Covariant Residual Entropy

TL;DR

This paper introduces covariantly-defined geometric constructs, the 'strip wedge' and 'rim wedge', to robustly characterize residual entropy in AdS/CFT, overcoming limitations of previous differential entropy approaches.

Contribution

It identifies and develops two covariant constructs, the 'strip wedge' and 'rim wedge', that generalize residual entropy beyond previous restrictive frameworks in AdS/CFT.

Findings

The 'strip wedge' and 'rim wedge' are well-defined in arbitrary time-dependent AdS spacetimes.

These constructs relate to residual entropy but do not always correspond directly to the regions of unknown.

The paper establishes conditions for when these constructs coincide and discusses implications for holographic quantum information.

Abstract

A recently explored interesting quantity in AdS/CFT, dubbed 'residual entropy', characterizes the amount of collective ignorance associated with either boundary observers restricted to finite time duration, or bulk observers who lack access to a certain spacetime region. However, the previously-proposed expression for this quantity involving variation of boundary entanglement entropy (subsequently renamed to 'differential entropy') works only in a severely restrictive context. We explain the key limitations, arguing that in general, differential entropy does not correspond to residual entropy. Given that the concept of residual entropy as collective ignorance transcends these limitations, we identify two correspondingly robust, covariantly-defined constructs: a 'strip wedge' associated with boundary observers and a 'rim wedge' associated with bulk observers. These causal sets are well-defined in arbitrary time-dependent asymptotically AdS spacetimes in any number of dimensions. We discuss their relation, specifying a criterion for when these two constructs coincide, and prove an inclusion relation for a general case. We also speculate about the implications for residual entropy. Curiously, despite each construct admitting a well-defined finite quantity related to the areas of associated bulk surfaces, these quantities are not in one-to-one correspondence with the defining regions of unknown. This has nontrivial implications about holographic measures of quantum information.