Twice Upon a Time: Timelike-Separated Quantum Extremal Surfaces

TL;DR

This work extends the Python's Lunch program by demonstrating timelike-separated quantum extremal surfaces (QES) in explicit JT gravity setups that include bulges, throats, and a central bounce. By formulating both a transverse-symmetric and a general Hessian-based classification of QESs, it shows that timelike configurations are robust and that outer-minimal QESs play a key role in reconciling bulk reconstruction with entanglement wedge concepts. The authors propose a fully general Python's Lunch conjecture that uses maximinimax constructions to define a bulge- versus throat-driven complexity difference, arguing that the gravitational analogue of a tensor network need not lie on a time-reflection-symmetric slice. They also provide explicit spacetime constructions and discuss the implications for reconstructions and boundary duals, emphasizing that gravity alone can reveal the appropriate degree of post-selection. Overall, the paper broadens the domain of the Python's Lunch framework to covariant, timelike-surface settings and clarifies when tensor-network intuition remains valid versus when a more general, covariant prescription is required.

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.

Figures (10)

• Figure 1: A spatial slice of a Python's lunch geometry. The boundary theory lives on a copy of the asymptotic boundary on the right. Figure reproduced from our previous work EngPen21b.
• Figure 2: A tensor network representation of the python's lunch. The black dots are bulk legs, and the increase and decrease in the total bond dimension of the shown cuts introduces post-selection into the bulk-to-boundary map, from the bulk legs to the left of $\gamma_{\text{min}}$ to CFT$_R$. According to the Python's lunch conjecture, this results in an exponential enhancement in the map's complexity. Figure reproduced from our previous work EngPen21b.
• Figure 3: A schematic representation of our JT solutions with time-like separated extremal surfaces homologous to the right (or the left) boundary. Both solutions contain a time-symmetric slice $\Sigma$ with a bounce QES in the middle.
• Figure 4: The profile of $\phi$ on the slice $\Sigma$ ($t=0$ axis) shows three extremal surfaces, at $x=0$ and $x \approx \pm 2$.
• Figure 5: By evolving the dilaton off of the time-symmetric initial data, we explicitly find bulges on the $x=0$ axis at times $t \approx \pm 0.06$.

Theorems & Definitions (47)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof : Proof sketch
• Corollary 1
• proof
• Theorem 2
• proof
• Definition 4