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(/spl rho/)/spl equiv/-tr/spl rho/log/spl rho/ be the Von Neumann entropy of a density operator p on a finite-dimensional space H, i/spl rarr//spl rho//sub i/ the state modulation map on an alphabet I, /spl rho/~/spl equiv//spl Sigma//sub i/p/sub i//spl rho//sub i/ the average state with respect to a prior distribution p/sub i/ on I, and S~(/spl rho//sub i/)/spl equiv//spl Sigma//sub i/p/sub i/S(/spl rho//sub i/). The minimum quantum state dimension per symbol needed to represent {/spl rho//sub i/} under {p/sub i/} with arbitrarily small error is shown to be S(/spl rho/~)-S~(/spl rho//sub 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(/spl rho/~)-S~(/spl rho//sub i/), to classical (nonquantum) information transfer are elaborated.