quantum bits) composing it; that is, the base-2 logarithm of the dimension.

Though there has been some preliminary work on quantum Huffman coding [@schpri], the most well-known quantum source coding scheme is a block coding scheme [@schumacher; @jozsa]. The converse of this coding theorem was proven rigorously in [@fuchs]. In block coding, if the signals are drawn from an ensemble with density matrix $\rho =\sum q_j |u_j\rangle \langle u_j |$, Schumacher coding, which is almost lossless, compresses $N$ signals into $N S(\rho)$ qubits, where $S(\rho) = -
{\rm tr}\ \rho \log \rho$ is the von Neumann entropy. To encode $N$ signals [*sequentially*]{}, it requires $O(N^3)$ computational steps [@cleve]. The encoding and decoding processes are far from instantaneous. Moreover, the lengths of all the codewords are the same.

Difficulties In A Quantum Generalization
========================================

A notable feature of quantum information is that measurement of it generally leads to disturbance. While measurement is a passive procedure in classical information theory, it is an integral part of the formalism of quantum mechanics and is an active process. Therefore, a big challenge in quantum coding is: How to encode and decode without disturbing the signals too much by the measurements involved? To illustrate the 