Cryptographic tests of the python's lunch conjecture

TL;DR

The paper explores the holographic implications of the python's lunch conjecture by grounding it in a cryptographic framework—Conditional Disclosure of Secrets (CDS)—and translating bulk geometric constraints into boundary correlation bounds. By deriving a holographic CDS lower bound, it connects the area difference between lunch surfaces to mutual information I(ˆV1:ˆV2), yielding a universal bound I(ˆV1:ˆV2) ≥ α0(A[γ_bulge]−A[γ_app])/(4G_N) under suitable conditions, and it verifies weakened forms of this bound in AdS_2+1 geometries (including NEC-satisfying spacetimes) and in concrete examples (ETW branes, defects, BTZ). The analysis includes a broad set of results: a general CDS lower bound on correlation, geometric proofs that I(ˆV1:ˆV2) scales as Θ(1/G_N) when a lunch is present, and explicit calculations showing both support and limitations of the proposed bound across multiple spacetimes. The work also discusses a conjectured bound linking CDS complexity to boundary mutual information, provides counterexamples in vacuum AdS_2+1, and presents an extensive appendix detailing the complexity-correlation relationships in several geometries. Overall, the results offer nontrivial evidence for the tensor-network picture behind the python's lunch and lay groundwork for further probing the relationship between bulk geometry, complexity, and boundary correlations in AdS/CFT.

Abstract

In the AdS/CFT correspondence, a subregion of the CFT allows for the recovery of a corresponding subregion of the bulk known as its entanglement wedge. In some cases, an entanglement wedge contains a locally but not globally minimal surface homologous to the CFT subregion, in which case it is said to contain a python's lunch. It has been proposed that python's lunch geometries should be modelled by tensor networks that feature projective operations where the wedge narrows. This model leads to the python's lunch (PL) conjecture, which asserts that reconstructing information from past the locally minimal surface is computationally difficult. In this work, we use cryptographic tools related to a primitive known as the Conditional Disclosure of Secrets (CDS) to develop consequences of the projective tensor network model that can be checked directly in AdS/CFT. We argue from the tensor network picture that the mutual information between appropriate CFT subregions is lower bounded linearly by an area difference associated with the geometry of the lunch. Recalling that the mutual information is also computed by bulk extremal surfaces, this gives a checkable geometrical consequence of the tensor network model. We prove weakened versions of this geometrical statement in asymptotically AdS$_{2+1}$ spacetimes satisfying the null energy condition, and confirm it in some example geometries, supporting the tensor network model and by proxy the PL conjecture.

Figures (25)

• Figure 1: Spatial slice of a spacetime with a python's lunch. The light green curve denotes a boundary region $\hat{\mathcal{R}}$, the solid blue curve denotes the corresponding Ryu-Takayanagi surface, the dashed blue curve denotes a locally minimal "appetizer" surface, and the dashed red curve denotes the "bulge" surface. The region between the Ryu-Takayanagi surface and the appetizer surface is the "python's lunch" for the region $\hat{\mathcal{R}}$.
• Figure 2: A tensor network model of a python's lunch geometry. Each vertex corresponds to a tensor, with the edges corresponding to tensor indices. Vertices are connected by edges according to the pattern in which tensors are contracted. Figure from brown2020python.
• Figure 3: An asymptotically AdS$_{2+1}$ geometry. The boundary region $\hat{\mathcal{R}}$ features a lunch, bounded by an extremal surface called the appetizer (dashed blue) which is locally minimal with respect to spacelike deformations, and an extremal surface called the RT surface (solid blue) which is globally minimal with respect to such deformations. Between them is a bulge surface (dashed red), which is extremal but not locally minimal. We argue that, assuming the python's lunch conjecture, when signals from $c_1$ and $c_2$ can meet in the bulk and then travel to either side of the appetizer surface, associated boundary regions $\hat{\mathcal{V}}_1$ and $\hat{\mathcal{V}}_2$ (shaded grey) must have mutual information satisfying a lower bound of the form \ref{['eq:lowerboundintro']}, and in particular they must have a connected entanglement wedge.
• Figure 4: Two intervals $\hat{\mathcal{R}}_1$ and $\hat{\mathcal{R}}_2$ in the boundary of vacuum AdS$_{2+1}$. The RT surface $\gamma_{\hat{\mathcal{R}}_1\cup \hat{\mathcal{R}}_2}$ defining the entanglement wedge of $\hat{\mathcal{R}}_1\cup \hat{\mathcal{R}}_2$ is shown in solid blue; the appetizer surface $\gamma_{\hat{\mathcal{R}}_1}\cup \gamma_{\hat{\mathcal{R}}_2}$ is in dashed blue. The lunch region sits between the appetizer and RT surfaces. We observe that situations in which the region $\hat{\mathcal{R}}$ consists of two intervals can furnish counter-examples to the bound \ref{['eq:conjlowerbound']}.
• Figure 5: (a) General form of a CDQS protocol. Alice, on the bottom left, holds input $x$ and a secret $s$. Bob, on the bottom right, holds input $y$. Alice and Bob may share entanglement, represented as the lower curved wire. Alice and Bob perform quantum channels on their locally held systems and send the resulting outputs to the referee. The referee knows $x,y$ and should be able to determine $s$ if and only if $f(x,y)=1$, where $f$ is a function known to all parties. (b) A (classical) CDS protocol for the equality function on single bit inputs. Alice and Bob share two random bits, $r_0$ and $r_1$. Alice sends the XOR of the secret $z$ with $r_x$. Bob sends $r_y$. If $x=y$, then the referee can compute $m_a\oplus m_b=s$, while if $x\neq y$ then $m_a$ and $m_b$ are independent random bits which reveal nothing about $s$.

Theorems & Definitions (6)

• Definition 1
• Theorem 2
• Conjecture 3
• Theorem 4
• Theorem 5
• Theorem 6