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

Technical Summary: Out-of-Time-Order-Correlators in Holographic EPR pairs

Problem Statement
This paper investigates the dynamics of quantum information scrambling and entanglement disentanglement in a holographic framework describing Einstein-Podolsky-Rosen (EPR) pairs. Specifically, the authors aim to characterize the influence of shock waves on the quantum entanglement of a holographic EPR pair by computing Out-of-Time-Order Correlators (OTOCs). While previous work established the connection between wormhole geometry and EPR pairs (ER=EPR) and calculated two-point correlators in shock wave backgrounds, it remained unclear whether the equivalence between OTOCs calculated via shock wave backgrounds and those derived from eikonal scattering amplitudes holds in the context of string worldsheet theories. Unlike bulk gravity where such equivalences are proven via gravitational actions, the worldsheet theory is not strictly gravitational, necessitating a re-examination of this relationship.

Methodology
The authors employ two distinct but related approaches within the AdS/CFT correspondence, utilizing a string moving in $AdS_{d+1}$ space with endpoints anchored on the boundary and accelerating in opposite directions. This setup induces a metric on the string worldsheet that describes an AdS wormhole.

1. Holographic Influence Functional (Shock Wave Background):
   Building on previous work [6], the authors analyze the worldsheet wormhole distorted by massless energy quanta (shock waves) propagating along the horizon. They utilize the "double scaling limit," where the shock wave is generated by patching two worldsheets with slightly different accelerations. The cross-correlator between fields coupled to the left and right EPR particles is calculated in this background. The authors demonstrate that these correlators, when analytically continued, correspond to thermal OTOCs in the boundary theory.

2. Worldsheet Scattering in the Eikonal Approximation:
   To provide an alternative derivation and test the equivalence, the authors compute four-point and six-point OTOCs using the eikonal approximation for worldsheet scattering. They treat the operators $\hat{W}$ (perturbing the system) and $\hat{F}$ (probing the system) as dual to worldsheet fields $q_W$ and $q$. The scattering amplitude is modeled as a phase factor $e^{i\delta(s)}$, where $\delta(s)$ is the eikonal phase depending on the center-of-mass energy $s$.

   • Key Identification: The authors establish a direct link between the two methods by identifying the eikonal scattering phase $\delta(s)$ with the shock wave shift parameter $\gamma$. Specifically, they argue that $\delta(s) = 2p_U \gamma$ (or $\delta(s) = \gamma' s$), effectively mapping the shock wave background calculation to the scattering amplitude.

Key Contributions and Results

• Equivalence of Approaches: The primary contribution is the demonstration that the holographic influence functional method (shock wave backgrounds) and the eikonal scattering amplitude method yield consistent results for OTOCs in the holographic EPR setup. The authors show that the two-to-two eikonal scattering amplitude can be identified with the doubling scattering parameter, clarifying the connection between wormhole geometry and quantum information scrambling.
• Four-Point OTOCs:
  • The authors derive the four-point OTOC $\langle \hat{W}(t_0) \hat{F}(0) \hat{W}(t_0) \hat{F}(0) \rangle_\beta$.
  • The result exhibits an initial phase of exponential growth followed by decay.
  • The scrambling time is identified as $\tau^* = b \ln(b\bar{\gamma})$, where $b$ is related to the inverse temperature ($\beta = 2\pi b$) and $\bar{\gamma}$ is a renormalized shock wave strength parameter.
  • For times $t_0 > \tau^*$, the OTOC decays exponentially with a Lyapunov exponent $\lambda_L = 1/b = 2\pi/\beta$, consistent with maximal chaos.
  • The results match the analytic forms obtained from the shock wave method in the appropriate parameter regions.
• Six-Point OTOCs:
  • The authors extend the calculation to six-point OTOCs of the form $\langle \hat{W} \hat{F} \hat{W} \hat{W} \hat{F} \hat{W} \rangle_\beta$, which probe scattering behind the worldsheet horizons.
  • The six-point OTOC shows a marginally longer scrambling time, $\tau'_* = b \ln(b^2\bar{\gamma})$, compared to the four-point case.
  • The difference in scrambling times ($\tau'_* - \tau^* = b \ln 2$) is attributed to the fact that six-point correlators probe "finer-grained" chaos.
  • Like the four-point case, the six-point OTOC exhibits exponential decay with the same Lyapunov exponent but different proportional constants.
• Late-Time Behavior: Both four-point and six-point OTOCs decay as $t^{-2}$ at late times ($t \gg b$), consistent with the behavior of two-point correlators in the absence of significant shock wave effects.

Significance and Claims
The paper claims to solidify the theoretical framework connecting wormhole geometry to quantum information scrambling in holographic EPR pairs. By establishing the equivalence between shock wave background calculations and eikonal scattering amplitudes in a non-gravitational worldsheet theory, the authors validate the use of scattering amplitudes to probe entanglement dynamics in these systems.

The results confirm that the holographic EPR pair exhibits fast scrambling behavior characteristic of black holes, with a scrambling time logarithmically dependent on the entropy (or shock wave strength). The paper also notes that the six-point OTOC provides a more sensitive probe of the system's chaotic structure, revealing a slightly delayed scrambling time compared to four-point correlators.

The authors modestly suggest that while their results align with the ER=EPR conjecture, further investigation is needed to determine if the wormhole entropy corresponds directly to the entanglement entropy of the EPR pair and how decoherence (due to coupling with the ambient CFT) manifests in these correlators. They also note that the divergence observed when the shock wave parameter is negative (rendering the wormhole traversable) warrants further study regarding information exchange through the black hole interior.