Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 2 / d / iii / Solution

Extend the normalized vector $|\varphi ⟩$ to an orthonormal basis $\{|f_j⟩{\}}_{j=1}^d$ with $|f_1⟩=|\varphi ⟩$, and extend $|\psi ⟩$ to an orthonormal basis $\{|e_j⟩{\}}_{j=1}^d$ with $|e_1⟩=|\psi ⟩$. The Gram-Schmidt process provides such extensions. DefineThen $U^{†}U={\sum }_j|f_j⟩⟨f_j|=I$ and $U{U}^{†}=I$, so $U$ is unitary and $U|\varphi ⟩=|\psi ⟩$. Construct $V$ on the second subsystem in the same way, sending $|{\varphi }^{\prime }⟩$ to $|{\psi }^{\prime }⟩$. ConsequentlyThus any two normalized product states on the same pair of subsystem spaces are local-unitary equivalent. The dimension and normalization requirements matter because a unitary preserves both its domain and vector norms.