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

The Schmidt decomposition states that every finite-dimensional bipartite pure state haswhere $s_j>0$, ${\sum }_j{s}_j^2=1$, and the two displayed families are orthonormal. The integer $r$ is the Schmidt rank.

To prove it, choose product bases and writeApply the singular value decomposition $C=US{V}^{†}$. Absorbing the columns of $U$ and the complex conjugates of the columns of $V$ into new orthonormal bases gives the stated sum, with the nonzero singular values as the Schmidt coefficients. Equivalently, $s_j^2$ are the common nonzero eigenvalues of the two reduced density matrices, so $r=\mathrm{rank}{\rho }_A=\mathrm{rank}{\rho }_B$.

For a purification $|{\Psi }_{\rho }{⟩}_{RA}$ of $\rho$, the entanglement fidelity isUsing a Kraus representation $\Lambda (X)={\sum }_k{A}_{k}X{A}_k^{†}$ givesIn a Schmidt decomposition of the purification,Therefore

Embed the input qubit as the span of $|0⟩,|1⟩$ in the three-dimensional output. Because the input is pure, its only purification has a trivial reference and entanglement fidelity reduces to its survival probability:Since $|e⟩$ is orthogonal to the embedded qubit subspace,

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