Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 323 / 2 / ii

Memorylessness means that the ensemble's average density operator is ${\rho }^{(n)}={\pi }^{\otimes n}$. Its spectral decomposition isThe product eigenvalues giveZero eigenvalues contribute zero to the Von Neumann entropy and are excluded from logarithmic typicality tests.

The quantum typical subspace is the span of eigenvectors whose eigenvalues satisfyIts orthogonal projection $P_{\epsilon }^{(n)}$ selects precisely the classical typical set for the eigenvalue distribution $q$. The typical-set cardinality bounds and the weak law of large numbers therefore givefor any fixed $\delta >0$ and all sufficiently large $n$.

A compression-decompression scheme consists of quantum channels ${\mathcal{E}}_n:\mathcal{D}({\mathcal{H}}^{\otimes n})\to \mathcal{D}({\mathcal{K}}_n)$ and ${\mathcal{D}}_n:\mathcal{D}({\mathcal{K}}_n)\to \mathcal{D}({\mathcal{H}}^{\otimes n})$, where ${\mathrm{limsup}}_n{n}^{-1}{\log }_2\dim {\mathcal{K}}_n\le R$. For the emitted pure-state ensemble, reliability means that the mean squared quantum fidelity tends to one:The average is taken over the actual source distribution, rather than the worst possible input vector.

A standard stronger formulation of reliable quantum source compression requireswhere entanglement fidelity tests preservation of a purification and its reference-system correlations. It implies the mean-fidelity condition for every pure-state ensemble of ${\pi }^{\otimes n}$. Schumacher compression achieves this at any rate $R>S(\pi )$: choose $0<\epsilon <R-S(\pi )$, encode the quantum typical subspace, and map atypical outcomes to a fallback state. The composed channel has a Kraus term $P_{\epsilon }^{(n)}$, so its entanglement fidelity is at least $[\mathrm{Tr}({\pi }^{\otimes n}{P}_{\epsilon }^{(n)}){]}^2\to 1$.