On the Page curve under final state projection

TL;DR

The paper investigates how postselection and final-state projection affect quantum correlations in a two-dimensional CFT and their holographic duals, aiming to connect to Page-curve physics. It introduces the pseudo-entropy $S_A^{1|2}$ as a transition-based generalization of entanglement entropy and studies its time evolution under homogeneous, inhomogeneous, and partial postselection scenarios. For holographic CFTs, the real part $\mathrm{Re}[S_A^{1|2}]$ exhibits Page-curve-like behavior, governed by a competition between connected and disconnected geodesic saddles, with the gravity dual incorporating an End-of-the-World brane in AdS$_3$ and geodesic lengths $L_A^{\mathrm{con}}$ and $L_A^{\mathrm{dis}}$. The results suggest that postselection reshapes correlations throughout evaporation-like processes, offering a calculable bridge between final-state proposals and holographic descriptions of information flow.

Abstract

The black hole singularity plays a crucial role in formulating Hawking's information paradox. The global spacetime analysis may be reconciled with unitarity by imposing a final state boundary condition on the spacelike singularity. Motivated by the final state proposal, we explore the effect of final state projection in two dimensional conformal field theories. We calculate the time evolution under postselection by employing the real part of pseudo-entropy to estimate the amount of quantum entanglement averaged over histories between the initial and final states. We find that this quantity possesses a Page curve-like behavior.

Figures (6)

• Figure 1: Final state projection in an evaporating black hole (left) and its field theory simplification (right).
• Figure 2: The gravity dual of the Lorentzian BCFT. The blue surface describes the EOW brane defined in Eq. \ref{['eq:adsthmetq']}. For a subsystem A (red curve) located on the asymptotic boundary, the black and green curves denote the corresponding disconnected geodesics and the connected geodesic, respectively.
• Figure 3: A sketch of Euclidean path-integral on the $w$-sheet for inhomogenous postselection and its conformal transformation to the upper half-plane in terms of the $z$ coordinate. For $x<0$, the path-integral is terminated at $\text{Im}\,w=0$ describing the projection to $\ket{B}_{x<0}$, while for $x>0$ it extends to $\text{Im}\,w=\infty$, corresponding to the projection to the vacuum state $\ket{0}_{x>0}$ on the right half.
• Figure 4: The plot for the real part of pseudo-entropy Re$[S_A]$ for subsystems $A=[0,\infty]$ (left) and $A=[0,5]$ (right) as a function of time $t$. We choose $T=5$, $\delta=\epsilon=0.01$ and $\tilde{S}_{\rm bdy}=0$.
• Figure 5: A sketch of the path-integral description of partial postselection (left) and its conformal map to the upper half-plane (right).