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

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.