Past exam of the computer science course of the University of Oxford / 2023 / Quantum information / 1 / c

Solution

A

M_{0} = (α 0 0 β), M_{1} = (β 0 0 α) .

The Kraus representation completeness condition holds even for complex

α, β

M_{0}^{†} M_{0} + M_{1}^{†} M_{1} = (∣ α ∣^{2} + ∣ β ∣^{2}) I = I .

To realize this using the permitted quantum gates and a basic measurement, Alice initializes an ancilla qubit

C

∣0 ⟩

. Conditional on her computational-basis state being

j

, she applies to

C

V_{0} = (α β - \overline{β} \overline{α}), V_{1} = (β α - \overline{α} \overline{β}) .

Each has orthonormal columns. Therefore

W = ∣0 ⟩ ⟨ 0 ∣_{A} \otimes V_{0} + ∣1 ⟩ ⟨ 1 ∣_{A} \otimes V_{1}

W (∣ u ⟩_{A} ∣0 ⟩_{C}) = M_{0} ∣ u ⟩_{A} ∣0 ⟩_{C} + M_{1} ∣ u ⟩_{A} ∣1 ⟩_{C} .

C

implements the desired quantum instrument. Its two unnormalized branches on

A B

are

(M_{0} \otimes I) ∣ ϕ_{+} ⟩ = \frac{α ∣00 ⟩ + β ∣11 ⟩}{2}, (M_{1} \otimes I) ∣ ϕ_{+} ⟩ = \frac{β ∣00 ⟩ + α ∣11 ⟩}{2} .

1/2

. Alice sends her outcome

m

as one classical bit. For

m = 0

, both parties keep their qubits unchanged. For

m = 1

(X \otimes X) (β ∣00 ⟩ + α ∣11 ⟩) = α ∣00 ⟩ + β ∣11 ⟩ .

Thus both outcomes produce the exact target using one classical bit and local operations. The construction includes

α = 0

β = 0

and arbitrary relative quantum phases; it requires no quantum channel between the parties.

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