Mixed Qubit Cannot Be Universally Broadcast

We show that there does not exist any universal quantum cloning machine 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 $\rho_s(\theta,\omega,\lambda)$ is regarded as constant in the fidelity. For the case of constant $\omega$, we establish the $1\to2$ optimal symmetric dependent cloner with a fidelity 1/2. It is also shown that the $1\to M$ optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when $\lambda$ is the unique parameter in the fidelity. For general $N\to M$ broadcasting of mixed qubits, the situation is very different.