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

The state is a convex combination of product density operators and is therefore a separable quantum state. Every product density operator has a decomposition into product pure states, so

The Bell state $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$ has two nonzero Schmidt coefficients. Since the density operator is pure, every ensemble decomposition uses vectors in the same one-dimensional support, and hence

The four Bell states form an orthonormal basis, so their uniform mixture isIt is a product state and therefore

The linear extension of the stated k-reduction map isIt suffices to consider a pure state $|v⟩$ of Schmidt rank $r\le k$, because positivity is preserved by sums. WriteThenFor every $|x⟩={\sum }_{ij}{x}_{ij}|ij⟩$, the Cauchy-Schwarz inequality givesThus the operator is a positive semidefinite operator. Applying this to every vector in a Schmidt-number-$k$ ensemble proves