Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel

Certain bounds are derived for the quantity of information transmitted by a quantum channel. It is proved that if at least two of the set of density operators $\rho_0,\dots,\rho_n$ do not commute, then $J(\pi)<\mathscr H(\sum_\alpha\pi_\alpha\rho_\alpha)-\sum_\alpha\pi_\alpha\mathscr H(\rho_\alpha)$, where $J(\pi)$ is the upper bound of the quantity of information with respect to all generalized measurements at the channel output for a fixed distribution $\pi=(\pi_0,\dots,\pi_n)$ at the input. A sharper upper bound for $J(\pi)$ is explicitly stated.