Existence and construction of simultaneous cloning machines for mixed states

It is a well-known fact that the no-cloning theorem forbids the creation of identical copies of an arbitrary unknown quantum state. In other words, there does not exist a quantum cloning machine that can clone all quantum states. However, it is possible to clone given quantum states under certain conditions, for instance, k distinct pure states |Ψ 1〉, |Ψ 2〉, …, |Ψ k 〉 can be cloned simultaneously if and only if they are orthogonal. This paper discusses the existence and construction of simultaneous cloning machines for mixed states. It is proved that k distinct mixed states ρ 1, ρ 2, …, ρ k of the n-dimensional quantum system ℂ n can be cloned simultaneously, that is, there exists a quantum channel Φ on M n ⊗ M n and a state Σ in M n , such that Φ(ρ i ⊗ Σ) = ρ i ⊗ ρ i for all i, if and only if ρ i ρ j = 0 (i ≠ j). Also, the constructing procedure of the desired simultaneous cloning machine is given.