  Symmetry, irreversibility, and quantum coherence are foundational concepts in
physics. Here, we present a universal trade-off relation that builds a bridge
between these three concepts. This trade-off particularly reveals that (1)
under a global symmetry, any attempt to induce local dynamics that change the
conserved quantity will cause inevitable irreversibility, and (2) such
irreversibility could be mitigated by quantum coherence. Our fundamental
relation also admits broad applications in physics and quantum information
processing. In the context of thermodynamics, we derive a trade-off relation
between entropy production and quantum coherence in arbitrary isothermal
processes. We also apply our relation to black hole physics and obtain a
universal lower bound on how many bits of classical information thrown into a
black hole become unreadable under the Hayden-Preskill model with the energy
conservation law. This particularly shows that when the black hole is large
enough, under suitable encoding, at least about $m/4$ bits of the thrown $m$
bits will be irrecoverable until 99 percent of the black hole evaporates. As an
application to quantum information processing, we provide a lower bound on the
coherence cost to implement an arbitrary quantum channel. We employ this bound
to obtain a quantitative Wigner-Araki-Yanase theorem that comes with a clear
operational meaning, as well as an error-coherence trade-off for unitary gate
implementation and an error lower bound for approximate error correction with
covariant encoding. Our main relation is based on quantum uncertainty relation,
showcasing intimate connections between fundamental physical principles and
ultimate operational capability.
