Reflected entropy for an evaporating black hole
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Springer
Received: July 6, Revised: September 21, Accepted: October 26, Published: November 27,
2020 2020 2020 2020
Reflected entropy for an evaporating black hole
Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China
E-mail: [email protected], [email protected], [email protected] 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. Keywords: AdS-CFT Correspondence, Black Holes, Holography and condensed matter physics (AdS/CMT), 2D Gravity ArXiv ePrint: 2006.10846
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)155
JHEP11(2020)155
Tianyi Li, Jinwei Chu and Yang Zhou
Contents 1 Introduction
1
2 Reflected entropy in simple wormhole model 2.1 Reflected entropy between R1 and R2 2.2 Reflected entropy between R2 and B
4 5 6 8 8 11 12
4 Quantum extremal cross section
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5 Reflected entropy in 2d eternal black hole + CFT model 5.1 Review of the model 5.2 A formula of reflected entropy 5.3 Radiation and black hole 5.4 Black hole and black hole 5.5 Radiation and radiation
16 16 18 19 19 21
6 Conclusion and discussion
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A The quotient space construction of wormholes
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B The length formulas of L1 and L2
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C The length formula of LI
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D The length formula of Lp
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E Bounds on reflected entropy
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1
Introduction
The black hole evaporation process is expected to be unitary, therefore the Von Neumann entropy of the outgoing radiation should follow Page curve [1–3], in which the entropy first incr
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