Out-of-Time-Order-Correlators in Holographic EPR pairs

TL;DR

This work analyzes information scrambling in a holographic EPR pair by computing four-point and six-point OTOCs for a string-based wormhole in $AdS_{d+1}$. It develops and compares two approaches: holographic influence functionals with shock waves on the worldsheet and a worldsheet eikonal scattering framework, showing they yield consistent OTOCs and identifying the late-time Lyapunov exponent as $\lambda_L = 1/b$. Four-point OTOCs scramble on a timescale $\tau_* = b\ln(b\bar{\gamma})$, while six-point OTOCs scramble later with $\tau'_* = b\ln(2b\bar{\gamma})$, indicating finer-grained chaotic dynamics. The results reinforce the ER=EPR perspective by linking wormhole geometry to quantum information scrambling and demonstrate the practical equivalence of bulk eikonal scattering and shock-wave calculations on the holographic worldsheet. The discussion outlines implications for entanglement, wormhole entropy, decoherence, and potential extensions beyond the eikonal limit, including non-equal temperatures and traversable wormhole regimes.

Abstract

In this note, we investigate the out-of-time-order correlators (OTOCs) for quantum fields in a holographic framework describing Einstein-Podolsky-Rosen (EPR) pairs. We compute the four-point and six-point OTOCs using the gravity dual, represented by the string worldsheet theory in Anti-de Sitter (AdS) space. These correlators quantify the rate at which information is scrambled, leading to the disentanglement of the EPR pair. We demonstrate consistency between two approaches for calculating OTOCs: the holographic influence functional on worldsheets perturbed by shock waves, and the worldsheet scattering in the eikonal approximation. We show that the OTOCs exhibit an initial phase of exponential growth, with six-point correlators indicating a marginally longer scrambling time compared to four-point correlators.

Figures (2)

• Figure 1: The worldsheet metric in $(U,V)$-coordinate. Both $U$ and $V$ have their ranges from $-\infty$ to $\infty$. The left (right) boundary is identified with the left (right) quark trajectory, and are described by $UV=-1$. A shock wave trajectory is illustrated by the dashed straight line.
• Figure 2: We plot the four-point OTOC ($F_4$) and six-point OTOC ($F_6$) in (\ref{['F4']}) and $(\ref{['F6']})$ respectively. We rescale the saturated value to be 1, and set $b=1$ , $\tau^*=-2$, (This choice of parameters is just for illustration. In reality, we have$|\tau^*|\gg b$) and $\Delta_F=2$.