Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 2 / b

Table of contents
- Solution b

Solution

For every ancillary system,

(Λ \otimes id) (X) = \sum_{k} (A_{k} \otimes I) X (A_{k}^{†} \otimes I) \geq 0

when

X \geq 0

, proving complete positivity. Cyclicity of trace gives

Tr Λ (ρ) = Tr (ρ \sum_{k} A_{k}^{†} A_{k}) .

This equals

Tr ρ

for every

ρ

exactly when

k \sum A_{k}^{†} A_{k} = I,

the trace-preserving condition in the Kraus representation.

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