Sphere Packing and Zero-Rate Bounds to the Reliability of Classical-Quantum Channels

In this paper, the sphere packing bound of Fano, Shannon, Gallager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The upper bound for the reliability function obtained from the sphere packing coincides at high rates, for the case of pure state channels, with a lower bound derived by Burnashev and Holevo [21]. Thus, for pure state channels, the reliability function at high rates is now exactly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. Finally, the obtained zero-rate upper bound to the reliability function of a general classical-quantum channel with no zero-error capacity coincides with a lower bound obtained by Holevo, thus determining the exact expression.