Reflected Entropy for an Evaporating Black Hole

TL;DR

This paper investigates reflected entropy as a measure of correlations in black hole evaporation, revealing its behavior in various models and proposing a quantum extremal cross section concept, with results resembling the Page curve.

Contribution

It introduces a quantum extremal cross section for reflected entropy and computes its behavior across different black hole models, connecting it to the island formula.

Findings

Reflected entropy between radiation and black hole peaks then decreases, similar to the Page curve.

Reflected entropy between radiation and radiation increases and saturates.

Reflected entropy between left and right black holes decreases to zero at late times.

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.