[*fixed*]{} state $| 0 \cdots 0 \rangle$.

$\forall \epsilon ,\delta > 0, \exists N_0 > 0$ such that $\forall
N > N_0$, $F \ge 1 -
\epsilon$ where $F$ is the fidelity between the true state $\rho$ of the $N$ quantum signals and the projection of $\rho$ on the typical subspace $S_\delta$ in our quantum Huffman coding scheme.

Proof: The proof is identical to the case of Schumacher’s noiseless quantum coding theorem [@schumacher; @jozsa; @fuchs].

Therefore, the truncation and subsequent replacement of the discarded portion by $| 0 \cdots 0 \rangle$ still lead to a high fidelity in the decoding.

In conclusion, we have constructed an explicit parallel encoding and decoding scheme for the storage of $N$ independent and identically distributed quantum signals that asymptotically has only $O ( (\log N)^a)$ depth and uses $N ( \bar{L} + \delta + \lceil \log l_{\rm max} \rceil )$ qubits for storage where $ \bar{L}$ is the average length of the Huffman coding for the classical coding problem for the set of probabilities given by the eigenvalues of the density matrix of each signal. Here $ \delta $ can be any positive number and $l_{\rm max}$ is the length of the longest Huffman 