Abstract
We study reflected entropy as a mixed state 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, and even black hole and black hole. 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 section. 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.
Highlights
We propose a quantum extremal surface for reflected entropy, which we call quantum extremal cross section
We study reflected entropy as a mixed state correlation measure in black hole evaporation
For 3-side wormhole model, we find that reflected entropy is dual to island cross section
Summary
We follow [44] to model the black hole evaporation using a multi-boundary wormhole (figure 1). We can view the CFT states on the boundary R1 and R2 (figure 1) as two parts of the Hawking radiation, which are entangled with the CFT states of the black hole on the boundary B. While [44] increases the number of legs of the radiation to simulate the black hole evaporation, here we choose to keep the number of the legs fixed but increase the size of the horizons corresponding to the radiation states. Let the length of the horizon M3 of the original black hole be L0 and m1 = m2 = 0 as the initial condition. At any moment during evaporation, the horizon length of the black hole B is determined by m3 = L20 − 2m21. As shown in figure 1, the entanglement wedge of radiation covers the shared interior after the transition between different RT surfaces, the shared interior is considered as the island in this model
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