Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 50 / 3 / d

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- Solution d

Solution

Take the rank-one orthogonal projections

P_{i} = ∣ ϕ_{i} ⟩ ⟨ ϕ_{i} ∣

. Their pinching map gives

P (ρ) = \sum_{i} P_{i} ρ P_{i} = \sum_{i} ⟨ ϕ_{i} ∣ ρ ∣ ϕ_{i} ⟩ ∣ ϕ_{i} ⟩ ⟨ ϕ_{i} ∣ = ρ_{d} .

This has exactly the specified diagonal entries and no off-diagonal entries. Applying the entropy increase under nonselective projective measurement proved in part (c) therefore gives

S (ρ) \leq S (ρ_{d}) .

q_{i} = ⟨ ϕ_{i} ∣ ρ ∣ ϕ_{i} ⟩

, the output Von Neumann entropy is the Shannon entropy

S (ρ_{d}) = - \sum_{i} q_{i} lo g q_{i}

. Equality holds precisely when

ρ

was already diagonal in this basis.

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