  We prove a theorem for coding mixed-state quantum signals. For a class of
coding schemes, the von Neumann entropy $S$ of the density operator describing
an ensemble of mixed quantum signal states is shown to be equal to the number
of spin-$1/2$ systems necessary to represent the signal faithfully. This
generalizes previous works on coding pure quantum signal states and is
analogous to the Shannon's noiseless coding theorem of classical information
theory. We also discuss an example of a more general class of coding schemes
which {\em beat} the limit set by our theorem.
