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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-34/2/a/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

A compression scheme consists of quantum channels

C_{n} : B (H^{\otimes n}) \to B (K_{n}), D_{n} : B (K_{n}) \to B (H^{\otimes n}),

(1)

where each is a CPTP map. The Hilbert space

K_{n}

is the compressed register. Its block rate in qubits per source qubit is

R_{n} = n^{- 1} lo g_{2} dim K_{n}

, with asymptotic rate

R = lim sup_{n} R_{n}

(or the limit when it exists).

Use the squared quantum fidelity convention for the ensemble overlap:

F_{n} = k \sum p_{k}^{(n)} ⟨ Ψ_{k}^{(n)} ∣ D_{n} C_{n} (∣ Ψ_{k}^{(n)} ⟩ ⟨ Ψ_{k}^{(n)} ∣) ∣ Ψ_{k}^{(n)} ⟩ .

(2)

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.

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