Phase conjugation of continuous quantum variables

The phase conjugation of an unknown Gaussian state cannot be realized perfectly by any physical process. A heuristic argument is used to derive a tight lower bound on the noise that must be introduced by an approximate phase conjugation operation. A universal transformation achieving the optimal imperfect phase conjugation is then presented, which is the continuous counterpart of the universal-NOT transformation for quantum bits. In addition, it is shown that more information can be encoded into a pair of conjugate Gaussian states than by using the same state twice.