Reliable quantum source 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.