Twice upon a time: timelike-separated quantum extremal surfaces

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.

Geometric surprises in the Python's lunch conjecture

“…However, it is important to note that, even restricted to time-reflection symmetric states, this is not always true. Explicit states for which time reflection symmetry is spontaneously broken by the bulge and other minimal surfaces have been constructed recently, in near-extremal black hole interiors in [31]. 2 We do not believe that this phenomenon occurs in any of the spacetimes we study in this paper, although we have not proven that this is the case.…”

“…In our analysis, we have restricted ourselves to time reflection-symmetric states, where all of the extremal surfaces lie on the time reflection-symmetric Cauchy slice Σ. Situations in which the time-reflection symmetry is spontaneously broken by the bulge and other locally minimal surfaces have been reported in [31] for specific spherically symmetric initial data in near-extremal black hole interiors. The specific data has been constructed using the two-dimensional description of these systems given by JT gravity with additional matter fields coupled to the metric.…”

“…An interesting open problem that we leave for future work is to investigate the nature of more general extremal surfaces, such as the surfaces called bounces (cf. [31]), that are expected to arise in these situations. near each puncture θ p and θ 0 ≪ 1.…”