Entanglement Holonomies

TL;DR

The paper proposes entanglement holonomies as a quantum analogue of spacetime connections, captured by transporting quantum frames through entangled states. In holographic settings, boundary entanglement holonomies correspond to gravitational Wilson lines threading wormholes, while for subregions of a single CFT they arise from the modular Berry connection and reflect AdS curvature on transported minimal surfaces. By analyzing simple teleportation-based transports and modular-frame transports across multiple subregions, the authors connect boundary quantum data to bulk geometric data, illustrating a concrete facet of GR=QM and the spacetime-entanglement duality. This framework offers a novel diagnostic for emergent bulk geometry and prompts further exploration of how quantum information transport encodes gravitational holonomies.

Abstract

We introduce a quantum notion of parallel transport between subsystems of a quantum state whose holonomies characterize the structure of entanglement. In AdS/CFT, entanglement holonomies are reflected in the bulk spacetime connection. When the subsystems are a pair of holographic CFTs in an entangled state, our quantum transport measures Wilson lines threading the dual wormhole. For subregions of a single CFT it is generated by the modular Berry connection and computes the effect of the AdS curvature on the transport of minimal surfaces. Our observation reveals a new aspect of the spacetime-entanglement duality and yet another concept shared between gravity and quantum mechanics.

Figures (4)

• Figure 1: The orientation of a pair of gyroscopes can be used to define local Cartesian frames. Each of them is a quantum system described by the conjugate pair $(\Theta_i,L_i)$, $i=A,B$ which controls the direction in which they point.
• Figure 2: Transporting a probe gyroscope (C) around a closed loop. The solid blue line corresponds to standard spacetime transport of the gyroscope in the lab. The dashed light blue lines represent quantum transport of the gyroscope from reference frame B to A via quantum teleportation with reference state $|AB\rangle$ (\ref{['refstate']}). The angle between the gyroscopes transported along the two different paths measures the holonomy of the loop.
• Figure 3: A minimal surface $\Gamma_A$ in AdS and the associated pair of Rindler wedges $W_A, W_{\bar{A}}$. The red arrows represent Rindler time translations in the two wedges. The combined transformation $(t_A, t_{\bar{A}})\rightarrow (t_A+\epsilon, t_{\bar{A}}-\epsilon)$ is an isometry of AdS and acts as a boost in the vicinity of $\Gamma_A$. Fixing the Rindler frame of $\Gamma_A$ amounts to choosing an origin for the Rindler clocks. The clocks on $W_A$, $W_{\bar{A}}$ are synced, as dictated by the vacuum state (\ref{['vacuum']}).
• Figure 4: Three infinitesimally minimal surfaces in AdS, $\Gamma_A,\Gamma_B,\Gamma_C$, can be used to close a loop of modular Hamiltonians and compute the precession of the modular frame. The first leg of the closed path is transport of a Rindler clock directly from $A$ to $\bar{C}$ (LEFT). The invariance of the state under (\ref{['envariance']}) can be utilized to sync the Rindler clocks ($t_A=t_C$ and $t_{\bar{A}}=t_{\bar{C}}$) and trivialize the transport. The second leg is transport from $A$ to $\bar{B}$ and then to $\bar{C}$ (RIGHT). The clock of $\bar{B}$ is now misaligned with both $A$ and $\bar{C}$ as can be seen by the kinks in the red and blue AdS time-slices. The modular Berry holonomy (\ref{['BerryHolonomy']}) depends on the hyperbolic angles of these kinks.