Finding Pythons in Unexpected Places

TL;DR

This paper argues that strongly nonclassical quantum extremal surfaces generate hidden Python's lunches within maximally mixed code subspaces of isolated black holes, making interior reconstruction exponentially complex even without evaporation. It extends the Python's Lunch framework to scenarios with multiple bulges, and provides a concrete mechanism—via restricted maximin and quantum expansion calculations—for the appearance of nonminimal QESs behind the horizon. By constructing an explicit code subspace from near-horizon Hawking wave-packets and analyzing entropy gradients, the authors show how a lunch can be nucleated in the maximally mixed state, linking geometric data to decoding complexity. The work also connects these lunches to BFV pseudorandomness and discusses implications for firewalls, state dependence, and extensions beyond spherical symmetry, underscoring the broad role of nonminimal QESs in the holographic dictionary.

Abstract

We argue that novel (highly nonclassical) quantum extremal surfaces play a crucial role in reconstructing the black hole interior even for isolated, single-sided, non-evaporating black holes (i.e. with no auxiliary reservoir). Specifically, any code subspace where interior outgoing modes can be excited will have a quantum extremal surface in its maximally mixed state. We argue that as a result, reconstruction of interior outgoing modes is always exponentially complex. Our construction provides evidence in favor of a strong Python's lunch proposal: that nonminimal quantum extremal surfaces are the exclusive source of exponential complexity in the holographic dictionary. We also comment on the relevance of these quantum extremal surfaces to the geometrization of state dependence in the typicality arguments for firewalls.

Figures (13)

• Figure 1: A time-symmetric slice of a one-sided Python's lunch geometry involving two quantum extremal surfaces $\gamma_{\text{aptz}}$ (or the outermost quantum extremal surface) and $\gamma_{\text{bulge}}$. The slice asymptotes to the boundary of AdS on the right. Since the quantum minimal extremal surface is empty, the entire geometry is in the entanglement wedge of the boundary CFT, but the region behind $\gamma_{\text{aptz}}$ is encoded in it with exponential complexity. The exponent of the complexity is given by half of the difference between the generalized entropies of $\gamma_{\text{bulge}}$ and $\gamma_{\text{aptz}}$.
• Figure 2: A quantum extremal surface $\gamma$ in the future of the asymptotic boundary in an isolated black hole is forbidden by the generalized second law. When we have spherical symmetry, the quantum extremal surface $\gamma$ would lie on a past horizon $\mathcal{H}^-$, which by the generalized second law must generically have positive quantum expansion towards the boundary -- in contradiction with the vanishing quantum expansion which would follow from the quantum extremality of $\gamma$. The non-spherically-symmetric argument works analogously EngWal14.
• Figure 3: The Python had fish for lunch. The tensor network prepares a boundary state on left and right CFTs. The fish (triangles) are isometries while the squares involve postselection on one of the legs and the out-of-plane legs (shown with dots) represent bulk degrees of freedom. The network in particular generates an (approximate) isometry from the $\gamma_{\text{min}}$ cut, together with the bulk legs to its right, into the right CFT. The bulk legs between $\gamma_{\text{min}}$ and $\gamma_{\text{aptz}}$ are expected to be encoded on the CFT with exponential complexity. The conjectured exponent is given by half of the difference between the total bond dimensions cut through by $\gamma_{\text{bulge}}$ and $\gamma_{\text{aptz}}$, plus the bulk legs in between.
• Figure 4: An illustration of the Grover-search algorithm. Acting with $U$ takes $\left| \psi \right\rangle\left| 0 \right\rangle^{\otimes n}$ to a state where the $m$ ancilla qubits have a very small probability of being in the desired state $V \left| \psi \right\rangle\left| 0 \right\rangle^{\otimes m}$, i.e. at an angle $\pi/2-\theta$ with the green axis with $\theta \sim 1/\sqrt{A}\sim 2^{-m/2}$. One "back-and-forth" iteration of the algorithm results in reflections shown by the curved arrows. This sequence of operations moves $U \left| \psi \right\rangle\left| 0 \right\rangle^{\otimes n}$ closer to the green axis by an angle $2 \theta$. Therefore to get close to the green axis one needs $\sim 2^{m/2}$ iterations of this procedure, hence an exponential complexity.
• Figure 5: Two types of multiple bulge scenarios are depicted. The salient difference between them is that in the left (right) figure, the throats get bigger (smaller) as we move towards the boundary. For simplicity, the lunches shown are one-sided, i.e. $\gamma_1 = \varnothing$.