  Quantum information is scrambled via chaotic time evolution in many-body
systems. The recovery of initial information embedded locally in the system
from the scrambled quantum state is a fundamental concern in many contexts.
From a dynamical perspective, information recovery can measure dynamical
instability in quantum chaos, fault-tolerant quantum computing, and the black
hole information paradox. This article considers general aspects of quantum
information recovery when the scrambling dynamics have conservation laws due to
Lie group symmetries. Here, we establish fundamental limitations on the
information recovery from scrambling dynamics with arbitrary Lie group
symmetries. We show universal relations between information recovery, symmetry,
and quantum coherence, which apply to many physical situations. The relations
predict that the behavior of the Hayden-Preskill black hole model changes
qualitatively under the assumption of the energy conservation law.
Consequently, we can rigorously prove that under the energy conservation law,
the error of the information recovery from a small black hole remains
unignorably large until it completely evaporates. Moreover, even when the black
hole is very large, the recovery of information thrown into the black hole is
not completed until most of the black hole evaporates. The relations also
provide a unified view of the symmetry restrictions on quantum information
processing, such as the approximate Eastin-Knill theorem and the
Wigner-Araki-Yanase theorem for unitary gates.
