|Ψ>=a|0>+b|1>

U(|Ψ>|0>)=|Ψ>|Ψ>=(a|0>+b|1>)(a|0>+b|1>)=aa|0>|0>+ab|0>|1>+ba|1>|0>+bb|1>|1>)

U(|Ψ>|0>)=U(a|0>|0>+b|1>|0>)=aU(|0>|0>+bU(|1>|0>)=a|0>|0>+b|1>|1>