Disentangling quantum states while preserving all local properties

We consider here a disentanglement process which transforms a state rho of two subsystems into an unentangled (i.e., separable) state, while not affec ting the reduced density matrix of either subsystem. Recently, Terno [Phys. Rev. A 59, 3320 (1999)] showed that an arbitrary state cannot be disentang led, by a physically allowable process, into a tensor product of its reduce d density matrices. In this Letter we show that there are sets of states wh ich can be disentangled, but only into separable states other than the prod uct of the reduced density matrices, and other sets of states which cannot be disentangled at all. Thus, we prove that a universal disentangling machi ne cannot exist.