A sharp bound on spacetime distance from quantum entanglement

Technical Summary: A Sharp Bound on Spacetime Distance from Quantum Entanglement

Problem Statement
While the Ryu-Takayanagi (RT) formula and its quantum extremal surface (QES) extension have established a precise dictionary between boundary entanglement entropy and bulk extremal surface area, a corresponding quantitative relationship between boundary correlations and bulk point-to-point geodesic distance has remained elusive. Entropy-area relations constrain the areas of extremal surfaces but do not inherently fix the distances between points in the bulk. Consequently, it is unclear how much boundary correlation is required to support a connected bulk geometry or how spatial separation grows as correlations diminish. This paper addresses the gap in the "entanglement builds geometry" paradigm by deriving a rigorous, non-perturbative inequality that bounds bulk geodesic distance from boundary mutual information.

Methodology
The authors derive a "metric-from-information" (MfI) bound by combining two controlled relations:

1. Information-Theoretic Input: Utilizing the quantum Pinsker inequality, the authors establish that mutual information $I(A:B)$ bounds all connected correlators of bounded observables across a bipartition. Specifically, for observables $O_A$ and $O_B$ with operator norms $\leq 1$, the connected correlator is bounded by $\sqrt{2 \ln 2 I(A:B)}$.
2. Holographic Input: In a semiclassical bulk, the two-point function of a heavy scalar primary operator $O_\Delta$ is governed by the renormalized length $L_{ren}$ of the corresponding bulk geodesic, decaying exponentially as $e^{-\Delta L_{ren}}$.

By restricting observables to a holographic code subspace (where operators are bounded and mean-subtracted) and combining the Pinsker bound with the heavy-probe dictionary, the authors invert the exponential decay to obtain a lower bound on $L_{ren}$ in terms of $I(A:B)$.

To extend this pairwise bound to global connectivity, the authors employ a multiscale iteration. They tile the boundary at various resolutions and construct an adjacency graph. By chaining the pairwise distance bounds along paths in this graph and controlling the alignment error (the deviation of consecutive edge estimates from a single geodesic), they derive a global obstruction to bulk connectivity.

Key Contributions and Results

• The Metric-from-Information Bound (Theorem 1):
  The paper derives an explicit lower bound on the renormalized bulk geodesic distance $L_{ren}(x, y)$ between boundary points $x \in A$ and $y \in B$:
  $$L_{ren}(x, y) \geq \frac{1}{2\Delta} \ln \left( \frac{\kappa_\Delta^2 (1-\epsilon_\star)^2}{2 \ln 2 \, I(A:B)} \right)$$
  where $\Delta$ is the scaling dimension of the probe, $\kappa_\Delta$ is a calibration constant, and $\epsilon_\star$ represents controlled semiclassical errors.

  • Divergence: As mutual information $I(A:B) \to 0$, the lower bound on the distance diverges logarithmically. This implies that vanishing boundary correlations force the allowed bulk separation to become infinite.
  • Universality: The logarithmic rate $(2\Delta)^{-1}$ is universal within the code subspace, independent of specific gravitational or stringy corrections, which only affect the finite calibration constants.

• Global Connectivity Obstruction (Theorem 2):
  The authors promote the pairwise inequality to a global constraint on the bulk diameter. If neighboring boundary cells at any resolution scale lack sufficient mutual information, the minimum allowed bulk diameter grows. This establishes that bulk connectivity is a quantitative property of the boundary's multiscale correlation structure, not merely a topological feature of a chosen saddle.

• Resolution of the RT Transition:
  Applied to parallel strips in $AdS_5/CFT_4$, the bound reveals a contradiction at the classical RT phase transition. Classically, mutual information vanishes for disconnected wedges ($s > s_c$), while the bulk geodesic distance remains finite. The MfI bound dictates that a finite distance requires strictly positive mutual information.

  • Implication: This forces a "quantum smoothing" of the transition. The classical zero in mutual information must be replaced by a strictly positive quantum contribution (via QES corrections) to maintain a finite semiclassical distance. The bound thus turns QES smoothing from a refinement into a consistency condition.

• Asymptotic Scaling:
  For large separations, the bound reproduces the correct logarithmic scaling of the semiclassical geodesic distance ($L_{ren} \sim 2 \ln s$) when the mutual information decays algebraically, demonstrating that the bound is asymptotically saturated in pure AdS.

• Distinction from Entropy:
  The paper clarifies that subsystem entropy (e.g., the Page curve) is insufficient to determine metric connectivity. States can share identical subsystem entropies but possess distinct mutual information structures, leading to different geometric connectivities. Conditional mutual information is identified as the complementary criterion for reconstruction, while mutual information specifically forbids short distances.

Significance
The paper claims to transform the qualitative paradigm "entanglement builds geometry" into a quantitative theorem at the metric level. By providing an explicit, non-perturbative inequality, it establishes that boundary mutual information acts as a rigorous lower bound on bulk geodesic separation.

The significance lies in three features:

1. Explicit Inequality: Unlike order-of-magnitude estimates, the bound is a rigorous inequality with controlled errors, excluding any semiclassical geometry where insufficient boundary information supports a short bulk distance.
2. Global Criterion: It elevates pairwise correlations to a global connectivity criterion, showing that a connected bulk requires sufficient mutual information between neighboring boundary cells at every resolution.
3. Quantum Consistency: It exposes the necessity of quantum corrections (QES) to resolve classical phase transitions where mutual information vanishes but geometry remains connected.

The authors suggest this framework offers a diagnostic for when and how quantum correlations give rise to connected spacetime, with potential applications to island transitions in JT gravity, multi-boundary wormholes, and evaporating black holes, though these are presented as directions for future application rather than results derived within the paper.