One-shot holography

TL;DR

The paper develops a covariant, operational framework for bulk reconstruction in holography by introducing max-EW and min-EW, defined via one-shot entropies and governed by conjectured one-shot QFCs. It extends one-shot quantum Shannon theory to finite-dimensional von Neumann algebras with centers and formulates generalized entropies that couple quantum information with geometric area terms, yielding UV-finite quantities in gravity. It proves key structural properties of the wedges, connects state merging to gravity, and discusses continuum limits via Type II algebras, offering a path to rigorous, covariant statements about information flow and the emergence of bulk time. The framework unifies entanglement-wedge reconstruction with one-shot information theory, providing a robust, general mechanism for understanding holographic encoding beyond time-symmetric settings and across regulator regimes.

Abstract

Following the work of [2008.03319], we define a generally covariant max-entanglement wedge of a boundary region $B$, which we conjecture to be the bulk region reconstructible from $B$. We similarly define a covariant min-entanglement wedge, which we conjecture to be the bulk region that can influence the state on $B$. We prove that the min- and max-entanglement wedges obey various properties necessary for this conjecture, such as nesting, inclusion of the causal wedge, and a reduction to the usual quantum extremal surface prescription in the appropriate special cases. These proofs rely on one-shot versions of the (restricted) quantum focusing conjecture (QFC) that we conjecture to hold. We argue that these QFCs imply a one-shot generalized second law (GSL) and quantum Bousso bound. Moreover, in a particular semiclassical limit we prove this one-shot GSL directly using algebraic techniques. Finally, in order to derive our results, we extend both the frameworks of one-shot quantum Shannon theory and state-specific reconstruction to finite-dimensional von Neumann algebras, allowing nontrivial centers.

Figures (8)

• Figure 1: This figure depicts the deformation of a wedge. The undeformed wedge is $a$ with edge $\eth a$ drawn with the solid line. We deform the region $a$ by deforming $\eth a$ in the null direction by the bump $V^+_{\mathcal{A},\delta, y_0}$ at transverse coordinate $y_0$ with width $\mathcal{A}$ and height $\delta$. This takes $\eth a$ to the dashed line. The new, deformed wedge $a(V^+_{\mathcal{A},\delta, y_0})$ then has edge given by $V^+_{\mathcal{A},\delta, y_0}$. Expansions are then defined via limits as $\mathcal{A}, \delta\to 0$.
• Figure 2: An illustration of $V$ as a tensor "network" composed of a single, random tensor from $b$ to outputs $B$ and $C$. We then feed the state $\ket{\psi}\in \mathcal{H}_a \otimes \mathcal{H}_b \otimes \mathcal{H}_c$ into this random tensor on $b$.
• Figure 3: An illustration of a random tensor network described in the text. Each square or triangle represents a single random tensor with a dangling bulk leg (in blue), denoted by $a_i$, with local Hilbert space $\mathcal{H}_{a_i}$. The network maps the tensor product of $\mathcal{H}_{a_i}$ over all $i$ into the boundary Hilbert space $\mathcal{H}_{B} \otimes \mathcal{H}_{\gamma}$. In the analogy to AdS/CFT, we can think of $\mathcal{H}_B$ as being associated to some CFT subregion and $\mathcal{H}_{\gamma}$ as associated to degrees of freedom localized to the entangling surface of the bulk legs $\mathcal{H}_a = \otimes_i \mathcal{H}_{a_i}$.
• Figure 4: An illustration of a random tensor network as described in the text. This time we denote a candidate surface $\gamma_{\tilde{\mathbf{a}}}$ which bounds all the bulk sites $\tilde{\mathbf{a}}$ between $\gamma_{\tilde{\mathbf{a}}}$ and $B$. The dimension $\dim \gamma_{\tilde{\mathbf{a}}}$ is then the product of dimensions of the black legs cut by the dashed green line.
• Figure 5: A tensor network with the regions $\text{max-EW}[B]$ and $\text{min-EW}[B]$ labeled. As discussed in the main text, the max-EW is conjectured to be the largest bulk region that can be state-specifically reconstructed from $B$. The min-EW is the bulk region whose state possibly affects the state of $B$. The vN-EW, which we discuss in the next subsection, is bounded by the minimal generalized entropy surface. The vN-EW lies between the min- and max-EWs.

Theorems & Definitions (206)

• Definition 2.1: Conditional entropies
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5: Purified distance
• Definition 2.6: Smooth conditional one-shot entropies
• Theorem 2.7: Duality between min- and max-entropies
• Remark 2.8
• Theorem 2.9: Quantum asymptotic equipartition principle
• Theorem 2.10: Chain rule