 Solution

Construct ensemble purificationsThey have the same reduced state exactly when ${\rho }_1={\rho }_2$. By the unitary freedom of purification, this holds exactly when $|{\Phi }_2⟩=(I\otimes U)|{\Phi }_1⟩$ for a unitary $U$. Comparing reference-basis coefficients gives the Hughston–Jozsa–Wootters theorem relationConversely, substituting this relation and using $U^{†}U=I$ immediately gives ${\rho }_2={\rho }_1$.