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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-323/1/v/solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 323 / 1 / v / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

Apply the measure-and-prepare channel locally to an arbitrary bipartite density operator

ρ_{A R}

. The resulting state is

(Λ \otimes id_{R}) (ρ_{A R}) = \sum_{a} σ_{a} \otimes R_{a},

(1)

where

R_{a} = Tr_{A} [(E_{a} \otimes I) ρ_{A R} (E_{a} \otimes I)] \geq 0.

(2)

To see that this is the correct output, the partial trace is cyclic for operators acting only on the traced subsystem, so the displayed expression equals

Tr_{A} [(E_{a} \otimes I) ρ_{A R}]

. Put

q_{a} = Tr R_{a}

. The POVM completeness relation implies

\sum_{a} q_{a} = 1

, and terms with

q_{a} = 0

vanish. Therefore

(Λ \otimes id_{R}) (ρ_{A R}) = \sum_{a : q_{a} > 0} q_{a} σ_{a} \otimes \frac{R _{a}}{q _{a}}

(3)

is a separable quantum state. Thus the channel is an entanglement-breaking channel, for every reference-system dimension.

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