Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 2 / a / ii / 3

The states

ρ_{A}^{i} = ∣ a_{i} ⟩ ⟨ a_{i} ∣

are pure, so a purification is

∣Ψ ⟩ = i = 1 \sum n j = 1 j \neq = i \sum n p_{ij} ∣ i ⟩_{X_{1}} ∣ j ⟩_{X_{2}} ∣ a_{i} ⟩_{A_{1}} ∣ a_{j} ⟩_{A_{2}} ∣ ij ⟩_{R} .

(1)

Tracing out the orthonormal reference labels removes all cross terms and recovers the stated classical-quantum mixture. Its rank is the number of nonzero

p_{ij}

, so that is the minimum reference dimension for a particular state. The smallest dimension that can purify every state of the stated form is

dim H_{R} = n (n - 1) .

(2)

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