A theorem is proven for quantum information theory that is analogous to the noiseless coding theorem of classical information theory. In the quantum result, the von Neumann entropy S of the density operator describing an ensemble of pure quantum signal states is equal to the number of spin-1/2 systems (quantum bits or "qubits") necessary to represent the signal faithfully. The theorem holds whether or not the signal states are orthogonal. Related results are also presented about the fidelity of quantum coding and about representing entangled quantum states. PACS numbers: 03.65, 05.30 I. Entropy and Information In the classical information theory developed by Shannon and others, the central problem is coding [1]. For example, suppose that A is a message source that produces the message a with probability p(a), and further suppose that we wish to represent the messages with sequences of binary digits (bits) that are as short as possible. It can be shown that the mean length ¯ L of th...