Bulk Locality and Entanglement Swapping in AdS/CFT

TL;DR

The paper tackles how bulk operators in AdS/CFT can be represented on the boundary and how gravitational dressing affects entanglement. It introduces entanglement-swapping boundary operators as a simple mechanism that reproduces qualitative features of bulk backreaction and the Ryu–Takayanagi entropy formula, while naturally supporting multiple boundary representations of a bulk excitation. Through explicit qubit and free-field models, it demonstrates that such swaps leave certain boundary subsystems unchanged yet decrease the entanglement of the region containing the dressing, and it shows that nonperturbative effects implied by Reeh–Schlieder limit exact equivalences. The work provides a tractable, entanglement-centric framework for connecting boundary degrees of freedom to bulk dynamics and motivates further nonperturbative developments, including potential simplifications in AdS$_3$/CFT$_2$.

Abstract

Localized bulk excitations in AdS/CFT are produced by operators which modify the pattern of entanglement in the boundary state. We show that simple models--consisting of entanglement swapping operators acting on a qubit system or a free field theory--capture qualitative features of gravitational backreaction and reproduce predictions of the Ryu-Takayanagi formula. These entanglement swapping operators naturally admit multiple representations associated with different degrees of freedom, thereby reproducing the code subspace structure emphasized by Almheiri, Dong, and Harlow. We also show that the boundary Reeh-Schlieder theorem implies that equivalence of certain operators on a code subspace necessarily breaks down when non-perturbative effects are taken into account (as is expected based on bulk arguments).

Figures (2)

• Figure 1: Schematic diagram of a bulk excitation localized around the point $P$ with gravitational dressing (dashed lines) focused into the boundary region $B$. The entire bulk excitation---including gravitational dressing---extends beyond the entanglement wedge of $B$ but is contained in the entanglement wedges of $AB$ and $BC$. The boundaries of the entanglement wedges are indicated with solid lines.
• Figure 2: (left) Diagram of the qubit "vacuum state" \ref{['eq:qubitVacuum']} with dotted lines signifying Bell paris. (right) The resulting state after any of the three unitary operators \ref{['eq:qubitOps']} act on the vacuum state. Note that the resulting state can be obtained by swapping qubits $a$ and $c$ or $b$ and $d$. In this diagram the entanglement entropy of any region is equal to $\log(2)$ times the number of dotted lines crossing the boundary of the associated entanglement wedge.