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 |Ψ1>, |Ψ2>,..., |Ψk> 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 ρ1, ρ2,..., ρk of the n-dimensional quantum system Cn can be cloned simultaneously, that is, there exists a quantum channel Φ on Mn ⊗ Mn and a state ∑ in Mn, such that Φ (ρi ⊗ ∑) = ρi ⊗ ρi for all i, if and only if ρiρj= 0 (i≠j). Also, the constructing procedure of the desired simultaneous cloning machine is given.
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