Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 5 / c

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- Solution c

Solution

Discarding the outcome of the complete projective measurement gives

σ = i \sum P_{i} ρ P_{i} .

For

m

projectors, let

ω = e^{2 πi / m}

and

U = \sum_{j} ω^{j} P_{j}

. Then the pinching identity is

σ = \frac{1}{m} \sum_{k = 0}^{m - 1} U^{k} ρ U^{- k} .

The Concavity of Von Neumann entropy and its invariance under unitary operators imply

S (σ) \geq \frac{1}{m} k \sum S (U^{k} ρ U^{- k}) = S (ρ) .

Equality holds exactly when every conjugate in the average is the same, equivalently

[ρ, P_{i}] = 0 for every i .

Thus equality holds when the input already has no coherence between distinct measurement subspaces.

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