Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 10D / a

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- Solution a

Solution

A pure bipartite state is entangled when it is not a product state

∣ u ⟩_{A} \otimes ∣ v ⟩_{B}

.

Since

H \otimes H ∣11 ⟩ = \frac{1}{2} (∣00 ⟩ - ∣01 ⟩ - ∣10 ⟩ + ∣11 ⟩),

we have

∣ α ⟩ = \frac{1}{2 3} (3∣00 ⟩ + ∣01 ⟩ + ∣10 ⟩ - ∣11 ⟩) .

Its coefficient matrix is proportional to

(31 1 - 1),

whose determinant is

- 4 \neq = 0

. By the two-qubit product-state determinant criterion, a product state would have determinant zero. Therefore $∣ α ⟩$ is entangled.

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