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

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 quantum thermodynamics, we derive a universal trade-off between the coherence cost of an arbitrary quantum channel and the entropy production (thermodynamic irreversibility) of the channel. We also apply our relation to the Hayden-Preskill black hole model, we rigorously show that the rate of the escape of classical/quantum information from a black hole varies significantly with and without energy conservation. 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, our trade-off also unifies and extends various restrictions on measurements, quantum computation gates, and quantum error corrections imposed by symmetry, including the Wigner-Araki-Yanase theorems and the Eastin-Knill theorems.