A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix. Those cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized: let { p j , | ψ ψ j ⟩ ⟩ } be an ensemble. Then for any complex matrix U such that U † † U = I (a partial isometry), the ensemble { q i , | φ φ i ⟩ ⟩ } defined by