Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 1 / b / ii

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- Solution ii

Solution

The linear extension of the stated k-reduction map is

Λ_{k} (X) = k Tr (X) I - X .

It suffices to consider a pure state

∣ v ⟩

of Schmidt rank

r \leq k

, because positivity is preserved by sums. Write

∣ v ⟩ = \sum_{j = 1}^{r} λ_{j} ∣ j ⟩ ∣ j ⟩, ρ_{A} = \sum_{j = 1}^{r} λ_{j} ∣ j ⟩ ⟨ j ∣.

Then

(id \otimes Λ_{k}) (∣ v ⟩ ⟨ v ∣) = k ρ_{A} \otimes I - ∣ v ⟩ ⟨ v ∣.

For every

∣ x ⟩ = \sum_{ij} x_{ij} ∣ ij ⟩

, the Cauchy-Schwarz inequality gives

∣ ⟨ v ∣ x ⟩ ∣^{2} = \sum_{j = 1}^{r} λ_{j} x_{jj}^{2} \leq r \sum_{j = 1}^{r} λ_{j} ∣ x_{jj} ∣^{2} \leq k ⟨ x ∣ ρ_{A} \otimes I ∣ x ⟩ .

Thus the operator is a positive semidefinite operator. Applying this to every vector in a Schmidt-number-

k

ensemble proves

SN (σ) \leq k ⟹ (id_{n} \otimes Λ_{k}) (σ) \geq 0 .

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