Finite IID quantum source and compression code

Let $H_A$ be a nonzero finite-dimensional complex inner-product space and let $\rho$ be a density operator (Density operator) on $H_A$. The finite IID quantum source with state $\rho$ emits the block state ${\rho }^{\otimes n}$ on $H_A^{\otimes n}$ at blocklength $n\ge 1$.

An $(n,M)$ quantum compression code for $\rho$ consists of a finite-dimensional code space $H_C$ with $\dim H_C=M\ge 1$, a quantum channel (Finite-dimensional quantum channel) $\mathcal{E}:\mathcal{L}(H_A^{\otimes n})\to \mathcal{L}(H_C)$, and a quantum channel $\mathcal{D}:\mathcal{L}(H_C)\to \mathcal{L}(H_A^{\otimes n})$. Its squared entanglement fidelity is

$F_e^{(n)}:=⟨\varphi \mathrm{\mid }({\mathrm{id}}_R\otimes (\mathcal{D}\circ \mathcal{E}))(\mathrm{\mid }\varphi ⟩⟨\varphi \mathrm{\mid })\mathrm{\mid }\varphi ⟩,$

where $H_R$ is any finite-dimensional reference space supporting the chosen unit-vector purification $\mathrm{\mid }\varphi ⟩$ of ${\rho }^{\otimes n}$ on $H_R\otimes H_A^{\otimes n}$. The value is independent of the chosen finite purification by the isometric equivalence of Purifications exist and are unitarily equivalent after padding. A sequence of codes is reliable when $F_e^{(n)}\to 1$. The code's rate is $n^{-1}\ln M$ nats per source letter, equivalently $n^{-1}{\log }_{2}M$ qubits per source letter. The entropy threshold in these units is $S(\rho )$ nats or $S_2(\rho ):=S(\rho )\mathrm{/}\ln 2$ qubits, where $S$ is the entropy of Finite-dimensional von Neumann entropy.