difficulties involved, we shall first attempt a naive generalization of Huffman coding to the quantum case. Consider the density matrix for each signal $\rho =\sum q_j
| u_j \rangle \langle u_j |$ and diagonalize it into $$\rho=\sum_i p_i | \phi_i \rangle \langle \phi_i | \;,$$ where $| \phi_i \rangle$ is an eigenstate and the eigenvalues $ p_i$’s are arranged in decreasing order. Huffman coding of a corresponding classical source with the same probability distribution $p_i$’s allows one to construct a one-to-one correspondence between Huffman codewords $h_i $ and the eigenstates $ | \phi_i \rangle $. Any input quantum state $| u_j\rangle$ may now be written as a sum over the complete set $|\phi_i\rangle$. Remarkably, this means that, for such a naive generalization of Huffman coding, the length of each signal is a quantum mechanical variable with its value in a superposition of the length eigenstates. It is not clear what this really means nor how to deal with such an object. If one performs a measurement on the length variable, the statement that measurements lead to disturbance means that irreversible changes to the $N$ signals will be introduced which disastrously reduce the fidelity.

Therefore, to encode the signals faithfully, the 