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Symmetry, irreversibility, and quantum coherence are foundational concepts inphysics. Here, we present a universal trade-off relation that builds a bridgebetween these three concepts. This trade-off particularly reveals that (1)under a global symmetry, any attempt to induce local dynamics that change theconserved quantity will cause inevitable irreversibility, and (2) suchirreversibility could be mitigated by quantum coherence. Our fundamentalrelation also admits broad applications in physics and quantum informationprocessing. In the context of thermodynamics, we derive a trade-off relationbetween entropy production and quantum coherence in arbitrary isothermalprocesses. We also apply our relation to black hole physics and obtain auniversal lower bound on how many bits of classical information thrown into ablack hole become unreadable under the Hayden-Preskill model with the energyconservation law. This particularly shows that when the black hole is largeenough, 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 anapplication to quantum information processing, we provide a lower bound on thecoherence cost to implement an arbitrary quantum channel. We employ this boundto obtain a quantitative Wigner-Araki-Yanase theorem that comes with a clearoperational meaning, as well as an error-coherence trade-off for unitary gateimplementation and an error lower bound for approximate error correction withcovariant encoding. Our main relation is based on quantum uncertainty relation,showcasing intimate connections between fundamental physical principles andultimate operational capability.

Universal trade-off structure between symmetry, irreversibility, and quantum coherence in quantum processes