Universal limitation of quantum information recovery: symmetry versus coherence

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. Here, we establish fundamental limitations on the information recovery that hold whenever the scrambling dynamics satisfy an arbitrary Lie group symmetry. We show universal relations between information recovery, symmetry, and quantum coherence, which apply to many physical situations. The relations predict that the behaviour 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, quantum black holes do not return an unignorable part of information until the black holes completely evaporate. Furthermore, the relations 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.