Twice Upon a Time: Timelike-Separated Quantum Extremal Surfaces

TL;DR

This paper extends the Python's Lunch conjecture to include spacetimes with timelike-separated quantum extremal surfaces, introducing new classifications and proposing a more general framework for bulk reconstruction complexity.

Contribution

It constructs explicit examples of spacetimes with timelike-separated bulges and throats, and proposes an updated conjecture accommodating these features and bounces.

Findings

Constructed spacetimes with timelike-separated bulges and throats.

Identified a third QES type resembling de Sitter bifurcation surfaces.

Proposed an extended Python's Lunch conjecture for general timelike-separated QESs.

Abstract

The Python's Lunch conjecture for the complexity of bulk reconstruction involves two types of nonminimal quantum extremal surfaces (QESs): bulges and throats, which differ by their local properties. The conjecture relies on the connection between bulk spatial geometry and quantum codes: a constricting geometry from bulge to throat encodes the bulk state nonisometrically, and so requires an exponentially complex Grover search to decode. However, thus far, the Python's Lunch conjecture is only defined for spacetimes where all QESs are spacelike-separated from one another. Here we explicitly construct (time-reflection symmetric) spacetimes featuring both timelike-separated bulges and timelike-separated throats. Interestingly, all our examples also feature a third type of QES, locally resembling a de Sitter bifurcation surface, which we name a bounce. By analyzing the Hessian of generalized entropy at a QES, we argue that this classification into throats, bulges and bounces is exhaustive. We then propose an updated Python's Lunch conjecture that can accommodate general timelike-separated QESs and bounces. Notably, our proposal suggests that the gravitational analogue of a tensor network is not necessarily the time-reflection symmetric slice, even when one exists.