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We re-analyse the Bu\v{z}ek-Hillery Universal Quantum Cloning machineprotocol and show that it allows better values for fidelity and Hilbert-Schmidtnorm than hitherto reported. This higher value for the fidelity is identical tothe maximum fidelity of phase covariant quantum cloning of Bru\ss-Cinchetti-D'Ariano-Macchiavello. This value of fidelity has also been obtainedby Niu and Griffiths in their work without machine states. This is the maximumpossible fidelity obtainable in $1\rightarrow 2$ qubits cloning. We thendescribe a different and new state dependent cloning protocol with four machinestates where all non-exact copies of input states are taken into account in theoutput and we use the Hessian method of determining extrema of multivariatefunctions. The fidelity for the best overall quantum cloning in this protocolis $\bar{F}=0.847$ with an associated von-Neumann entropy of $\bar{S}=0.825$.

Maximum possible fidelity in $1\rightarrow 2$ qubits cloning