 Solution

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

Conjugation by one Pauli matrix fixes that same matrix and reverses the other two. For example,

XXX = X

Y X Y = - X

, and

ZXZ = - X

. Applying this to each component of the Bloch vector gives

\sum_{j = x, y, z} σ_{j} ϱ σ_{j} = \frac{1}{2} (3 I - s \cdot σ) .

E_{p} (ϱ) = \frac{1}{2} [I + (1 - \frac{4 p}{3}) s \cdot σ], s^{'} = \frac{3 - 4 p}{3} s .

1 - p I

and

p /3 σ_{j}

0 \leq p \leq 1

. At

p = 3/4

every state becomes maximally mixed. For

p > 3/4

the factor is negative: it reverses the vector as well as reducing its length. Its most negative value is

- 1/3

, consistent with the complete-positivity range of the isotropic qubit map.

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