Entanglement Wedge Reconstruction and Entanglement of Purification

TL;DR

This paper explores the limits of reconstructing bulk regions in holography using entanglement entropy and introduces a new concept, differential purification, to fully recover spacelike curves within entanglement wedges.

Contribution

It identifies the limitations of entanglement entropy in reconstructing certain bulk curves and proposes differential purification as a new tool for complete reconstruction.

Findings

Entanglement shadows prevent full reconstruction with entanglement entropy alone.

Differential purification encodes information about nonreconstructible curve segments.

Combining differential entropy and differential purification enables complete bulk curve reconstruction.

Abstract

In the holographic correspondence, subregion duality posits that knowledge of the mixed state of a finite spacelike region of the boundary theory allows full reconstruction of a specific region of the bulk, known as the entanglement wedge. This statement has been proven for local bulk operators. In this paper, specializing first for simplicity to a Rindler wedge of AdS$_3$, we find that generic curves within the wedge are in fact not fully reconstructible with entanglement entropies in the corresponding boundary region, even after using the most general variant of hole-ography, which was recently shown to suffice for reconstruction of arbitrary spacelike curves in the Poincare patch. This limitation is an analog of the familiar phenomenon of entanglement shadows, which we call 'entanglement shade'. We overcome it by showing that the information about the nonreconstructible curve segments is encoded in a slight generalization of the concept of entanglement of purification, whose holographic dual has been discussed very recently. We introduce the notion of 'differential purification', and demonstrate that, in combination with differential entropy, it enables the complete reconstruction of all spacelike curves within an arbitrary entanglement wedge in any 3-dimensional bulk geometry.