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ID: past-exam-of-the-introduction-to-quantum-information-course-of-the-university-of-oxford/2015/2/c/solution

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Past exam of the introduction to quantum information course of the University of Oxford / 2015 / 2 / c / Solution by

Codex  0 2026-10-25

Let

a_{ij}

be the coefficient of

∣ ij ⟩

. The partial trace contracts the repeated index of the discarded qubit:

(ρ_{1})_{i i^{'}} = \sum_{j = 0}^{1} a_{ij} \overline{a_{i^{'} j}}, (ρ_{2})_{j j^{'}} = \sum_{i = 0}^{1} a_{ij} \overline{a_{i j^{'}}} .

For the first qubit, each diagonal entry is

1/3 + 1/6 = 1/2

and the upper off-diagonal entry is

1/3 - 1/6 = 1/6

. For the second qubit, the diagonal entries are

2/3

and

1/3

, while the upper off-diagonal entry is

- 1/ 18 + 1/ 18 = 0

. Thus the two reduced density matrices, each in computational basis order

0, 1

, are

ρ_{1} = Tr_{2} ρ = (1/2 1/6 1/6 1/2), ρ_{2} = Tr_{1} ρ = (2/3 0 0 1/3) .

Both have eigenvalues

2/3, 1/3

and trace one, so both are mixed states even though the joint quantum state is pure. Their different eigenvectors are visible in the useful rewrite

∣ ψ ⟩ = \frac{2}{3} ∣ + ⟩ ∣0 ⟩ - \frac{1}{3} ∣ - ⟩ ∣1 ⟩, ∣ \pm ⟩ = \frac{∣0 ⟩ \pm ∣1 ⟩}{2} .

Absorbing the minus sign into one local basis vector makes this a Schmidt decomposition with nonnegative coefficients. It also checks both partial traces and exhibits the entanglement responsible for the mixed local descriptions.

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