``Bekenstein-hawking entropy and strange metals''. ``Remarks on the sachdev-ye-kitaev model''. 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. 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. We further discover novel microscopic-macroscopic correspondences that directly connect quantum information and symmetry of quantum black holes. 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. 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. This is known as the Hayden-Preskill recovery. 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. A central question is how information in a black hole leaks out as the black hole evaporates by the Hawking radiation. The black hole information paradox is a long-standing problem in fundamental physics, highlighting a gap between general relativity and quantum mechanics. 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. 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. Developing a partial decoupling approach, we first show that the symmetry induces a delay of leakage and an information remnant. We especially focus on the conservation of the number of up-spins. 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. 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. The Hayden-Preskill protocol is a qubit-toy model of the black hole information paradox.
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