Coding theorems of quantum information theory

Summary form only given. Shannon´s original proof of the channel coding theorem via typical input/output sequences is presented in a sphere-packing form to determine the minimum quantum source coding rate. Let S(ρ)≡-trρlogρ be the Von Neumann entropy of a density operator p on a finite-dimensional space H, i→ρ_i the state modulation map on an alphabet I, ρ¯≡Σ_ip_iρ_i the average state with respect to a prior distribution p_i on I, and S¯(ρ_i)≡Σ_ip_iS(ρ _i). The minimum quantum state dimension per symbol needed to represent {ρ_i} under {p_i} with arbitrarily small error is shown to be S(ρ¯)-S¯(ρ_i) for a whole class of error measures. This generalizes to arbitrary mixed states the pure state result. Further generalizations of this result to arbitrary alphabet I, infinite dimensional H, as well as rate-distortion coding are presented. Channel coding for restoring quantum states is discussed with the conclusion that for typical noisy channels nonzero channel capacity cannot be obtained. Relations of these results, in particular the quantity S(ρ¯)-S¯(ρ_i), to classical (nonquantum) information transfer are elaborated