Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 34 / 2

A compression scheme consists of quantum channelswhere each is a CPTP map. The Hilbert space ${\mathcal{K}}_n$ is the compressed register. Its block rate in qubits per source qubit is $R_n=n^{-1}{\log }_2\dim {\mathcal{K}}_n$, with asymptotic rate $R={\mathrm{limsup}}_n{R}_n$ (or the limit when it exists).

Use the squared quantum fidelity convention for the ensemble overlap:This definition tests preservation of the emitted vectors themselves, including nonorthogonal ones; preserving only their classical labels would be a different task. The scheme is reliable when $F_n\to 1$ as $n\to \infty$. This is the ensemble-average criterion for reliable quantum source compression. A square root is not taken in this definition, even though an unsquared convention is also used for quantum fidelity.

Write the spectral decomposition of the one-qubit density operator as $\pi ={\sum }_{j=0}^1{\lambda }_j|e_j⟩⟨e_j|$. For a word $\mathbf{j}=(j_1,\dots ,j_n)$, the tensor product vectoris an eigenvector of ${\pi }^{\otimes n}$, with eigenvalueThese product vectors form an orthonormal eigenbasis, allowing arbitrary orthonormal choices within degeneracies. Using ${\sum }_j{\lambda }_j=1$ and $0\log 0=0$,Thus the Von Neumann entropy isThis calculation uses the spectral ensemble of the memoryless quantum information source; the originally emitted vectors need not equal these eigenvectors.

For fixed $\epsilon >0$, let the quantum typical subspace ${\mathcal{T}}_{\epsilon }^{(n)}$ be spanned by the product eigenvectors with nonzero eigenvalues satisfyingEquivalently their spectral information $-n^{-1}{\log }_2{\lambda }_{\mathbf{j}}$ lies within $\epsilon$ of $S(\pi )$. Denote its orthogonal projection by $P_{\epsilon }^{(n)}$.

The typical subspace theorem states that for every $\delta >0$ there is $n_0$ such that for $n\ge n_0$,Also the operator bounds on its range areTo see the concentration statement, sample $j_s$ independently with probabilities ${\lambda }_j$. The mean of $-{\log }_2{\lambda }_{j_s}$ is $S(\pi )$, so the weak law of large numbers gives typical probability tending to one. Zero-eigenvalue letters have probability zero and are excluded. The dimension bounds follow by summing the upper and lower typical eigenvalue bounds and using the typical mass between $1-\delta$ and $1$.

For a block with nonzero typical dimension, write $P=P_{\epsilon }^{(n)}$ and $Q=I-P$. Let the compressed register be a copy of the typical space plus a one-dimensional orthogonal failure flag. Choose an isometry $V:P{\mathcal{H}}^{\otimes n}\to {\mathcal{K}}_n$ onto the successful code sector, extended by zero on $Q$, and a normalized flag $|f⟩$ orthogonal to its range. Choose any normalized vector $|\phi ⟩$ in the source Hilbert space. DefineThese are CPTP maps: the first performs the projective test and either encodes or prepares a flag; the second embeds the successful sector and prepares a fixed state on failure. Their trace identities follow from $V^{†}V=P$ and $V{V}^{†}+|f⟩⟨f|=I_{{\mathcal{K}}_n}$. Thus typical-subspace compression with a failure flag implementsFor a source vector $|{\Psi }_k⟩$, let $a_k=⟨{\Psi }_k|P|{\Psi }_k⟩={\alpha }_k^2$. Its overlap with the decoded state iswhere the final step is $(1-a_k{)}^2\ge 0$. Averaging proves average pure-source fidelity after typical projection:No orthogonality of the emitted vectors was used. The code dimension is $\dim {\mathcal{T}}_{\epsilon }^{(n)}+1$; the extra flag has vanishing asymptotic rate cost. Exceptional small block lengths with empty typical space can use any channel, since they do not affect reliability.

Choose $0<\epsilon <R-S(\pi )$. The preceding construction and the typical subspace theorem give, for all sufficiently large $n$,Embed this code into $m_n=\lceil nR\rceil$ qubits. The unused subspace can be decoded to a fixed source state, making the extended decoder trace preserving; encoded states never enter it. Hence its rate is $m_n/n\to R$.

For every $\delta >0$, sufficiently large $n$ has $\mathrm{Tr}({\pi }^{\otimes n}{P}_{\epsilon }^{(n)})\ge 1-\delta$. The proved fidelity bound yieldsSince $\delta$ is arbitrary, $F_n\to 1$. ThereforeThis establishes the achievability part of Schumacher compression, including the failure flag and without presuming the source ensemble is orthogonal. If $R$ exceeds one, the same rate-budget construction simply uses spare code qubits; no claim of reducing the physical register size is then needed.