The Entanglement Wedge of Unknown Couplings

TL;DR

This work shows that the black hole interior is highly sensitive to the precise couplings of the boundary theory. By coupling the system to a journal that records boundary couplings and applying quantum extremal surface methods, the authors demonstrate that unknown couplings obscure interior reconstruction, while the exterior entanglement wedge remains accessible. They establish that replica wormholes and island formations restore unitarity in both JT gravity with BCFT matter and SYK models, supported by analytic calculations and numerical simulations. The results illuminate how non-perturbative, theory-dependent effects—captured by islands and wormholes—shape bulk reconstruction and chaos diagnostics, with broad implications for higher-dimensional AdS/CFT and the role of the Loschmidt echo.

Abstract

The black hole interior is a mysterious region of spacetime where non-perturbative effects are sometimes important. These non-perturbative effects are believed to be highly theory-dependent. We sharpen these statements by considering a setup where the state of the black hole is in a superposition of states corresponding to boundary theories with different couplings, entangled with a reference which keeps track of those couplings. The entanglement wedge of the reference can then be interpreted as the bulk region most sensitive to the values of the couplings. In simple bulk models, e.g., JT gravity + a matter BCFT, the QES formula implies that the reference contains the black hole interior at late times. We also analyze the Renyi-2 entropy of the reference, which can be viewed as a diagnostic of chaos via the Loschmidt echo. We find explicitly the replica wormhole that diagnoses the island and restores unitarity. Numerical and analytical evidence of these statements in the SYK model is presented. Similar considerations are expected to apply in higher dimensional AdS/CFT, for marginal and even irrelevant couplings.

Figures (17)

• Figure 1: Generalized entropy $S_m + \phi - \phi_0$ as a function of the real coordinate $z$. Here $S_m$ is the matter entropy of an interval $[-1+\epsilon^{1/2},z]$. For large values of $\phi_h$, there is a QES near the horizon. As $\phi_r$ gets smaller, the QES shifts closer to the left side.
• Figure 2: The conditional Renyi $\tilde{R}_2$ as a function of Lorentzian time evolution $J T$, obtained by numerically diagonalizing the SYK Hamiltonian for $N=20$, $\beta J = 16$ and $n_\mathrm{trials} = 3000$. $1 \sigma$ error bars are displayed. The solid curve is the full answer while the dashed curve is the "disk" approximation $\tilde{R}_2^\mathsf{disk}$. We see that the disk approximation is good for early times, but decays too rapidly at late times. Even without computing the full answer (in orange), the unitarity bound $\tr \rho^2 \ge 2^{-N}$ would rule out the disk answer at large times.
• Figure 3: $N = 20$, $n_\mathrm{trials} = 1000$, $\beta J = 16, \epsilon = 0.1$. We show $q_2 = 4,6$ deformations and $1\sigma$ error bars. Note that $q_2=6$ is an irrelevant deformation, but the curves seem qualitatively similar to the marginal case.
• Figure 4: The 2-pt correlator $|G_{LL}|$ as a function of Lorentzian time for the $\lambda^2=0$ disk and wormhole for the case of many marginal couplings. We take $N = 20$, $n_\mathrm{trials} = 600, \beta \mathcal{J} = 1, JT=10^3$. Notice that as $t \to T$ we reach the second quench site. For the disk solution, a prediction from the large q analysis is that the correlator becomes large on the disk, whereas it remains small on the wormhole. This seems to be in rough agreement with the modest $N,q= 4$ numerics. In computing the correlator, we divide by the average norm $\braket{\beta+2iT,J}{\beta+2iT,J'}$ for the disk and $|\braket{\beta+2iT,J}{\beta+2iT,J'}|^2$ for the wormhole.
• Figure 5: The 2-pt correlator $|G_{LL}|$ as a function of Lorentzian time, where a single irrelevant $q=6$ parameter is varied, as in Figure \ref{['fig:q4q6']}. We take $N = 20$, $n_\mathrm{trials} = 2000, \beta \mathcal{J} = 1, JT=10^5$. The correlators are qualitatively similar to the case with many marginal parameters in Figure \ref{['fig:l0c']}.