The Python's Lunch: geometric obstructions to decoding Hawking radiation

TL;DR

This work reconciles the mismatch between holographic complexity (volume/action) and the Harlow-Hayden result by introducing restricted complexity as the holographic quantity dual to a geometric obstruction—the Python's Lunch—in wormholes. It shows that one-sided (restricted) decoding tasks incur exponential complexity tied to the lunch's bulge, while unrestricted complexity remains polynomial, explaining when geometry mirrors complexity. The authors develop a covariant generalization using quantum extremal surfaces, analyze evaporating black holes, and extend the lunch concept to non-evaporating and classical spacetimes, arguing that the tensor-network perspective (with allowed post-selection) is essential for capturing the correct dual. They also explore post-selected state complexity, arguing that even with post-selection there exist exponentially hard states, thus refining the proposed holographic duals for complexity.

Abstract

According to Harlow and Hayden [arXiv:1301.4504] the task of distilling information out of Hawking radiation appears to be computationally hard despite the fact that the quantum state of the black hole and its radiation is relatively un-complex. We trace this computational difficulty to a geometric obstruction in the Einstein-Rosen bridge connecting the black hole and its radiation. Inspired by tensor network models, we conjecture a precise formula relating the computational hardness of distilling information to geometric properties of the wormhole - specifically to the exponential of the difference in generalized entropies between the two non-minimal quantum extremal surfaces that constitute the obstruction. Due to its shape, we call this obstruction the "Python's Lunch", in analogy to the reptile's postprandial bulge.

Figures (28)

• Figure 1: A spatial slice through a 'Python's Lunch' geometry. On the far left, the wormhole opens up to one asymptotic region with infinite cross-sectional area; on the far right, the wormhole opens up to the other asymptotic region also with infinite cross-sectional area. In AdS-Schwarzschild black holes the cross-sectional area reaches a minimum in the middle of the wormhole, and increases on either side. By contrast, in the Python's Lunch geometry the cross-sectional area first shrinks, then grows, then shrinks, then grows again, giving rise to a bulge in the middle of the wormhole---the eponymous Lunch. $\mathcal{A}_L$ and $\mathcal{A}_R$ are the areas of the minimal surfaces on each side and $\mathcal{A}_\textrm{max}$ is the area of the luncheon bulge.
• Figure 2: Successive spatial slices through the wormhole. Since the two sides are maximally entangled, Alice is able to shorten the wormhole by unitary operations $U_A \otimes \mathds{1}$ that act only on her side.
• Figure 3: Left: the quantum circuit that prepares $U_A(t) \otimes U_B(t) | \textrm{TFD} \rangle$. Right: the quantum circuit that prepares $U_A(2t) \otimes \mathds{1} | \textrm{TFD} \rangle$. The two states are the same.
• Figure 4: By acting with $U_A(t)^{\dagger} \otimes \mathds{1}$ in a series of incremental $k$-local steps, Alice can undo time evolution, thus mapping $U_A(t) \otimes \mathds{1} | \textrm{TFD} \rangle$ back to $| \textrm{TFD} \rangle$.
• Figure 5: A unitary $U_{AB}$ cannot in general be decomposed as $U_A \otimes U_B$. In the example in this figure, the horizontal red links between the left and right sides represent gates which couple qubits on the two sides.