Entanglement island, miracle operators and the firewall

TL;DR

The paper probes how interior information of an evaporating AdS black hole can be retrieved by exterior baths using entanglement islands and the quantum extremal surface, deriving a one-norm state-reconstruction formula. It identifies a paradox when interior-information-revealing measurements are allowed in the bath, which cannot be reconciled within a smooth replica geometry. The authors introduce miracle operators, constructed via replica wormholes, to show that such measurements nontrivially alter the bulk-boundary geometry and invalidate the reconstruction in certain cases. They apply this framework to firewall scenarios, demonstrating both entanglement-breaking and entanglement-checking measurements can produce firewall-like effects or preserve detectable entanglement across replicas. The work highlights limits of island-based information retrieval under general bath dynamics and motivates further exploration of gravity as an ensemble theory and the role of QUALM-type measurements.

Abstract

In this paper, we obtain some general results on information retrieval from the black hole interior, based on the recent progress on quantum extremal surface formula and entanglement island. We study an AdS black hole coupled to a bath with generic dynamics, and ask whether it is possible to retrieve information about a small perturbation in the interior from the bath system. We derive a state reconstruction formula based on one norm. However, we show that a contradiction arises if we apply this result to a special situation when the bath dynamics includes a unitary operation that carries a particular measurement to a region $A$ and send the result to another region $W$. Physically, the contradiction arises between transferability of classical information during the measurement, and non-transferability of quantum information which determines the entanglement island. We propose that the resolution of the contradiction is to realize that the state reconstruction formula does not apply to the special situation involving interior-information-retrieving measurements. This implies that the assumption of smooth replica AdS geometry with boundary condition set by the flat space bath has to break down when the particular measurement operator is applied to the bath. Using replica trick, we introduce an explicitly construction of such operator, which we name as "miracle operators". From this construction we see that the smooth replica geometry assumption breaks down because we have to introduce extra replica wormholes connecting with the "simulated blackholes" introduced by the miracle operator. We study the implication of miracle operators in understanding the firewall paradox.

Figures (7)

• Figure 1: (a) Illustration of the replica manifold with branch-covering at $A$ and possibly an island region $I$. (b) In the analytic continuation to $n\rightarrow 1$ limit, illustration of the quantum extremal surface (\ref{['eq:QES']}) which is the boundary of $I$.
• Figure 2: Illustration of two states $\rho,\sigma$ which are only different by a local unitary $U_P$ acting on a small region $P$ in the island. For example $U_P$ can flip the spin of a particle. The QFT states are defined on a Cauchy surface that includes $A$ and $I$.
• Figure 3: (a) Illustration of the ancilla $W$ which only couples with the rest of the bath $R_1$ through LOCC. (b) A quantum circuit representation of the same setup, with black hole $B$ couples with $R_1$ through quantum gates, while $R_1$ and $W$ are coupled only by LOCC.
• Figure 4: Illustration of the particular setup in subsection (\ref{['subsec:paradox']}) in Penrose diagram (a) and quantum circuit (b). We consider two states different by an interior unitary $U_P$, as was discussed in Fig. \ref{['fig:rhosigma']}. In addition, a unitary $U_M$ (defined by Eq. (\ref{['eq:measurement unitary']}) applied to region $A$ and two ancilla qubits $W,\tilde{W}$ measures $A$ and records the result on $W$. The green and red horizontal dashed lines indicate the time $t_1$ before applying $U_M$, and $t_2$ right after applying $U_M$. What is relevant to our discussion is the application of state reconstruction formula (\ref{['eq:one norm']}) for $A$ at time $t_1$ and $W$ at time $t_2$.
• Figure 5: Illustration of the replica calculation for (a) ${\rm tr}(\rho_A\hat{O})^n$ for a regular operator $\hat{O}$, (b) ${\rm tr}\left(\rho_A^n\right)$ and (c) ${\rm tr}\left(\rho_AZ_A^{(k)}\right)^n$ with $Z_A^{(k)}$ defined in Eq. (\ref{['eq:def Zk']}). The illustration shows the case $n=2,k=1$. The part of Penrose diagram below a Cauchy surface represents the QFT state at that surface, prepared by the QFT path integral on the given background geometry. The red bridge connecting different replicas in (b) and (c) is introduced by multiplying $\rho_A$ with $\rho_A$ or $Z_A^{(k)}$. The green bridge is the replica wormhole. Note that in (c) replica wormhole connects the physical copies $(1,1)$ and $(2,1)$ with simulated copies $(1,2)$ and $(2,2)$ in blue, which represent $Z_A^{(k)}$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2