[*quantum*]{} case, we are given a quantum source which emits a time sequence of independent identically distributed pure-state quantum signals each of which is chosen from $| u_1 \rangle , | u_2 \rangle, \cdots | u_m
\rangle$ with probabilities $q_1, q_2 \cdots, q_m$, respectively. Notice that $| u_i \rangle $’s are normalized (i.e., unit vectors) but not necessarily orthogonal to each other. Classical coding theory can be regarded as a special case when the signals $| u_i \rangle$ are orthogonal. The goal of quantum source coding is to minimize the number of dimensions of the Hilbert space needed for almost lossless encoding of quantum signals, while maintaining a high fidelity between input and output. For a pure input state $| u_i\rangle $, the fidelity of the output density matrix $\rho_i$ is defined as the probability for it to pass a yes/no test of being the state $|
u_i\rangle $. Mathematically, it is given by $\langle u_i | \rho_i |
u_i\rangle $ [@fidelity]. In particular we will be concerned with the average fidelity $F=\sum_i q_i \langle u_i | \rho_i | u_i\rangle $ It is convenient to measure the dimensionality of a Hilbert space in terms of the number of qubits (i.e., 