The Hayden-Preskill protocol is a quantum information theoretic model of the
black hole information paradox. Based on the protocol, it was revealed that
information scrambling and entanglement lead to an instant leakage of
information. In this paper, we study the information paradox with symmetry in
the framework of the Hayden-Preskill protocol. Symmetry is an important feature
of black holes that induces yet more conceptual puzzles in the regime of
quantum gravity. We especially consider an axial symmetry and clarify its
consequences in the information leakage problem. Using a partial decoupling
approach, we first show that symmetry induces a \emph{delay} of information
leakage and an \emph{information remnant}, both of which can be macroscopically
large for certain initial conditions. We then clarify the physics behind the
delay and the information remnant. By introducing the concept of \emph{clipping
of entanglement}, we show that the delay is characterized by thermodynamic
properties of the black hole associated with the symmetry. We also show that
the information remnant is closely related to the symmetry-breaking of the
black hole. These relations indicate the existence of non-trivial
microscopic-macroscopic correspondences in the information leakage problem.Comment: Ver.1: 7 pages + Supplementary. Ver.2: short version (14 pages) +
long version (47 pages). Presentation improved. Connection to thermodynamics
is much more elaborated. Ver.3: Substantial update of presentation and
others, including the title. 21 pages + 6 pages of appendice