Past exam of the computer science course of the University of Oxford / 2022 / Quantum information / 1 / c / i

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- Solution i

Solution

If Alice's basic measurement returns a normalized vector

∣ u ⟩_{A}

, form Bob's unnormalized conditional vector

∣ χ_{u} ⟩_{B} = (⟨ u ∣_{A} \otimes I_{B}) ∣ χ ⟩_{A B} .

The Born rule and normalization give

p_{u} = ∥ χ_{u} ∥^{2}, ∣ χ_{u} ⟩_{B} = \frac{∣ χ _{u} ⟩ _{B}}{p _{u}} (p_{u} > 0) .

The joint post-measurement state is

∣ u ⟩_{A} \otimes ∣ χ_{u} ⟩_{B}

. For

∣ u ⟩_{A} = α ∣0 ⟩_{A} + β ∣1 ⟩_{A}

with real coefficients and

α^{2} + β^{2} = 1

, contraction of the Bell state gives

(⟨ u ∣ \otimes I) ∣ ϕ_{+} ⟩ = \frac{α ∣0 ⟩ + β ∣1 ⟩}{2}, p_{u} = \frac{1}{2} .

Thus Bob's normalized conditional state is $α ∣0 ⟩_{B} + β ∣1 ⟩_{B}$ . For complex coefficients the contraction instead contains their complex conjugates; this is the same conjugation underlying conjugate-basis steering of a Bell pair.

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