  In this article we show that the Holevo capacity of a classical quantum
channel, can be reached not only by a POVM but by von Neumann measurement too.
Furthermore we show two use of this fact 1) We can generalize the classical
compound channel to quantum environment and with this new measure technique we
can give an optimal coding/decoding algorithm to it. 2) We give an algorithm,
with it a classical system can be decoded by quantum apparatus in linear time,
which is a surprising result, because best known optimal classical decoding
algorithm need n*log(n) time. This means that if a quantum machine can perform
arbitrarily von Neumann measurement with only two possible outcome, then it can
solve non-polynomial classical problems in linear time.
