Upper bounds on the Holevo quantity arising from the fundamental entropic inequality

We show how the fundamental entropic inequality proved recently in [arXiv:2408.15306] can be applied to obtain a useful relation for the Holevo quantity of discrete and continuous ensembles of quantum states. This relation gives a tight upper bound on the Holevo quantity of a given ensemble $\mu$ expressed in terms of the Holevo quantities of two auxiliary ensembles ${\mu }_{+}$ and ${\mu }_{-}$ produced by $\mu$. Among others, this implies quite accurate upper bounds on the Holevo quantity of a discrete ensemble of quantum states expressed via the probabilities and the metric characteristics of an ensemble.