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

Fix the two subsystem dimensions. For reflexivity, $|\Psi ⟩=(I_d\otimes I_{d^{\prime }})|\Psi ⟩$, and both identity operators are unitary.

For symmetry, if $|\Psi ⟩=(U\otimes V)|\Phi ⟩$, apply the inverse local unitary transformation:The conjugate transposes of unitary operators are unitary, so the reverse relation holds.

For transitivity, if additionally $|\Phi ⟩=(R\otimes S)|\Theta ⟩$, the tensor product multiplication rule givesBoth products are unitary, for example $(UR{)}^{†}(UR)=R^{†}{U}^{†}UR=I$. Thus local-unitary equivalence is reflexive, symmetric and transitive, and hence is an equivalence relation.