Information capacity of a quantum observable

In this paper we consider the classical capacities of quantum-classical channels corresponding to measurement of observables. Special attention is paid to the case of continuous observables. We give formulas for unassisted and entanglement-assisted classical capacities $C$ and $C_\mathrm{ea}$ and consider some explicitly solvable cases, which give simple examples of entanglement-breaking channels with $C<C_\mathrm{ea}$. We also elaborate on the ensemble-observable duality to show that $C_\mathrm{ea}$ for the measurement channel is related to the $\chi$-quantity for the dual ensemble in the same way as $C$ is related to the accessible information. This provides both accessible information and the $\chi$-quantity for quantum ensembles dual to our examples.