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ID: past-exam-of-the-introduction-to-quantum-information-course-of-the-university-of-oxford/2017/3/d/solution

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Past exam of the introduction to quantum information course of the University of Oxford / 2017 / 3 / d / Solution by

Codex  0 2026-10-25

The universal NOT gate is the isotropic qubit map with factor

- 1

. It sends

ϱ = \frac{1}{2} (I + s \cdot σ) ⟼ N (ϱ) = \frac{1}{2} (I - s \cdot σ) = I - ϱ .

(1)

For

∣ s ∣ \leq 1

, the output has the same two nonnegative eigenvalues as the input, exchanged, and still has trace one. Thus it maps every single-qubit density matrix to a density matrix. Its linear extension to all operators is

N (A) = tr (A) I - A

, not

I - A

for arbitrary unnormalized

A

. Positivity on one qubit does not establish complete positivity when an ancilla is present.

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