Schumacher compression

Schumacher compression is quantum source coding into the typical subspace of ${\rho }^{\otimes n}$. Asymptotically reliable compression is possible at every qubit rate above the Von Neumann entropy $S(\rho )$.

A pure-state source emits independent states from a fixed quantum state ensemble. Its average block density operator is the tensor power of the one-use average. Unassisted reliable quantum source compression preserves the emitted quantum states, rather than only their classical labels. Schumacher compression gives the optimum asymptotic qubit rate $S(\rho )$ for such a source, with every larger rate achievable. Nonorthogonal equally likely pure states need not have entropy one: their classical labels have entropy one but their average density operator can have lower Von Neumann entropy.

For a density operator $\pi ={\sum }_i{q}_i|{\varphi }_i⟩⟨{\varphi }_i|$, the quantum typical subspace is spanned by product eigenvectors of ${\pi }^{\otimes n}$ whose eigenvalues $q_{i_1}\cdots q_{i_n}$ satisfyIt applies the classical weakly typical sequence definition to the spectrum. If $P_{\epsilon }^{(n)}$ is its orthogonal projection, thenThe first statement follows from the typical-set cardinality bounds, and the second from the weak law of large numbers. This subspace holds nearly all the source probability while using exponentially fewer dimensions than the whole space when $S(\pi )<{\log }_{2}d$.

For each fixed $\delta >0$, the quantum typical subspace projector selects eigenvalues of ${\pi }^{\otimes n}$ between $2^{-n(S(\pi )+\delta )}$ and $2^{-n(S(\pi )-\delta )}$. Its probability tends to one by the weak law of large numbers for the spectral information variable. If its probability is at least $1-ϵ$, its dimension is between $(1-ϵ)2^{n(S(\pi )-\delta )}$ and $2^{n(S(\pi )+\delta )}$. These statements give the size and concentration estimates behind Schumacher compression.

A quantum source-compression scheme has encoding and decoding quantum channels with compressed space ${\mathcal{K}}_n$ and rate ${\mathrm{limsup}}_n{n}^{-1}{\log }_2\dim {\mathcal{K}}_n$. For a pure-state source ensemble, average reliability means that the mean squared quantum fidelity after decoding tends to one. A stronger coherent criterion is $F_e({\pi }^{\otimes n},{\mathcal{D}}_n\circ {\mathcal{E}}_n)\to 1$, where entanglement fidelity also tests a purification reference. Encoding the quantum typical subspace achieves both criteria above $S(\pi )$. Schumacher's original quantum coding paper establishes quantum source coding.

Measure the quantum typical subspace projector $\Pi$. Encode its successful sector isometrically and reserve one orthogonal compressed flag for failure; decode the flag to a fixed state. The composite channel is ${\mathcal{N}}_n(\tau )=\Pi \tau \Pi +\mathrm{Tr}[(I-\Pi )\tau ]|{\phi }_0⟩⟨{\phi }_0|$. It is trace preserving. The Kraus formula for entanglement fidelity gives the displayed lower bound, while convexity of the square gives the same bound for average pure-source fidelity. One flag adds only a vanishing asymptotic rate overhead.

For a quantum state ensemble of pure vectors and an orthogonal projection $P$, measure $\{P,I-P\}$, preserve the successful subspace, and decode failure to a fixed normalized vector $\phi$. The resulting quantum channel is $\Lambda (X)=PXP+\mathrm{Tr}[(I-P)X]|\phi ⟩⟨\phi |$. For ${\alpha }_k=\Vert P{\psi }_k\Vert$, its squared quantum fidelity on the $k$th source vector is ${\alpha }_k^4+(1-{\alpha }_k^2)|⟨{\psi }_k|\phi ⟩{|}^2$. The inequality $a^2\ge 2a-1$ on $[0,1]$ proves the displayed average bound. It depends only on the source density operator's weight in $P$ and does not require the emitted vectors to be orthogonal.

If encoding and decoding quantum channels factor through a $k$-dimensional space, their composite Kraus operators all have matrix rank at most $k$. The rank bound for a weighted operator trace and the Kraus formula for entanglement fidelity implyThus high operation fidelity requires the leading $k$ eigenvalues of the source density operator to carry almost all its probability. The bound applies to arbitrary completely positive trace-preserving encoders and decoders.