of the generalized Renyi entropies of order half. The second proposal involves the extremization of the sum of the effective entanglement negativity of quantum matter fields and the backreacted area of a cosmic brane spanning the entanglement wedge cross section which also extremizes the generalized Renyi reflected entropy of order half. These proposals are utilized to obtain the island contributions to the entanglement negativity of various pure and mixed state configurations involving the bath systems coupled to extremal and non-extremal black holes in JT gravity demonstrating an exact match with each other. Furthermore, the results from both the proposals match precisely with the island contribution to half the Renyi reflected entropy of order half providing a strong consistency check. We then allude to a possible doubly holographic picture of our island proposals and provide a derivation of the first proposal by determining the corresponding replica wormhole contributions.

Figures (31)

• Figure 1: Schematic for the holographic construction of reflected entropy. Figure modified from Dutta:2019genLi:2020ceg.
• Figure 2: Schematic for the path integral representation of the replicated manifold for the generalied Renyi entropy. Figure modified from Dong:2020uxp.
• Figure 3: Schematic for the islands proposal-I for the entanglement negativity of the mixed state configuration for disjoint intervals in phase-I. The circled curves denote the backreacted cosmic branes in the double holographic picture of our proposal-I.
• Figure 4: Schematic for island proposal-II for the entanglement negativity of the mixed state configuration of disjoint intervals in phase-I. The gray region here and for the rest of the figures corresponds to the entanglement wedge of the subsystem-$AB$. The blue dotted line denotes the EWCS in the double holographic picture. The above figure is modified from Chandrasekaran:2020qtn.
• Figure 5: Schematic for the island proposal-II for the entanglement negativity of the disjoint intervals in phase-I. The emergent twist operators corresponding to the reflected entropy are depicted. Figure modified from Chandrasekaran:2020qtn.