# Black holes as clouded mirrors: the Hayden-Preskill protocol with symmetry

Yoshifumi Nakata1,2,3, Eyuri Wakakuwa4, and Masato Koashi2

1Yukawa Institute for Theoretical Physics, Kyoto university, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto, 606-8502, Japan
2Photon Science Center, Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
3JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan
4Department of Communication Engineering and Informatics, Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan

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### Abstract

The Hayden-Preskill protocol is a qubit-toy model of the black hole information paradox. Based on the assumption of scrambling, it was revealed that quantum information is instantly leaked out from the quantum many-body system that models a black hole. In this paper, we extend the protocol to the case where the system has symmetry and investigate how the symmetry affects the leakage of information. We especially focus on the conservation of the number of up-spins. Developing a partial decoupling approach, we first show that the symmetry induces a delay of leakage and an information remnant. We then clarify the physics behind them: the delay is characterized by thermodynamic properties of the system associated with the symmetry, and the information remnant is closely related to the symmetry-breaking of the initial state. These relations bridge the information leakage problem to macroscopic physics of quantum many-body systems and allow us to investigate the information leakage only in terms of physical properties of the system.

The black hole information paradox is a long-standing problem in fundamental physics, highlighting a gap between general relativity and quantum mechanics. A central question is how information in a black hole leaks out as the black hole evaporates by the Hawking radiation. Based on a qubit-toy-model of a quantum black hole, it was shown that, if the black hole is fully scrambling, information leaks out in a surprisingly quick manner. This is known as the Hayden-Preskill recovery.

In this paper, we further develop the information-theoretic approach to the information paradox by taking another important feature of physical systems, i.e., symmetry, into account. We show that the presence of symmetry leads to two significant deviations from the original Hayden-Preskill recovery: one is the delay of information leakage, and the other is the information remnant. We further discover novel microscopic-macroscopic correspondences that directly connect quantum information and symmetry of quantum black holes.

The micro-macro correspondences we discovered allow one to easily deduce how information leaks out from the black hole with symmetry in terms of physical quantities without referring to too much details of information-theoretic assumptions. This will be a stepping stone toward the full understanding of the information leakage in a realistic situation, such as in the situation with energy conservation.

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### Cited by

[1] Hiroyasu Tajima and Keiji Saito, "Universal limitation of quantum information recovery: symmetry versus coherence", arXiv:2103.01876, (2021).

[2] Hiroyasu Tajima, Ryuji Takagi, and Yui Kuramochi, "Universal trade-off structure between symmetry, irreversibility, and quantum coherence in quantum processes", arXiv:2206.11086, (2022).

[3] Linghang Kong and Zi-Wen Liu, "Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries", PRX Quantum 3 2, 020314 (2022).

[4] Yoshifumi Nakata, Da Zhao, Takayuki Okuda, Eiichi Bannai, Yasunari Suzuki, Shiro Tamiya, Kentaro Heya, Zhiguang Yan, Kun Zuo, Shuhei Tamate, Yutaka Tabuchi, and Yasunobu Nakamura, "Quantum Circuits for Exact Unitary t -Designs and Applications to Higher-Order Randomized Benchmarking", PRX Quantum 2 3, 030339 (2021).

[5] Kanato Goto, Masahiro Nozaki, Shinsei Ryu, Kotaro Tamaoka, and Mao Tian Tan, "Scrambling and Recovery of Quantum Information in Inhomogeneous Quenches in Two-dimensional Conformal Field Theories", arXiv:2302.08009, (2023).

[6] Zi-Wen Liu and Sisi Zhou, "Approximate symmetries and quantum error correction", arXiv:2111.06355, (2021).

[7] Pak Hang Chris Lau, Toshifumi Noumi, Yuhei Takii, and Kotaro Tamaoka, "Page curve and symmetries", Journal of High Energy Physics 2022 10, 15 (2022).

[8] Ryota Katsube, Masanao Ozawa, and Masahiro Hotta, "Limitations of Quantum Measurements and Operations of Scattering Type under the Energy Conservation Law", arXiv:2211.13433, (2022).

[9] Yoshifumi Nakata, Takaya Matsuura, and Masato Koashi, "Constructing quantum decoders based on complementarity principle", arXiv:2210.06661, (2022).

[10] Beni Yoshida, "Recovery algorithms for Clifford Hayden-Preskill problem", arXiv:2106.15628, (2021).

[11] Eyuri Wakakuwa and Yoshifumi Nakata, "One-Shot Randomized and Nonrandomized Partial Decoupling", Communications in Mathematical Physics 386 2, 589 (2021).

[12] Masahiro Fujii, Ryosuke Kutsuzawa, Yasunari Suzuki, Yoshifumi Nakata, and Masaki Owari, "Characterizing quantum pseudorandomness by machine learning", arXiv:2205.14667, (2022).

[13] Yoshifumi Nakata and Masaki Tezuka, "Hayden-Preskill Recovery in Hamiltonian Systems", arXiv:2303.02010, (2023).

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