The Markov gap for geometric reflected entropy

TL;DR

This work identifies the Markov gap $S_R(A:B)-I(A:B)$ as a fidelity-controlled obstruction to Markov recovery on the canonical purification, tying its size to the geometry of the entanglement wedge cross-section in holography. It proves a universal lower bound in time-symmetric pure AdS$_3$ gravity, showing $S_R-I\ge (\log 2)\ell_{ ext{AdS}}/(2G_N)$ times the number of cross-section endpoints, and provides evidence this bound persists with bulk matter and in higher dimensions. A tractable fixed-area-state model then yields explicit formulas for the Markov-recovery fidelity and related sandwiched Rényi entropies, offering quantitative links between boundary recovery and bulk generalized entropy differences between KRT and RT surfaces. Collectively, the results illuminate how multipartite holographic entanglement patterns underpin geometric connections, and suggest a deep link between cross-section boundaries and minimal tripartite entanglement structures in the boundary theory, with potential generalizations to broader holographic settings and dimensions.

Abstract

The reflected entropy $S_R(A:B)$ of a density matrix $ρ_{AB}$ is a bipartite correlation measure lower-bounded by the quantum mutual information $I(A:B)$. In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-$N^2$ gap between $S_R$ and $I$. We provide an information-theoretic interpretation of this gap by observing that $S_R - I$ is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity $S_R - I$ the Markov gap. We then prove that for time-symmetric states in pure AdS$_3$ gravity, the Markov gap is universally lower bounded by $\log(2) \ell_{\text{AdS}}/2 G_N$ times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling $S_R - I$ using fixed area states. This analysis involves deriving a formula for the quantum fidelity -- in fact, for all the sandwiched Rényi relative entropies -- between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.

Figures (24)

• Figure 1: (a) A boundary state $\rho$ whose bulk dual is a Rindler wedge in AdS$_3$. (b) The initial data for the canonical purification $\left.\left| \sqrt{\rho} \right\rangle \right\rangle$, formed by pasting $\operatorname{W}(\rho)$ to its CPT conjugate along the quantum extremal surface that forms the spatial boundary of $\operatorname{W}(\rho)$. The canonical purification is the AdS$_3$ vacuum state defined on a boundary circle whose circumference is twice the spatial extent of the domain of dependence on which $\rho$ was initially defined.
• Figure 2: (a) A time slice of the AdS$_3$ vacuum with two boundary intervals $A$ and $B$ labeled. The intervals have been chosen large enough so that their minimal surface is $\gamma_{AB}$. The entanglement wedge cross-section is labeled $\sigma_{A:B}$. (b) A time slice of the canonical purification, formed by pasting $\operatorname{W}(\rho_{AB})$ to its CPT conjugate along the surface $\gamma_{AB}$. The top and bottom edges of this figure are identified. The minimal surface of boundary region $AA^*$ is the image of $\sigma_{A:B}$ under the symmetry $A \leftrightarrow A^*, B \leftrightarrow B^*$.
• Figure 3: A sketch of the surfaces $\operatorname{RT}(A)$ and $\operatorname{KRT}(A)$ for the case where $A$ and $B$ are two equal-time intervals in the AdS$_3$ vacuum. $\operatorname{KRT}(A)$ is formed by taking the union of the entanglement wedge cross-section $\sigma_{A:B}$ with the portion of $\operatorname{RT}(AB)$ lying "towards $A$" from the cross-section. Both $\operatorname{KRT}(A)$ and $\operatorname{RT}(A)$ are homologous to $A$.
• Figure 4: The minimal surface $\gamma_{AB}$ and entanglement wedge cross-section $\sigma_{A:B}$ for the case where $A$ and $B$ are neighboring intervals on a time slice of the AdS$_3$ boundary. In this case, the entanglement wedge cross-section has only one boundary. It has no boundary at infinity, since that point is at infinite distance.
• Figure 5: A sketch of the canonical purification for the density matrix of two intervals in the AdS$_3$ vacuum; the top and bottom lines are identified, making the entire geometry a two-boundary wormhole. The $ABB^*$ entanglement wedge is shaded in violet, and the $AB$ and $BB^*$ entanglement wedges are indicated with crosshatching. The blue lines are minimal surfaces for $AB$ and $A^* B^*$. Jagged curves have been imposed over two minimal surfaces whose tubular neighborhoods are visible to $ABB^*$ but not to $AB$ or $BB^*$; we argue in the text of the paper that these surfaces must be supported by boundary entanglement contributing to $I(A:B^*|B).$

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof