Probabilistic Uncorrelated Cloning Requires Negative Probability

TL;DR

This paper investigates the limits of probabilistic quantum cloning, demonstrating that uncorrelated cloning is impossible under positivity constraints, highlighting the deeper physical significance of positivity in quantum operations.

Contribution

It proves that uncorrelated probabilistic cloning maps cannot exist if positivity is enforced, extending previous deterministic impossibility results.

Findings

Uncorrelated probabilistic cloning maps exist without positivity constraints.

Positivity condition excludes the possibility of uncorrelated probabilistic cloning.

Positivity has a richer physical meaning than traditionally assumed.

Abstract

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."