Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 2 / iv

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- Solution iv

Solution

The average of

z_{j} z_{k}^{*} z_{j^{'}}^{*} z_{k^{'}}

vanishes unless the exponent of every independent fourth root is balanced modulo four. The surviving index patterns are

j = k, j^{'} = k^{'}

and

j = j^{'}, k = k^{'}

. Their intersection

j = k = j^{'} = k^{'}

has been counted twice. Hence, writing

D = \sum_{j = 1}^{d} ∣ j ⟩ ⟨ j ∣ \otimes ∣ j ⟩ ⟨ j ∣,

the phase average is

σ = \frac{1}{d ^{2}} (I \otimes I + d ∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣ - D) .

Solving for the projector onto the maximally entangled state gives

∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣ = d σ - \frac{1}{d} I \otimes I + \frac{1}{d} D .

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