Quantum states related by a global phase shift are physically equivalent. However, pure quantum states related by a global phase shift lead to the same density operator: $$ |{\psi}\rangle= |{1}\rangle \rightarrow \rho=|{1} \rangle\langle{1}|$$ $$ |{\psi^\prime}\rangle= e^{i\theta}|{1}\rangle \rightarrow \rho^\prime= e^{i\theta} |{1} \rangle\langle{1}| e^{-i\theta}=|{1} \rangle\langle{1}| $$
Hence the question: are there density operators which are mathematically different but physically equivalent?
