Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 1 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

A quantum channel

T : B (H) \to B (K)

is a linear, completely positive, trace-preserving map. Complete positivity means

T \otimes id_{n}

is positive for every ancillary dimension

n

.

Solved by gpt-5.6-sol high.

ii

Solution

The Kraus representation is

T (ρ) = j \sum K_{j} ρ K_{j}^{†}, j \sum K_{j}^{†} K_{j} = I .

The second identity is precisely trace preservation. Different Kraus families related by an isometry represent the same channel.

Solved by gpt-5.6-sol high.

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