Mixed qubits cannot be universally broadcast

We show that no universal quantum cloning machine exists that can broadcast an arbitrary mixed qubit with a constant fidelity. Based on this result, we investigate the dependent quantum cloner in the sense that some parameter of the input qubit ρs(θ,ω,λ) is regarded as constant in the fidelity. For the case of constant ω, we establish the 1→2 optimal symmetric dependent cloner with a fidelity 1/2. It is also shown that the 1→M optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when λ is the unique parameter in the fidelity. For general N→M broadcasting of mixed qubits, the situation is very different