Probabilistic Uncorrelated Cloning Requires Negative Probability

It is known that to imperfectly clone a phase-set of states uncorrelatedly is impossible due to the linearity and the hermitian-preservation of quantum operations deterministically shown by D'Ariano et al. In this paper we address uncorrelated cloning in probabilistic cases. We show that there exists a linear and hermitian-preserving probabilistic map to imperfectly clone a phase-set of states uncorrelatedly. It is pointed out that the existence of such a map is due to the difference between non-unit-trace output operators and normalized (unit-trace) output operators inherent to probabilistic maps. We however prove that such a possibility of uncorrelated cloning is completely excluded by the condition of positivity in addition to the linearity and the hermitian-preservation of quantum operations. Our results implicate that the positivity of quantum operation is richer physical meaning than the one which we usually assume a priori for the necessity of "probability interpretation."