= Economical two-state quantum cloner
{title2=$\overline F=\cos^2(\pi/24)\quad\text{for }|0\rangle,|+\rangle$}
A two-<qubit> <unitary gate> can approximately clone the known pair $|0\rangle,|+\rangle$ with <global cloning fidelity> $\cos^2(\pi/24)$ for each input, using no additional workspace. To construct it, put $a=|00\rangle$, $b=|++\rangle$, $u=(a+b)/\sqrt3$ and $v=a-b$. The vectors $u,v$ are <orthonormal>, and $a,b$ have coefficients $(\cos(\pi/6),\pm\sin(\pi/6))$ in that basis. Choose output vectors with coefficients $(\cos(\pi/8),\pm\sin(\pi/8))$. Their <inner product> is $\cos(\pi/4)=1/\sqrt2$, exactly the overlap of the inputs $|00\rangle,|+0\rangle$. Their linear matching therefore defines a <linear isometry> on the input span and extends to a two-<qubit> <unitary operator> by <unitary extension of a finite-dimensional isometry>. Each output's overlap with its ideal copy is $\cos(\pi/6-\pi/8)=\cos(\pi/24)$, proving the stated <squared quantum fidelity>.
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