Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 34 / 4 / b

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Solution

Write

∣ ϕ_{A} ⟩ = \sum_{j} c_{j} ∣ j ⟩_{A}

and choose the conjugate index vector

∣ ϕ_{B}^{*} ⟩ = \sum_{j} c_{j}^{*} ∣ j ⟩_{B}, ⟨ ϕ_{B}^{*} ∣ = \sum_{j} c_{j} ⟨ j ∣_{B} .

(1)

With

∣ Ψ_{A B} ⟩ = d ∣ Ψ_{A B} ⟩ = \sum_{j} ∣ j ⟩_{A} ∣ j ⟩_{B}

, the partial inner product gives

(I_{A} \otimes ⟨ ϕ_{B}^{*} ∣) ∣ Ψ_{A B} ⟩ = j \sum c_{j} ∣ j ⟩_{A} = ∣ ϕ_{A} ⟩ .

(2)

This is the relative state construction. The coefficient conjugation is defined relative to the fixed paired bases; it is essential, because the bra conjugates those coefficients again. Using the normalized entangled vector instead would give

∣ ϕ_{A} ⟩ / d

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 34 / 4 / b

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