Sphere packing bound for quantum channels

In this paper, the Sphere-Packing-Bound of Fano, Shannon, Gallager and Berlekamp is extended to general classical-quantum channels. The obtained upper bound for the reliability function, for the case of pure-state channels, coincides at high rates with a lower bound derived by Burnashev and Holevo [1]. 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.