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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-50/2/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 50 / 2 / b / Solution by

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

Fix a basis for the transposition map

T

. If

M \geq 0

, write

M = B B^{†}

. Then

T (M) = M^{T} = \overline{B} B^{T} = \overline{B} (\overline{B})^{†} \geq 0.

Thus transposition is a positive linear map and also preserves the trace.

D \geq 2

, apply it to one subsystem of the maximally entangled state from part (a):

(id \otimes T) (∣Ω ⟩ ⟨ Ω∣) = \frac{1}{D} \sum_{i, j} ∣ i ⟩ ⟨ j ∣ \otimes ∣ j ⟩ ⟨ i ∣ = \frac{F}{D},

where

F

is the swap operator,

F (∣ u ⟩ \otimes ∣ v ⟩) = ∣ v ⟩ \otimes ∣ u ⟩

. It satisfies

F^{†} = F

and

F^{2} = I

, so its eigenvalues are

+ 1

on symmetric vectors and

- 1

on antisymmetric vectors. In particular,

∣ η ⟩ = \frac{∣1 ⟩ ∣2 ⟩ - ∣2 ⟩ ∣1 ⟩}{2}, ⟨ η ∣ \frac{F}{D} ∣ η ⟩ = - \frac{1}{D} < 0.

A positive input has therefore acquired a negative expectation value after adjoining an identity channel. Transposition is positive but not completely positive in dimension at least two. The one-dimensional case is the identity map and is completely positive; the dimension qualification is necessary.

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