Economical two-state quantum cloner ($\overline{F}={\cos }^2(\pi /24)\text{for }|0⟩,|+⟩$)

A two-qubit unitary gate can approximately clone the known pair $|0⟩,|+⟩$ with global cloning fidelity ${\cos }^2(\pi /24)$ for each input, using no additional workspace. To construct it, put $a=|00⟩$, $b=|++⟩$, $u=(a+b)/\sqrt{3}$ 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 $\cos (\pi /4)=1/\sqrt{2}$, exactly the overlap of the inputs $|00⟩,|+0⟩$. 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.