Operator Spreading and Information Propagation: Equivalence and Beyond

We investigate the quantitative relationship between operator spreading and classical information propagation in quantum systems. Focusing on a bi-partite quantum channel, we derive new upper and lower bounds on the Holevo capacity, a typical information measure, in terms of the trace norm distance between output states, sharpening earlier results by Bravyi $\mathit{\text{et}}\mathit{\text{al}}$. Our results clarify the extent to which operator growth governs information flow.