Reflected Entropy for an Evaporating Black Hole

TL;DR

This work extends the study of information flow in black hole evaporation by focusing on reflected entropy, a correlation measure for bipartite mixed states. It introduces a quantum extremal cross section (QECS) formula and demonstrates its holographic realization across three models: a simple 3-side wormhole, a 3d EOW brane setup, and a 2d JT gravity + CFT system. In all cases, the reflected entropy between radiation and black hole shows Page-like behavior with a later transition, while radiation–radiation correlations grow early and saturate, and BH–BH correlations decay to zero, highlighting island and cross-section structures as the dual geometric objects. The results establish a unified, geometrical picture linking reflected entropy to island cross sections and QECS, with implications for understanding correlations during black hole evaporation and for generalizations to other gravitational settings.

Abstract

We study reflected entropy as a correlation measure in black hole evaporation. As a measure for bipartite mixed states, reflected entropy can be computed between black hole and radiation, radiation and radiation. We compute reflected entropy curves in three different models: 3-side wormhole model, End-of-the-World (EOW) brane model in three dimensions and two-dimensional eternal black hole plus CFT model. For 3-side wormhole model, we find that reflected entropy is dual to island cross sections. The reflected entropy between radiation and black hole increases at early time and then decreases to zero, similar to Page curve, but with a later transition time. The reflected entropy between radiation and radiation first increases and then saturates. For the EOW brane model, similar behaviors of reflected entropy are found. We propose a quantum extremal surface for reflected entropy, which we call quantum extremal cross section. In the eternal black hole plus CFT model, we find a generalized formula for reflected entropy with island cross section as its area term by considering the right half as the canonical purification of the left. Interestingly, the reflected entropy curve between the left black hole and the left radiation is nothing but the Page curve. We also find that reflected entropy between the left black hole and the right black hole decreases and goes to zero at late time. The reflected entropy between radiation and radiation increases at early time and saturates at late time.

Figures (29)

• Figure 2.1: The 3-side wormhole has three asymptotic boundaries $R_1$, $R_2$ and $B$. The entanglement wedge of $B$ is the blue shaded region in the figure. We have $m_1+m_2<m_3$ at early time of the evaporation and $m_1+m_2>m_3$ at late time.
• Figure 2.2: The purified geometry is a 4-side wormhole which is symmetric with respect to the horizon $M_3$. The brown geodesic $L_1 \cup L_1'$ intersects $M_3$ at two points $s_1$ and $s_2$. The minimal cross section can be obtained by minimizing the length of $L_1 \cup L_1'$ or equivalently, $L_1$, over $s_1$ and $s_2$.
• Figure 2.3: The covering space construction of a 2-side wormhole. The geodesics $g_1$ and $g_2$ are identified. The blue dashed circle $M$ is the causal horizon and $R_1$ and $R_2$ are two asymptotic boundaries.
• Figure 2.4: A fundamental region of the quotient space of the 4-side wormhole in Fig.\ref{['4side']}. It is symmetric with respect to the horizon $M_3$. The geodesics $g_a$ and $g_b$ are identified in an orientation-reversed way.
• Figure 2.5: The brown geodesic in Fig.\ref{['4side']} with two intersection points $s_1, s_2$ on the horizon $M_3$ is depicted in this covering space. This covering space has a $Z_2$ symmetry with respect to the vertical dashed blue line $M_3$, so we only have to consider the right half. $L_1$ consists of two arcs. The larger one is part of a large semicircle which is the image of $\gamma_2$ (\ref{['gamma2']}) applied to the smaller semicircle that includes the smaller arc of $L_1$. The two brown arcs are joint smoothly under the identification of the green semicircles. This type of geodesic is unique once we fix the intersection points $s_1$ and $s_2$, so we can move $s_1,s_2$ to obtain the minimal one.